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# Settling behaviour of thin curved particles in quiescent fluid and turbulence

Authors:
• Bardehle Pagenberg

## Abstract and Figures

The motion of thin curved falling particles is ubiquitous in both nature and industry but is not yet widely examined. Here, we describe an experimental study on the dynamics of thin cylindrical shells resembling broken bottle fragments settling through quiescent fluid and homogeneous anisotropic turbulence. The particles have Archimedes numbers based on the mean descent velocity 0.75 × 10 4 Ar 2.75 × 10 4. Turbulence reaching a Reynolds number of Re λ ≈ 100 is generated in a water tank using random jet arrays mounted in a coplanar configuration. After the flow becomes statistically stationary, a particle is released and its three-dimensional motion is recorded using two orthogonally positioned high-speed cameras. We propose a simple pendulum model that accurately captures the velocity fluctuations of the particles in still fluid and find that differences in the falling style might be explained by a closer alignment between the particle's pitch angle and its velocity vector. By comparing the trajectories under background turbulence with the quiescent fluid cases, we measure a decrease in the mean descent velocity in turbulence for the conditions tested. We also study the secondary motion of the particles and identify descent events that are unique to turbulence such as 'long gliding' and 'rapid rotation' events. Lastly, we show an increase in the radial dispersion of the particles under background turbulence and correlate the time scale of descent events with the local settling velocity.
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J. Fluid Mech. (2021), vol.922, A30, doi:10.1017/jfm.2021.520
Settling behaviour of thin curved particles in
quiescent ﬂuid and turbulence
Timothy T.K. Chan1,2, Luis Blay Esteban2, Sander G. Huisman1,
John S. Shrimpton2and Bharathram Ganapathisubramani2,
1Physics of Fluids Group, Max Planck Center Twente for Complex Fluid Dynamics, Faculty of Science
and Technology, MESA+ Research Institute, and J. M. Burgers Centre for Fluid Dynamics, University of
Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2Aerodynamics and Flight Mechanics Group, Faculty of Engineering and Physical Sciences, University of
Southampton, Southampton SO17 1BJ, UK
(Received 22 December 2020; revised 21 April 2021; accepted 3 June 2021)
The motion of thin curved falling particles is ubiquitous in both nature and industry but
is not yet widely examined. Here, we describe an experimental study on the dynamics
of thin cylindrical shells resembling broken bottle fragments settling through quiescent
ﬂuid and homogeneous anisotropic turbulence. The particles have Archimedes numbers
based on the mean descent velocity 0.75 ×104Ar 2.75 ×104. Turbulence reaching
a Reynolds number of Reλ100 is generated in a water tank using random jet arrays
mounted in a coplanar conﬁguration. After the ﬂow becomes statistically stationary, a
particle is released and its three-dimensional motion is recorded using two orthogonally
positioned high-speed cameras. We propose a simple pendulum model that accurately
captures the velocity ﬂuctuations of the particles in still ﬂuid and ﬁnd that differences
in the falling style might be explained by a closer alignment between the particle’s pitch
angle and its velocity vector. By comparing the trajectories under background turbulence
with the quiescent ﬂuid cases, we measure a decrease in the mean descent velocity in
turbulence for the conditions tested. We also study the secondary motion of the particles
and identify descent events that are unique to turbulence such as ‘long gliding’ and ‘rapid
rotation’ events. Lastly, we show an increase in the radial dispersion of the particles under
background turbulence and correlate the time scale of descent events with the local settling
velocity.
Key words: particle/ﬂuid ﬂow
licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,
provided the original work is properly cited. 922 A30-1
T.T.K. Chan and others
1. Introduction
Solid particles settling through ﬂuids are all around us. Some of these processes occur in
natural environments, such as falling leaves, while others happen in engineering processes
or due to human activities. In fact, the latter often have detrimental effects on nature such as
water and air pollution. Differences in the inertial characteristics of solid materials are also
used in engineering applications to separate residues and reduce the ‘human footprint’ on
the environment. Standard and uniﬂow cyclones are extensively used to remove particulate
matter (up to 10 µm) from the carrier ﬂuid; e.g. remove sand and black powder in the
& Singh Brar 2017) and in solid–solid separation in the mineral processing industry
(Tripathy et al. 2015). Hydrodynamic separators based on similar physical principles are
also employed in the recycling industry (Esteban et al. 2016), where they classify materials
based on the inertial properties of the materials through interaction with turbulence. In
this type of device, comingled waste is introduced into a container where background
turbulence prevents plastics from sinking. In contrast, glass particles which struggle to
follow vortical structures drop to the bottom of the tank, where a strong mean ﬂow carries
them to the next stage for further treatment. In these facilities, different turbulent regimes
are found at various depths of the separator. Plastic-glass separation predominantly occurs
in the middle region of the tank, where particle concentration is low and the turbulence is
not modiﬁed by the solids. However, to improve the separation efﬁciency of these devices,
a thorough understanding of settling characteristics of irregular particles in turbulence is
required.
Much research has been conducted on axisymmetric solids settling in quiescent ﬂuid
(see Ern et al. (2012) for a detailed review), and it is much accepted that particle dynamics
are determined by three dimensionless numbers. These are: (1) the Reynolds number Re =
VzD,whereVzstands for the particle mean descent velocity, Dfor its characteristic
length scale and νfor the ﬂuid kinematic viscosity; (2) the dimensionless rotational inertia
I, deﬁned as the ratio of the moment of inertia of the particle over that of its solid of
revolution with the same density as the ﬂuid; and (3) the particle aspect ratio D/h,where
hdenotes the object’s thickness.
The most widely studied non-spherical particles are planar disks and rectangular plates
(Stringham, Simons & Guy 1969;Smith1971; Field et al. 1997; Mahadevan, Ryu &
Samuel 1999; Zhong, Chen & Lee 2011;Ernet al. 2012; Auguste, Magnaudet & Fabre
2013; Chrust, Bouchet & Dušek 2013;Leeet al. 2013; Zhong et al. 2013; Heisinger,
Newton & Kanso 2014), whose falling styles share the same dominant features. Still,
speciﬁc dynamics occur when the particle perimeter contains sharp edges (Esteban,
Shrimpton & Ganapathisubramani 2018,2019b,c). The four dominant regimes in both
disks and rectangular plates are ‘steady fall’, ‘zigzag motion’, ‘chaotic motion’ and
‘tumbling motion’ – these are shown in the ReIphase space in ﬁgure 1.When
Re is sufﬁciently small, a particle descends following a ‘steady fall’ independent of
its dimensionless moment of inertia. Under this mode, the solid falls vertically with
oscillation amplitudes much smaller than its characteristic length scale. As Re increases,
the swaying motion grows and the particle transits into a ‘zigzag motion’ caused by vortex
shedding. Various types of zigzag motions have been identiﬁed, ranging from ‘planar
zigzag’ to more three-dimensional ones such as ‘spiralling’ and ‘hula-hoop’ motion
(Zhong et al. 2011; Auguste et al. 2013). From this point, as Irises, the pitching motion
of the particle overcomes the ﬂuid torque damping it and the descent enters a ‘chaotic
regime’ where the particle ﬂips over intermittently while exhibiting a zigzag motion. As
Iincreases further, tumbling becomes more persistent and eventually continuous in the
922 A30-2
Settling behaviour of thin curved particles
I*
10–2
10–4
104105
Re
Plates
Disks
Zigzag (disks)
Chaotic (disks)
Tumbling (disks)
Tumbling
(plates)
Zigzag
(plates)
falling
101102103
102
100
Figure 1. The ReIphase space explored in the current study. The regime boundaries are taken from Field
et al. (1997)andSmith(1971). Markers denote the particles considered, whose properties are listed in table 1.
