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

MODULATION OF HOMOGENEOUS ISOTROPIC TURBULENCE BYDISPERSED FIBRES AND PARTICLES

Ianto Cannon
Complex fluids and flows
Okinawa Institute of Science & Technology
1919-1 Tancha, Onna-son, Okinawa, 904-0495, Japanianto.cannon@oist.jp

Stefano Olivieri and Marco E. Rosti
Complex fluids and flows
Okinawa Institute of Science & Technology
1919-1 Tancha, Onna-son, Okinawa, 904-0495, Japan
stefano.olivieri@oist.jp and marco.rosti@oist.jp

ABSTRACT
We compare how a homogeneous isotropic turbulent flow
(micro-scale Reynolds number Reλ ≈ 435) is modulated by finite diameter spheres and finite length fibres. Both spheres andfibres produce a similar bulk effect, characterized by a reduction of the micro-scale Reynolds number to around Reλ ≈ 220.
However, a spectral analysis of the flows reveals very different
stories for the two particle geometries. For spheres, the flow is
modulated on scales comparable with the sphere diameter. Little modulation occurs at smaller scales, and the canonical turbulent energy cascade is recovered in this region of the spectra.
While for fibres, modulation occurs over a much wider rangeof length-scales, (including the fibre thickness) and a new turbulent kinetic energy scaling E ∼ κ −1 is found in the place ofthe canonical cascade.

Figure 1. 3-D views of the fluid vorticity magnitude in thetri-periodic cubic domain with spheres (left), and fibres (right).
presence of the particles (Balachandar & Eaton, 2010; Maxey,2017; Brandt & Coletti, 2021). A few studies consider nondilute suspensions of droplets (Dodd & Ferrante, 2016; Freund & Ferrante, 2019; Rosti et al., 2019; Cannon et al.,2021), isotropic particles (Lucci et al., 2010; Gualtieri et al.,2013; Capecelatro et al., 2018; Ardekani et al., 2019; Sozzaet al., 2020), and anisotropic particles (Andersson et al., 2012;
Olivieri et al., 2020a,b, 2021). In these studies, authors report changes in the bulk flow properties, as well changes in theturbulent kinetic energy spectrum. In this work, we build onthese studies by investigating non-dilute suspensions of finite
sized particles1 . In particular we directly compare homogeneous isotropic flows laden with spheres and fibres in order toask the question, “how does the shape of the particles affectthe flow?”
Figure 1 shows two snapshots of setups with spheres andfibres that we use to answer the above question. We performed
direct numerical simulations of an incompressible Newtonian
fluid in a tri-periodic cubic box of size L, discretised to 1024
numerical grid points in each of the three Cartesian directions.
The fluid is coupled to particle dynamics using the immersedboundary method. See Podvigina & Pouquet (1994); PopeINTRODUCTION
Particles of many different shapes are found in turbulent flows, ranging from spherical pollen particles in the atmosphere to elongated plastic microfibres in the ocean (Kane& Clare, 2019) and cellulose fibres involved in papermaking(Lundell et al., 2011). Understanding how these different particle shapes affect the flow is of environmental and industrial
importance (De Lillo et al., 2014; Breard et al., 2016; Sengupta et al., 2017; Falkinhoff et al., 2020; Rosti et al., 2020b).
Here we investigate how the shape of the dispersed particles
affects the flow. We explore two extremes of particle shape;
perfectly isotropic spheres and highly anisotropic elongated fibres. Throughout this article, we will use “particle” as a general term to refer to both spheres and fibres.
Elghobashi (1994) gives a good overview of the dimensionless parameters that can be used to describe particle laden
turbulent flows, including Stokes number, Galileo number, andparticle volume fraction. In this work we focus on the effectof particle mass fraction M ≡ m p /(m p + m f ), where m p is thetotal mass of the particles, and m f is the total mass of the fluid.
The majority of previous numerical investigations of particle laden turbulent flows consider only dilute suspensions(M << 1), where it is assumed the flow is not altered by

1 finite sized particles have a size which is comparable to the inertial
range of the flow

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 10500
Reλ4002spheres10−1

κ − 5 /3E300

10−4200
0.00.20.4M

0.60.81.010−7 0
10Figure 2. The micro-scale Reynolds number Reλ as a function of the particle mass fraction M. Flows with spheres aremarked in blue and flows with fibres are marked in green. The
shaded blue and green regions show the standard deviation inReλ from the time averaged value.

(b) 10101

κ

102

2fibers10−1

κ − 5 /3E
10−4(2000); Olivieri et al. (2021); Kim & Peskin (2007); Hori et al.
(2022); Huang et al. (2007); Olivieri et al. (2020a,b); Rostiet al. (2020a); Rosti & Brandt (2020); Rosti et al. (2020b,2021) for details and validation of the numerical method.
The flow is sustained by deterministic Arnold-BeltramiChildress forcing (Podvigina & Pouquet, 1994), giving a
micro-scale Reynolds number Reλ = u0 λ /ν ≈ 435 in the single phase flow, where ν is the kinematic viscosity, λ is Taylor’s micro-scale, and u0 is the root mean square of the turbulent fluctuations in fluid velocity. For the particles, we use 300
rigid spheres of diameter c = L/(4π), or 104 fibres of identical length c = L/(4π). These parameter choices give particles which have a similar total surface area, as well as being the same proportions. The numerical method is such thatthe fibres are slightly flexible, but their stiffness was set to alarge value so their deformation was always negligible. The
diameter of the fibres is d ≈ 2∆x = L/512, determined by thesmoothing of the Dirac delta function used in the immersedboundary method. We vary the density of the particles to givea range of mass fractions M. Note that M = 0 refers to the single phase case, and M = 1 refers to cases with particles fixedat random locations in the domain. The plots discussed in thefollowing section were produced from time averaged simulations once they had been allowed to reach a statistically steadystate.

10−7 010101

κ

102

Figure 3. Energy spectra of the turbulent flows laden with (a)spheres and (b) fibres. The particle mass fraction M for eachcase is denoted by the brightness of the line colour, using thesame colours as the markers in figures 2 and 4. The single
phase flow is marked by a black line on both (a) and (b). Thecanonical Kolmogorov scaling is marked by a grey dotted line.
The region in which we observe power law scalings (10 < κ <60) is shaded in dark grey.

and fibres cause a turbulent kinetic energy increase relative
to the single phase, and the effect is most pronounced in thecase of fibres. As we shall see later, this is due to the fluid
solid coupling which injects energy to the flow at small scales.
The most pronounced difference between spheres and fibres
is found at intermediate wave-numbers (10 < κ < 60). Withspheres, the flow recovers to the single phase value of turbulent kinetic energy. Whereas with fibres, the energy is reduced
and the canonical inertial scaling is not observed. Clearly theisotropic and anisotropic particles are interacting differently
with the flow in this intermediate wave-number range. In allof the cases, we see clear power-law scalings (straight lines on
the log-log scale), so we made power law fits E = Aκ −β to theturbulent kinetic energy in the range 10 < κ < 60, where A andβ are fitted parameters.
Figure 4 shows the dependence of the fitted exponent β
on the mass fraction M of particles. The single phase flow
(M = 0) and the flows with spheres can be seen to follow theKolmogorov scaling (β = 5/3), whereas the flows with fibres
show a significant departure from Kolmogorov’s theory as themass fraction increases; and we see instead that β ≈ 1 for
M > 0.6. We explain the mechanism underlying this new scaling by considering the two point velocity correlations δ u(r) inthe flows; δ u(r) is defined as the difference in fluid velocities
at two points separated by a distance r. By simple dimensional
analysis we expect the energy spectrum to scale linearly withthe variance of the velocity correlation h(δ u(r))2 ;RESULTS AND DISCUSSION
Figure 2 shows the effect of increasing particle mass fraction M on the micro-scale Reynolds number Reλ of the flow.
We see that Reλ decreases in a similar fashion for increasinginertia of spheres and fibres alike. In addition, the standard
deviation of Reλ decreases with M, this is typical of a transition towards laminar flow, where oscillations in time are suppressed. Overall, from figure 2 we see that the isotropic andanisotropic particles have a similar effect on the bulk flow. To
see the difference between the back-reactions of spheres andfibres we must observe the flows at all length scales.
Figure 3 shows the turbulent kinetic energy spectrum ofeach flow, with the Kolmogorov scaling E ∼ κ −5/3 for comparison. As expected, the single phase flow follows the canonical Kolmogorov scaling in the inertial range and is dominatedby dissipation at the small scales (κ > 100). At the largest
scales (κ < 3) we see the spheres and fibres produce a similar
reduction in turbulent kinetic energy E relative to the single
phase, that is, E reduces as the mass fraction M increases.
At the smallest scales (κ > 100), we see that both spheres

E(κ) ∼ h(δ u(r))2 iκ −1(1)The fibres are slender, and they act as a barrier to the flow
between any two points with separation greater than the fibre2

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

β = 5/3
1.5

Ffs /Ffsmax

β1.0
0.0

0.2

0.4

M

0.60.8(b)Ffs /Ffsmax

Figure 4. Dependence of the exponent β in the turbulent kinetic energy scaling E ∼ κ −β on particle mass fraction M.
Flows with spheres are marked in blue, flows with fibres ingreen, and the single phase flow in black. The blue and green
shaded regions show the approximate error in β , estimated bymoving the time averaging window. The Kolmogorov scaling
is marked by a grey dotted line.

0.0101

κ102

κ102

κc1.00.50.0−0.5 010101

Figure 5. Spectra of the work done on the fluid by the particles Ffs , normalised by their peak values Ffsmax for ease ofcomparison. The work done by spheres is plotted in blue (a),and the work done by fibres is plotted in green (b). The particle mass fraction M for each case is denoted by the brightnessof the line colour, using the same colours as the markers in figures 2 and 4. The vertical dotted lines show the wave-number
κc corresponding to the sphere diameter and fibre length.

diameter d. Hence fibres decorrelate the flow by increasing
the value of δ u(r) for all r > d. This means the variance ofthe correlation has little dependence on the separation, and soh(δ u(r))2 i ∼ r0 is roughly constant. Plugging this expression
into equation 1, we obtain the energy scaling E(κ) ∼ κ −1 observed in figure 4. That is, the fibres decorrelate the flow andreduce the magnitude of the scaling exponent β . Spheres on
the other hand can only act as barriers to the flow for separations larger than their diameter c, so we see a departurefrom Kolmogorov’s scaling only at the largest length-scales(κ < κc = 12.6) in figure 3a.
To view the length-scales at which the particles act on
the flow, we show the spectrum of work done on the fluid by
the particles Ffs in figure 5. This is the fluid-solid coupling
contribution to the turbulent kinetic energy-spectrum balance,∂t E(κ,t) = T (κ,t) +V (κ,t) + Ftur (κ,t) + Ffs (κ,t)

0.5−0.5 0101.0

κc1.0

and a disruption of Kolmogorov’s turbulent cascade at large
to intermediate scales. These results have applications in turbulent flow control, E.g., they show that fibres can be added
to a fluid in order to reduce velocity correlations and enhancemixing at small scales.

(2)

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We have investigated finite sized particle-laden turbulent
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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