‘tumbling motion’ regime. Markers in ﬁgure 1 locate the solids investigated in this study
in the ReIphase space originally determined for disks and plates (Willmarth, Hawk &
Harvey 1964; Stringham et al. 1969;Smith1971; Field et al. 1997). Details on deﬁning the
dimensionless numbers of these particles are included in § 2.
Regarding three-dimensional particles with curvature, spheroids, spheres and cylinders
are the canonical geometries that have been investigated in more detail. Oblate spheroids
have the same principal falling styles as disks. However, as they become more spherical,
zigzag and chaotic descents vanish. Yet, when they are close to spheres, chaotic motion
returns (Zhou, Chrust & Dušek 2017). The dynamics of spheres is also very complex,
with steady fall, oblique descent, horizontal oscillations due to vortex shedding,as well as
helical and chaotic motions all being observed (Jenny, Dušek & Bouchet 2004; Veldhuis
& Biesheuvel 2007; Horowitz & Williamson 2010a; Zhou & Dušek 2015; Ardekani
et al. 2016). Fibre-like shapes such as prolate spheroids fall helically with no visible
zigzag motion (Ardekani et al. 2016). Still, as the aspect ratio increases and the particles
become long cylinders, they settle rectilinearly or with oscillations along its axial direction
(Horowitz & Williamson 2006,2010b; Toupoint, Ern & Roig 2019). We refer the reader
to the comprehensive review by Voth & Soldati (2017) for the orientation of ﬁbre-like
particles under different ﬂow conditions.
Despite these studies, there has been little research on the kinematics of thin curved
particles settling in quiescent ﬂuid or under background turbulence. Nonetheless, this
represents an interesting area of research not only for its fundamental signiﬁcance but
also for its industrial relevance.
The literature concerning solids settling or rising in turbulence is far sparser due to the
relative complexity of turbulence generation in a controlled environment. Studies generally
focused on two issues: (1) the settling styles of individual particles and (2) how turbulence
modiﬁes the mean descent velocities. Note that research on the alignment or rotation of
nearly buoyant solids with the carrier ﬂow are not included.
Experiments generally focused on large particles so that their characteristic length
scale lies within the range of turbulent inertial scales where solid–turbulence interactions
are richer. Rising spheres in turbulence with a downward mean ﬂow perform zigzag
922 A30-3
T.T.K. Chan and others
motion or tumbling motion with the transition triggered by changes in I(Mathai et al.
2018). Disks which undergo planar zigzag motion in quiescent ﬂuid settle differently
in statistically stationary homogeneous anisotropic turbulence (Esteban, Shrimpton &
Ganapathisubramani 2020). The dominant features of the planar zigzag mode in quiescent
ﬂuid are still observed. However, these are sometimes replaced by fast descents, tumbling
events, long gliding sections and hovering motions, among others. The variety of descent
scenarios demonstrate the complexity of the particle–turbulence interactions that occur
during settling.
Despite the consensus that turbulence with zero mean ﬂow changes the average settling
velocity of spherical and non-spherical particles, a full understanding of this phenomenon
has yet to be established. Four mechanisms that modify settling have been proposed
to date: the ‘preferential sweeping effect’ (Maxey & Corrsin 1986; Maxey 1987;Tom
&Bragg2019), nonlinear drag due to ﬂuid acceleration (Ho 1964), ‘loitering effect’
(Nielsen 1993)and vortex entrainment (Nielsen 1984,1992). Preferential sweeping effect
refers to the situation where particles are accelerated by the descending side of vortices
as they spiral outwards from the core, whereas loitering effect simply means they stay
relatively longer in upward ﬂows. These four processes affect the local descend velocity
Vzdifferently, with the ﬁrst increasing it and the others reducing it. In this framework,
the settling rate modiﬁcation depends on the relative importance of the competing
mechanisms.
The situation is further complicated as these effects may not be easily delineated, and
opposite results regarding the descent speed have been reported. For droplets in isotropic
turbulence, settling is enhanced when the ratio of the particle’s characteristic gravitational
velocity to the root mean square (r.m.s.) ﬂow velocity ﬂuctuations is smaller than unity
and hindered otherwise (Good et al. 2014). Nonlinear drag has been proven to be vital for
attenuating the descent in that case. On the other hand, simulations of ﬁnite rigid spheres
in Fornari, Picano & Brandt (2016a) found slower settling velocities in turbulence for all
the tested ratios of the mean descent velocity in quiescent ﬂuid to the r.m.s. ﬂow velocity
ﬂuctuations (Vq/u
rms). However, the reduction in the mean descent velocity is greater
when Vq/u
rms <1(Fornariet al. 2016b). There, the authors attributed hindered settling
to unsteady wake forces in addition to severe nonlinear drag due to horizontal oscillations.
Recently, Tom & Bragg (2019) argued in the context of preferential sweeping that the
parameter demarcating enhanced and hindered settling should account for the multiscale
nature of particle–turbulence interactions. It is possible that the apparent contradictions
can be reconciled with scale-dependent quantities, which have been employed to model
pair statistics in turbulence (Bec et al. 2008).
The above results are restricted to spheres in turbulence. Non-spherical solids with ﬁnite
size and particle inertia add more complexity to the problem. Nearly neutrally buoyant
cylinders of the order of the Taylor microscale show small slip velocities in isotropic
turbulence (Byron et al. 2019), which may suggest nonlinear drag is not so important.
Similarly, particles describing falling styles that reﬂect strong interactions with the media,
where particle orientation plays a crucial role, also show an inconsistent behaviour with
the velocity ratio proposed for small spherical particles. More speciﬁcally, disks falling
in anisotropic turbulence where Vq/u
rms >1 settle more rapidly than in quiescent ﬂuid
(Esteban et al. 2020). Focusing on the frequency content of the trajectories, Esteban et al.
(2020) found that as turbulence intensity increases, the dominant frequency of the particles
reduces, and this leads to enhanced settling. However, as different types of motions may
occur in a single trajectory, the relation between the dominant frequency and the descent
styles is not entirely clear.
922 A30-4
Settling behaviour of thin curved particles
Particle no. Symbol R(mm) θ()D(mm) I
119 29 22 5.62
217 73 39 0.36
321 82 48 0.20
419 115 51 0.09
Table 1. Dimensions of the particles dropped. Here Ris roughly constant at 19 mm while θand Dare varied.
The uncertainty is captured by the number of digits reported.
Given these contrasting results, it is obvious that a better understanding on how
turbulence affects settling particles is needed, especially for complex geometries like
non-spherical particles with curvature.
We therefore study the kinematics of freely falling curved particles resembling bottle
fragments. This paper is organised as follows. In § 2, we present the experimental details
of the quiescent ﬂuid cases, discuss the results and propose a simple model for the motions
observed. Next, we show the effects of background turbulence on the settling kinematics
of the curved particles and discuss the results obtained in § 3. Last, this paper concludes
in § 4with the main experimental ﬁndings and directions for future research.
2. Settling in quiescent ﬂuid
2.1. Methods
To analyse the settling behaviour of thin curved objects, we drop bottle-fragment-like
particles in a tank ﬁlled with tap water at room temperature (17C).
Figure 2(a) shows the geometry used to model a broken cylindrical bottle. The particle
has a parallelogramic projection and one non-zero principal curvature oriented along one
of the diagonals of the parallelogram. Hence it is completely deﬁned by the radius of
curvature of the original cylinder R, the subtended angle θ, the diagonal length Dand
the thickness h. The values of these parameters are selected to mimic the dimensions of
fragments processed in recycling plants (Esteban et al. 2016). To delineate the effect of the
different variables, Ris kept largely constant at approximately 19 mm and θ(thus D)is
varied between 29and 115. The thickness also remains the same for all cases at h=1
mm, resulting in aspect ratios D/h=22 to 51. It has been shown that the kinematics of
freely falling disks at low Re may differ even for very large aspect ratios near the ‘steady
fall–zigzag motion’ transition (Auguste et al. 2013). However, the Re of the particles
concerned are far from this boundary and small differences in the aspect ratio have little
effect on their kinematics. Furthermore, thinner particles are not sufﬁciently rigid to
withstand ﬂow perturbations without deformation. We 3D-print all particles (Formlabs
Form 2 printer) using a glass-reinforced rigid resin which results in a material with a
ﬂexural modulus E3.7 GPa. A print resolution of 0.05 mm is used and the objects are
wet sanded with P800 sandpaper for a smooth ﬁnish. Black spray paint, which amounts
to less than 5 % of the particle mass, is applied to aid object detection. Table 1 shows
the particle dimensions determined post-production. The density ratios are also measured
and found to be nearly constant across all the cases, with ρ=ρpf=1.70 ±0.05. For
the particle dimensionless moment of inertia, I=I/I0,whereIis the object’s moment of
inertia and I0is the reference moment of inertia. Their precise deﬁnitions will be discussed
below.
922 A30-5
T.T.K. Chan and others
(e)(a)(b)
(c)
R
D
Axis
rotation
of
(d)
12
34
Axis
rotation
of
45°
Wire mesh
Pumps
Cameras
1 m
2 m
0.85 m
1 m × 0.9 m
Mirror
To syringe
x
y
z
3 cm
β
α
θ
β
Figure 2. (a) Bottle-fragment-like particle considered in this study. The dash–dot line is the axis of revolution
used to obtain the dimensionless moment of inertia I, whereas αand βare the pitch and roll angles,
respectively. (b) Front view and (c) top view of particle with β=0. (d)Drawings (to scale) of the four tested
particles whose dimensions are listed in table 1.(e) Tank and release mechanism employed. Pumps and wire
meshes are installed on both sides for symmetry, though only those on the right are shown to reduce clutter.
The distance between the pumps is 165 cm. The tank rests on a steel frame with a rectangular window at the
bottom to allow optical access. The coordinate system is shown on the top left.
Choosing a suitable Iis challenging without employing any assumptions regarding the
particle behaviour, so past studies generally assume the particle concerned would mainly
oscillate about a predetermined axis. Disks are supposed to rotate about its diameter.
Presumably using this as an inspiration, for spheroids, Zhou et al. (2017) incorporated
the ratio between the moment of inertia about the equatorial and polar axes in Iso the
same axis of rotation is considered in the limit of disks. For n-sided polygons, Esteban
et al. (2018) adopted an analogous axis of rotation to disks when calculating the particle
moment of inertia, but considered the perimeter of the particle relative to a circumscribed
disk to correct for the characteristic length scale in the non-dimensionalisation. To select
the appropriate Iand I0, we made an educated guess of the particle motion. Due to the
presence of a dihedral, the particle should be more stable against rotations around its
uncurved axis. We therefore expect it to rotate about the ‘axis of rotation’ indicated in
ﬁgure 2(a). Hence, in our case, Iis the object’s moment of inertia about an axis passing
through its centre of gravity and parallel to the line marked ‘axis of rotation’ in ﬁgure 2(a)
and I0is the moment of inertia of a ﬂuid-ﬁlled ellipsoid-like object generated by rotating
the arc in ﬁgure 2(b) about its vertices.
We release the particles in the glass tank shown in ﬁgure 2(e). The tank, measuring
2m×1m×0.85 m, is mounted on a steel frame with a central 1 m ×0.9 m rectangular
window at the bottom to enable optical access. In preparation for experiments in
turbulence, the tank is equipped with an 8 ×6 bilge pump array (Rule 24, 360 GPH) on
either side with a 13 mm square wire mesh 3 cm downstream of the nozzles. The pumps
are turned off for experiments in quiescent ﬂuid, and the method of turbulence generation
will be introduced later.
To hold the particle, a pressure mechanism consisting of a syringe pump connected
to a suction cup is used. First, the particle is afﬁxed at 0 pitch angle (α,seeﬁgure 2a)
to the suction cup by imposing a pressure deﬁcit. Then, by slowly pushing the plunger
of the syringe, the pressure is equalised to the atmosphere and the particle is released.
Similar to the work by Lau, Huang & Xu (2018), surface–particle interaction is minimised
922 A30-6
Settling behaviour of thin curved particles
by adjusting the position of the suction cup to at least 1.5Dbelow the water level and
particle transient kinematics are discarded prior to the data post-processing. The object’s
surface is carefully veriﬁed to be bubble-free before release. Conﬁnement effects are also
negligible as the sidewalls of the tank remained at least 4Dfrom the object. A minimum
of eight minutes separate releases to allow any residual ﬂow to dampen, and each particle
is dropped at least 25 times to reduce random error.
During each descent, the motion is recorded by two cameras operating at 60 Hz using
AF Nikkor 50 mm objectives. While the top camera (JAI GO-5000M-USB; pixel size =
5µm) captures the front view, the lower one (JAI GO-2400M-USB; pixel size =5.86 µm)
records the bottom view through a mirror inclined 45. The camera aperture is set so
that the contrast and the depth of ﬁeld are sufﬁciently large for the entire descent,and
the exposure times are adjusted accordingly. To ensure the three-dimensional particle
motion reconstruction is accurate, the cameras are synchronised with a 5 V external signal
(National Instruments USB-6212), aligned with respect to the tank by a vertical post and
calibrated using a square grid. For the bottom camera, the resolution at ﬁve different
heights is calculated and a linear ﬁt is used to obtain the image resolution as a function
of depth. The resolution of both cameras is 0.2mmpixel1, which corresponds to a
magniﬁcation of 1/40.
The three-dimensional position and orientation of the particle are extracted using
MATLAB. The image processing protocol to obtain the particle’s centre of gravity is
similar to the one proposed in Esteban et al. (2020), where a background image is ﬁrst
subtracted from all frames. Then, a Gaussian ﬁlter with a standard deviation of 3 pixels
is applied and the resulting images are binarised before calculating the centres of gravity.
Doing so, the script gives us the (x,y)and zcoordinates of the object from the recordings
of the bottom and top cameras, respectively.
On the other hand, the pitch and roll angles of the particle, which are sketched in
ﬁgure 2(a), are evaluated by measuring the diagonal lengths in each frame. To calculate
them, the corners of the particle are detected ﬁrst in the binary image and then reﬁned
using the greyscale one. Finally, the position of one diagonal’s midpoint relative to the
other diagonal provides the signs of the pitch and roll angles. The high-resolution image
allows the pitch angle to be determined to 3. All the data have been smoothed by
Gaussian ﬁlters to reduce high-frequency noise.
In this study, we are interested in the non-transient particle kinematics. To remove the
transient motions, we ﬁrst construct the cumulative average of the instantaneous vertical
velocity Vzc. By examining this magnitude, we observe that the particle descent velocity
is stable after descending 2/3 the tank depth (26Dand 11Dfor the smallest and largest
fragments, respectively). Then, the cumulative average Vzcat each vertical location
is compared with the stabilised velocity, and the initial part of the trajectories where
the deviation is greater than ±10% discarded. This threshold is robust, since halving it
to ±5 % did not affect the results signiﬁcantly. Similarly, the last particle oscillation is
ignored to eliminate motions affected by interactions with the bottom of the tank.
2.2. Results and discussion
Figure 3 shows the three-dimensional reconstruction of all 25 trajectories recorded for
particle no. 2 in quiescent ﬂuid after transient removal. All descents show periodic motions
with a constant mean vertical velocity. However, the solid sometimes drifts horizontally in
an apparently random direction as it settles. Similar trajectories are obtained for all types
of particles tested.
922 A30-7
T.T.K. Chan and others
–2
0
2
12
0
2
4
6
8
z/D
x/D
–2 –1 0y/D
Figure 3. Reconstructed three-dimensional trajectories of particle no. 2 (θ=73,D=39 mm) in quiescent
ﬂuid.
Particle no. Vdrift (mm s1)Ag/Rg˙γ(s1)Iro ll (g mm2)
13.8±1.40.05 ±0.04 9.4±1.58.7×106
23.2±1.40.04 ±0.03 7.4±0.98.4×105
33.8±2.00.02 ±0.02 0.8±0.52.0×104
42.5±1.10.03 ±0.02 2.8±0.73.9×104
Table 2. The mean horizontal drift velocity Vdrift, the dimensionless gliding section amplitude Ag/Rg,the
precession rate ˙γof each particle and the moment of inertia of rotating it about an axis passing through its
centre of gravity and parallel to its uncurved diagonal Iroll.
To ensure this motion can be neglected, we obtain the velocity associated with the
horizontal drift Vdrift for all trajectories, see table 2. The velocity magnitude appears
to be insensitive to particle geometry and the horizontal drift has no obvious preferred
direction. This suggests the drift is probably not inherent to the descent behaviour and
may have originated from minute ﬂows in the tank which are difﬁcult to eliminate. This
motion is unlikely to have been caused by the release mechanism since the ﬂow induced by
capillary waves decays exponentially in space. Experiments involving heavy cylinders in
Toupoint et al. (2019) also found similar behaviour and the authors argued this was related
to large-scale ﬂuid motions inside the tank. For the subsequent analysis, the trajectories
are dedrifted assuming Vdrift to be the average drift velocity over a square window centred
about the current location and capturing one full period.
We then plot the settling behaviour of particle no. 2 in quiescent ﬂuid in ﬁgure 4 (see also
the supplementary movies available at https://doi.org/10.1017/jfm.2021.520). As the object
falls, it oscillates periodically in the xy-plane with a constant amplitude (ﬁgure 4ac).
At the beginning of each oscillation, the particle carries no horizontal velocity Vhand
shows a highly negative pitch angle α(pointing downwards). As the particle is not in
equilibrium, it accelerates both downwards and horizontally along a direction inside its
symmetry plane containing the uncurved diagonal until it reaches its maximum velocity,
which occurs roughly at the middle of each swing. The particle then decelerates as
922 A30-8
Settling behaviour of thin curved particles
z/D
z/Dα (°) rh /D
y/D
(a)
(b)
(c)
(d)(e)
2
1
0
–1
–2
2
1
0
–1
3
–2 1
0
–1 2 3–3 –2 1
0
–1 2 3–3
–1
0
1
0
5
10
0123456
–50
0
50
t (s)
x/D
Vy /Vz
Vx /VzVh /Vz
Vz /Vz
y/D
0.5 –0.5
0
0.5
–0.5
0
0.5
0
–0.5
0.5 0–0.5
x/D
10
8
6
4
2
0
Figure 4. Descent of particle no. 2 (θ=73,D=39 mm) after removing transient motions and dedrifting.
(a) Three-dimensional and (b) top view of the trajectory reconstruction. (c)(Fromtoptobottom)Theradial
displacement along the direction of motion rh, the depth zand the pitch angle αplotted against time t.No
rolling motions are observed. (d) Velocity in the y-direction Vyplotted against that in the x-direction Vx.(e)
Instantaneous vertical velocity Vzagainst the horizontal velocity Vh(V2
x+V2
y)1/2whose sign switches
every swing. All velocities and positions are normalised with the mean descent velocity Vzand diagonal
length Dof the particle, respectively.
αincreases, drawing an arc-like trajectory. This process repeats itself in the opposite
direction to complete one oscillation. In contrast to N-sided regular polygons (Esteban
et al. 2019c), the particles tested here always travel in a preferred orientation, that is, along
the ﬂat diagonal. Also, no rolling motions are detected, which agrees with our expectation
in the discussion on I.
While it is obvious that the particles fall in a zigzag fashion, whether the trajectories
observed are planar or three-dimensional is not evident. Here, we use an analogous
approach to the one proposed in Esteban et al. (2018) where each trajectory is split into
‘gliding’ and ‘turning’ sections by local extrema of the instantaneous descent velocity. The
amplitude of each gliding section Ag, deﬁned as half the planar displacement of the gliding
section, is compared with its radius of curvature Rgin the top-down view (ﬁgure 4b).
922 A30-9
T.T.K. Chan and others
All trajectories tested satisfy the criterion Ag/Rg<0.1(table 2), and therefore are
considered to be within the ‘planar zigzag’ mode.
Other features of ‘planar zigzag’ trajectories are also observed: oscillations in the
z-direction have twice the frequency of those in the horizontal (ﬁgure 4c) (Zhong et al.
2013), and the velocity phase plot describes a characteristic butterﬂy shape (ﬁgure 4e)
(Auguste et al. 2013). Despite these similarities, the motion of bottle-fragment particles
differ from disks in the sense that disks yaw almost 180at every horizontal extremum
(Zhong et al. 2013), but this does not occur for the particles tested.
While certain disks exhibit three-dimensional ‘hula-hoop’ descents which precess
(Auguste et al. 2013), and ﬁgure 4(d) somewhat resembles such a mode, it is clear that
the particles concerned do not fall this way. This is because ‘hula-hoop’ settling has an
ellipsoidal proﬁle of Vyagainst Vx. Instead, the precession observed here probably emerges
due to another reason.
To examine this feature, we further studied the gliding and turning sections. As
negligible rotation occurs in the gliding sections, they are approximated by straight lines
in the xy-plane. Therefore, rotations have to occur during the turning sections and the
precession rate ˙γcan be deﬁned as the rate at which the gliding sections rotate, see table
2. We observe that ˙γdecreases as the rotational inertia about an axis parallel to the ﬂat
diagonal Iroll increases. Thus, we hypothesise that tiny ﬂuid ﬂuctuations due to residual
ﬂows can explain the precession. These ﬂuctuations may imperceptibly cause the object to
roll, hence precess in the turning sections.
In ﬁgure 5 one can see the evolution of the mean descent velocity Vzwith the
characteristic length scale of the particles D. For smaller objects, Vzdecreases as D
increases, yet larger particles behave oppositely so a minimum at D38 mm appears.
To examine whether it is related to a change in descent style, the Reynolds number Re is
calculated and plotted against the Archimedes number in ﬁgure 6. The Archimedes number
is deﬁned as Ar =(gD3|1ρ|)1/2,wheregis the gravitational acceleration. Previous
research has usually observed a linear relation (Fernandes et al. 2005; Zhong et al. 2013;
Toupoint et al. 2019), and noted that a change in slope can suggest a transition to another
descent style (Auguste et al. 2013). The data in ﬁgure 6 indeed shows a linear relation
for the three smallest particles, but there is a modest increase in the slope for the largest
particle. This might reﬂect a physical transition in the particle dynamics, where the upper
vertices of the particle with θ>90may interact more with the wake generated by the
leading edge. Nonetheless, this feature does not match the local minimum in Vz, whose
origin remains unclear.
Since the descent styles of particles no. 1 to no. 3 appear the same based on the ArRe
plot, we further evaluate the descent velocity behaviour by comparing the radii of curvature
of the trajectories Lpend in the vertical cross-sections (after applying planar projection
and removing the mean descent velocity). We use the subscript ‘pend’ in allusion to
the pendulum model that will be introduced later. Similarly, the maximum pitch angles
αmax, the planar oscillation amplitudes Aand the dominant radial frequencies fare also
evaluated. These are made non-dimensional (except for αmax) and shown in ﬁgure 7,where
the particles are characterised by their Archimedes number Ar.NotethatAdiffers subtly
from Ag, which is shown in table 2, since Aincludes the turning sections as well. Both the
dimensionless radius of curvature of the particle gliding section Lpend /Dand amplitude
of the oscillations A/Dincrease with Ar. However, for the largest particle, these two
magnitudes appear to decrease considerably from the global trend. On the other hand,
αmax decreases with increasing Ar. The Strouhal number, deﬁned as St =fD/Vz, remains
nearly constant across the particles tested, which implies that fis highest for particle no. 1.
922 A30-10
Settling behaviour of thin curved particles
Vz(mm s–1)
D (mm)
25 30 35 40 45 5020
64
66
68
60
62
70
Figure 5. The mean settling velocities Vzof the particles. Unless further speciﬁed, the deﬁnitions of the
data markers follow table 1 and vertical error bars represent the standard deviation of the measurements.
0.5 1.0 1.5 2.0 2.5 3.0
×104
1500
2000
2500
3000
3500
1000
Re
Ar
Figure 6. Plot of Re against Ar. A linear relation is observed for the ﬁrst three points and a kink seen for the
last, which suggests a transition in settling style. As Re(Ar)is one-to-one in the investigated range, the two are
used interchangeably.
Thus, a picture where the smallest particle oscillates rapidly about the vertical axis while
descending, and where the larger ones settle more gently emerges. The smallest particle
might not be fully gliding, and descends faster with less lift produced. As further evidence,
we calculate the average vertical slip angle in the gliding sections deﬁned as the difference
between the pitch angle and the angle of inclination of the velocity vector, i.e.
Δα =tan1Vz
(V2
x+V2
y)1/2−|α|.(2.1)
922 A30-11
T.T.K. Chan and others
(a)(b)
(c)(d)
0.6
0.8
1.0
1.2
0.2
0.4
0.6
0.8
1.0
0
0.45
40
50
60
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
30
12
16
Ar
Ar
A/DL
pend /D
αmax (°)
α (°)
Ar
St
×104
×104×104
×104×104
0.5 1.5 2.5
8
0.50
0.55
0.60
0.65
0.70
Figure 7. The (a) dimensionless radius of curvature of the trajectory in a vertical cross-section Lpend/D,
(b) magnitude of the maximum pitch angle αmax, (inset) average vertical slip angle over gliding sections Δα,
(c) dimensionless radial oscillation amplitude A/Dand (d) Strouhal number St =fD/Vz,wherefis the radial
oscillation frequency, versus the Archimedes number Ar.
This is plotted in the inset of ﬁgure 7(b). The ﬁgure shows that Δα decreases slowly as
the particle diagonal length Dincreases for particles no. 1 to 3, therefore proving its pitch
attitude is more closely aligned with the velocity vector.
Indeed, such a difference in falling behaviour can explain the initial reduction of the
mean descent velocity at small Ar. When the particle’s curved surface area increases,
more lift is generated and the gliding motion is enhanced, leading to a reduction in Vz.
However, this argument alone cannot explain the minimum in Vz.
To understand why the descents become faster at larger Ar, we measure the maximum
horizontal speed in each swing Vh,max.Asﬁgure 8(a) illustrates, Vh,max/Vzgenerally
grows with Ar. This increase in Vh,max leads to a larger Vzbecause the particle pitches
down at the beginning of each swing, so the horizontal and vertical speeds are coupled to
each other. Therefore, the minimum in the descent velocity manifests through a delicate
balance between lift enhancement and a reduction of the particle’s horizontal speed during
the glide.
Settling behaviour at large Ar (or equivalently θ) is more complex. While Vzincreases
even for the largest object, the behaviour of particle no. 4 is different from the other ones.
Figure 7(a,c) demonstrate that Lpend/Dand A/Dare reduced as compared with the linear
extrapolation from the previous three. The trend in A/Dcould be related to the increase in
the slope of Re(Ar). If energy is conserved, a smaller A/Dimplies more potential energy
922 A30-12
Settling behaviour of thin curved particles
Ar Ar
×104×104
(a)(b)
1.8
2.1
2.4
2.7
1.5
600
800
1000
1200
400
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
500 700 900
30
40
50
Vh,max /Vz
az,max (mm s–2)
az,max (mm s–2)
αmax (°)
Figure 8. (a) Plot of the non-dimensional maximum horizontal speed in a swing Vh,max /Vz.(b) Behaviour
of the maximum vertical acceleration az,max. The grey symbols represent the descending pendulum model
introduced in § 2.3. Inset shows αmax versus az,max together with the best-ﬁt line: αmax =(0.03 ±0.01)az,max +
(20.8±6.8).
is converted to vertical velocity instead of horizontal velocity. Since Re is based on Vz,
the Re(Ar)relation becomes steeper, as argued in Auguste et al. (2013). We hypothesise
that the differences observed are due to stronger interactions between the leading edge
vortex and the upper vertices of the particle. Further work is required to understand this
behaviour.
Since αmax indirectly determines the position of the slowest descent, linking it to a
more experimentally accessible quantity might be useful. As discussed, the larger particles
oscillate with a smaller αmax and descend more smoothly. This is also reﬂected by the
maximum vertical acceleration az,max displayed in ﬁgure 8(b). The inset shows αmax is
linearly related to az,max. This is somewhat expected for the gliding particles since a larger
initial pitch angle would mean a steeper descent near the extrema. However, it is worth
noting that the same slope extends to even the smallest particle which settles without
generating signiﬁcant lift based on our interpretation.
2.3. Modelling the settling behaviour
As the particles oscillate periodically while settling, pendulums whose pivots descend at
constant speeds are chosen to model their motions, as also proposed for freely falling disks
(Esteban 2019). Motivated by the fact that the amplitude of the motion does not vary in
time as the particle settles (see ﬁgure 4b), an idealised pendulum model is constructed
assuming that the system is non-dissipative. Thus, their equation of motion ignoring the
d2φ
dt2=−
g1)C
Lsin φ, (2.2)
where ρ>1. Here, φis the angular displacement from the vertical, Lis the (virtual)
pendulum length – i.e. the length from the swinging particle to the virtual origin
falling vertically with the particle – and Cis a constant to account for all accelerations
apart from gravity. By deﬁnition, Land the initial angular position correspond to
the measured quantities Lpend and αmax, respectively. This leaves only Cas a ﬁtting
parameter, whose value is found by matching the oscillation frequencies to experiments.
Although previous studies (Tanabe & Kaneko 1994; Belmonte, Eisenberg & Moses
1998) have used pendulums to describe the dynamics of settling particles, they focus
922 A30-13
T.T.K. Chan and others
–0.8 –0.4 0 0.4 0.8
rh/D
z/D
0
2
4
6
8
Figure 9. Model pendulum trajectory (grey line) overlaid on the experimental data of particle no. 2.
on the quasi-two-dimensional scenario involving ﬂat plates, as opposed to our fully
three-dimensional case with curved particles.
Figure 9 shows the pendulum trajectory overlaid on the experimental data of particle
no. 2. Although not shown, similar plots are obtained for all four particles. In view of
the reasonably good agreement between the experimental data and the model proposed,
‘planar zigzag’ descents can be viewed as simple harmonic motions superposed on
uniform descents, though higher-order quantities such as az,max are not accurately captured
by the model as indicated by the grey symbols in ﬁgure 8(b).
To examine whether the ﬁtted parameter Ccan be determined without frequency
matching, it is plotted against Ar in ﬁgure 10. Interestingly, a nearly linear relation exists,
meaning this model allows one to predict the particle velocity ﬂuctuations simply by
computing Ar without any a priori knowledge. In the context of undamped underwater
pendulums, C=+m
a)1,wherem
ais the added-mass coefﬁcient characterising the
energy spent accelerating the surrounding ﬂuid. This can be obtained by comparing (2.2)
with the equation of motion of underwater pendulums as in Mathai et al. (2019),
d2φ
dt2=− g1)
L+m
a)sin φ. (2.3)
The inset of ﬁgure 10 shows that m
adecreases with increasing Ar, suggesting that
enhanced gliding means less effort is required to move the neighbouring liquid. The
magnitude of this parameter is much larger than in objects like cylinders, since particle
volume and ρfare used to non-dimensionalise the added mass.
To better understand the behaviour of bottle fragments in industrial facilities, the same
objects are dropped in the water tank with background turbulence. In the following section,
the ﬂow characteristics are presented and the dynamics of the particles discussed.
3. Settling in turbulent ﬂow
The experiments are conducted in a random jet array facility (ﬁgure 2e), where turbulence
is generated by the continuous action of submerged water pumps as in Esteban, Shrimpton
922 A30-14
Settling behaviour of thin curved particles
0.5 1.0 1.5 2.0 2.5 3.0
0.16
0.20
0.24
0.28
C
0.12
123
6
4
2
×104
×104
Ar
Ar
ma
*
Figure 10. Fitting parameter Cas a function of Ar. In the context of underwater pendulums, Cis related to
a. Inset shows m
aversus Ar.
& Ganapathisubramani (2019a). However, the addition of a pulse-width modulation
system allows us to control turbulence intensity while the facility is in operation. The
characteristics of the turbulence generated are in table 3, and further details can be
found in Appendix A. Turbulence is produced so that all the particles’ characteristic
length scales are smaller than the horizontal integral length scale Lx. As mentioned in
§2.1, these particles have sizes comparable to the ones processed in actual recycling
facilities. The experimental procedure to release particles in this section is analogous
to the one previously presented. However, as turbulent ﬂow quantities can only be
predicted statistically, the number of repeated experiments per particle is increased to at
least 49. The minimum waiting time between releases is reduced to three minutes since
the background turbulence washes residual ﬂows away rapidly. Nonetheless, to ensure
statistical stationarity, the pumps are switched on for no less than 10 minutes before the
ﬁrst drop. We position the lower camera further back which resulted in a resolution of
0.35 mm pixel1and a magniﬁcation of 1/60. We also monitor the water temperature
for accurate estimation of the dimensionless parameters.
Data analysis is very similar to the cases in quiescent ﬂuid, with the main differences
being the identiﬁcation of the transients, and that the trajectories are no longer detrended
to account for horizontal drifts. The presence of background turbulence means any
transient effects are conﬁned to an even smaller section of the trajectory. Despite this,
for each descent in turbulence, we still remove the mean length of the transients for the
corresponding quiescent experiments from the trajectory.
3.1. Results and discussion
Several particle descents in turbulence are plotted in ﬁgure 11 (see the supplementary
videos). The ‘planar zigzag’ mode found in quiescent ﬂuid is still present, with the
dominant oscillation frequency over each trajectory nearly unchanged in all particles
tested. However, their motions are diversiﬁed by ﬂow ﬂuctuations and therefore trajectories
are no longer repeatable. Still, four types of special events are identiﬁed across all the
particles investigated: (1) ‘slow descents’, where the quiescent settling style remains but
922 A30-15
T.T.K. Chan and others
Turbulence statistics Values
ux/uz1.34
u
rms =(ux+2uz)/315.9mms
1
MFF 0.48
HD 0.07
Lturb 45.0mm
(Lx,Lz)(65.3,34.8)mm
(Lxx,Lzx )(93.9,45.4)mm
(Lzz,Lxz )(44.3,27.4)mm
(λf,λg)(6.8,6.5)mm
(λxx,λzx )(8.1,7.8)mm
(λzz,λxz )(6.1,5.8)mm
Reλ98
(Reλ,x,Reλ,y)(139,77)
Table 3. Statistics of the background turbulence such as the r.m.s. velocity ﬂuctuations, the mean ﬂow factor
(MFF), homogeneity deviation (HD), integral length scales and Taylor microscales. Here λfand λgdenote the
longitudinal and transverse Taylor microscales, respectively. The reader is referred to Appendix A for the full
deﬁnitions. The values in brackets correspond to the respective quantities in the column on the left.
0
2
4
6
x/D0
2
y/D
024
–1
0
2
4
6
x/D
0
1
y/D
01
–1
y/D
012
x/D
0
1
z/D
z/D
z/D
z/D
0
2
4
6
8
(a)
(b)
(c)
(d)
y/D
x/D
8
6
1
4
2
0
01
0
–1 –1
Figure 11. Trajectories of particles in turbulent ﬂow. Four special types of motions are observed though the
underlying zigzag mode seen in quiescent ﬂuid remains: (a) ‘slow descent’, (b) ‘rapid rotation’, (c) ‘vertical
descent’ and (d) ‘long gliding motion’. The positions of the events within the trajectories are marked by square
brackets on the side. Square markers denote locations corresponding to local minima of Vz.
vertical velocity is attenuated (ﬁgure 11a); (2) ‘rapid rotation’, where the direction of the
oscillations changes rapidly at the end of a swing (ﬁgure 11b); (3) ‘vertical descents’,
where the planar motion diminishes and the particle essentially falls straight down
(ﬁgure 11c); and (4) ‘long gliding motions’, where the gliding section in the ‘planar zigzag’
motion is especially long (4.8Din the illustrated case) and is sometimes preceded by a
large α(ﬁgure 11d). Apart from vertical descents, which we do not observe for particle
no. 1, these events occur for all the particles. Multiple types of the motions listed may
occur in a single descent. Remarkably, the particles never ﬂip over, possibly due to their
dihedral conﬁguration.
Slow descents probably occur when the object encounters strong incident ﬂows that
enhance lift. As the smallest particle does not generate sufﬁcient lift to fully glide in
quiescent ﬂuid, it indeed rarely exhibits this behaviour. Rapid rotations can emerge when
the solid enters a region of horizontal shear, causing it to rotate and sometimes roll slightly.
922 A30-16
Settling behaviour of thin curved particles
Efluc(mm2 s–2)
0
0.2
0.4
0.6
0.8
(a)(b)
p.d.f.
Ar
123456 0.5 1.0 1.5 2.0 2.5 3.0
900
800
700
600
500
Ar ×104
Turbulent
Quiescent
///
///
1234
rh/D
rh/D
0
0.1
0.2
0.3
0.4
p.d.f.
Figure 12. (a) The p.d.f.s of rh/Dalong the descent. The solid line and solid data points are for quiescent ﬂuid
while the dotted one and hollow data points are for turbulent settling. The p.d.f.s of particles no. 1 (turbulent
case only) and no. 3 are displayed. The inset shows the same quantity but for all the objects dropped. The
symbols follow those introduced in table 1.(b) Vertical ﬂuctuation kinetic energy per unit mass Eﬂuc of the
various particles.
This kind of motion becomes more likely the smaller the Iroll or the larger the distance
between the centre-of-gravity and the centre-of-pressure (i.e. a longer moment arm).
Heuristically, assuming the centre-of-pressure coincides with the centre of the solid’s
circle of curvature when viewed at the front (ﬁgure 2b), the smallest particle has the
longest moment arm. Either way, the smallest object should be the most sensitive to
such shear. Long gliding motions appear possibly as the local background ﬂow has a
signiﬁcant component along the particle’s direction of motion, pushing it along. Finally,
we hypothesise that vertical descents happen when the object encounters a downdraught.
Slow descents and long gliding motions have also been found for disks falling under
background turbulence (Esteban et al. 2020). However, we noticed key distinctions in the
settling characteristics between these two geometries. First, rapid rotations have not yet
been observed for disks. Second, fast descents of disks differ from vertical descents of the
particles tested here. This type of motion for disks is always preceded by an especially large
α, so the disks are aligned with the direction of motion. However, this is not necessarily
the case for the bottle-fragment-like particles.
To assess the effect of turbulence on all the descents collectively, the height-integrated
radial probability density functions (p.d.f.s) and the speciﬁc kinetic energy ﬂuctuations
of Vz(i.e. half of the variance of ﬂuctuations of Vz), Eﬂuc, are shown in ﬁgure 12.To
accurately capture the radial displacement rh, non-transient parts of the trajectories are
centred so the origin coincides with the mean position of the ﬁrst swing.
The diversiﬁcation of the settling dynamics by background turbulence is also evident
here. For the horizontal motion, focusing ﬁrst on ﬁgure 12(a), the radial p.d.f. in quiescent
and turbulent ﬂows of particle no. 3 reveal that the most likely radial position remains
unchanged. This conﬁrms that the quiescent zigzag motion is still signiﬁcant at the
current turbulence level. Yet, the p.d.f. is now much broader, with particle dispersion
reaching multiple Dinstead of only rh/D1. The inset in ﬁgure 12(a) shows how the
radial dispersion of the particles in turbulence reduces as Ar increases. However, the
vertical component of the velocity ﬂuctuations are modiﬁed differently. These are shown in
ﬁgure 12(b), and demonstrate a strong increase in velocity ﬂuctuations about Vz. Hence,
the motion is destabilised to a similar extent over most Ar tested. This difference may be
922 A30-17
T.T.K. Chan and others
(a)(b)
×104
56
60
64
68
72
D (mm)
25 35 45 55 0.5 1.0 1.5 2.0 2.5 3.0
1000
1500
2000
2500
3000
3500
Turbulent
Quiescent
///
///
Ar
Re
Vz(mm s–1)
Figure 13. (a) Mean descent velocities Vzof the various particles in turbulence and quiescent ﬂuid. (b)
Dimensionless version of the descent velocity plot, Re(Ar).
attributed to gravity, which has been used by Byron et al. (2019) to explain an identical
trend for slip velocities of nearly neutrally buoyant cylinders in turbulence.
The effect of turbulence on the mean descent velocity Vzhas long been an area
of great interest. Figure 13(a) plots Vzagainst the particle characteristic length scale,
showing Vzreduces compared with the quiescent case, although the data lies within
the statistical deviation of the turbulent one. We note this result is congruent with the
slip velocity of nearly neutrally buoyant cylinders (Byron et al. 2019), and opposite
to Vzof inertial disks falling in background turbulence (Esteban et al. 2020). As
mentioned in § 1, Good et al. (2014) found that settling is hindered by turbulence when the
characteristic gravitational velocity is greater than the typical ﬂow velocity ﬂuctuations
u
rms. To compare this with our results, we formed an analogous quantity by replacing
the characteristic gravitational velocity with the mean descent velocity in quiescent ﬂuid
Vq.Here Vq/u
rms is found to lie in between 3.92 and 4.27. Hence, our results are in
agreement with the prediction by Good et al. (2014) which suggests that the mean descent
velocity would be reduced when Vq/u
rms >1. We recognise Vq/u
rms does not reﬂect
the multiscale nature of particle–turbulence interactions, and it may be more insightful to
employ a scale-dependent quantity instead. However, theoretically deriving such a quantity
for our particle geometry is highly non-trivial and is beyond the scope of this study.
To further investigate the cause of the hindered settling, the relation between Re and Ar
is shown in ﬁgure 13(b). The general trend observed is the same as in quiescent ﬂuid – with
an approximately linear relation for the three smallest particles and an increase in slope
for the last one – and an identical interpretation is employed. As considering quantities
averaged over entire trajectories do not seem to help explain the change in Vz, particle
motions are examined over trajectory sections. Esteban et al. (2020) studied the correlation
between Vzand the dominant frequency of each trajectory. Instead of following this
approach, where the existence of a single ‘weak’ event might be hidden by the presence of
more severe ones, we propose an alternative method to capture the effect of all the events,
the average descent velocities Vevent and the characteristic frequencies fevent conditioned
on each type of event. However, this leads to a practical question on the deﬁnition of an
‘event’.
Classifying events using the instantaneous vertical velocity provides reasonable results.
The positions corresponding to local minima of Vz(squares in ﬁgure 11) also match those
of the radial extrema reasonably well, and these are used to separate events. Each event
922 A30-18
Settling behaviour of thin curved particles
(a)(b)
Vevent /Vq
Vevent/Vq
Vevent /Vq
fevent/fqfevent/fq
fevent/fq
0.4
0.8
1.2
1.6
VD
SD
Particle no. 1 Particle no. 2
Particle no. 3 Particle no. 4
0
0.5
1.0
1.5
1230
0.5
1.0
1.5
LG
RR
0123
1230
Figure 14. (a) Scatter plots showing the relationship between the average descent velocities Vevent and
frequencies fevent per event. To better visualise the effect of background turbulence, these quantities are
normalised by the mean descent velocity Vqand the dominant vertical oscillation frequency fqin quiescent
ﬂuid. The points corresponding to the special events shown in ﬁgure 11 –slowdescent, rapid rotation, long
gliding motion and vertical descent – are annotated. (b) The average Vevent against fevent of all the events in
panel (a).
then essentially corresponds to a half-swing, with fevent being the inverse of its duration.
Figure 14(a) shows the mean descent velocity of each event Vevent versus fevent ,both
normalised by the corresponding mean values in quiescent liquid.
In general, events with small frequencies fevent can increase the descent velocity Vz,
while those with large fevent have the opposite effect. This is quantitatively illustrated by
ﬁgure 14(b) where the mean event velocity Veventis plotted against fevent. The same was
also found for disks in Esteban et al. (2020), although the trend here is less prominent
due to the moderate particle inertia. Also, particle no. 2 exhibits a wider variety of events
compared with particle no. 1 as reﬂected by the scatter in the data, in agreement with the
initial observation that certain types of motions are less frequent for smaller particles.
Contrary to the variation in the horizontal displacement (see ﬁgure 12a), turbulence
introduces more extreme events for the larger particles.
To correlate the four types of events with the modulation in frequency, ﬁgure 15 displays
the variation of Vzover their durations, the corresponding Vevent and fevent. Each type
of descent behaviour modiﬁes Vzdifferently: ‘slow descents’ have Vevent 0.5Vz
(ﬁgure 15a); rapid rotations have no signiﬁcant effects on Vz(ﬁgure 15b), meaning
the rotation is not coupled to the vertical motion; long gliding motions (ﬁgure 15c)
could considerably enhance settling, regardless of the initial pitch angle; vertical descents
(ﬁgure 15d) increase Vz. The behaviour of vertical descents is as expected since the
distance travelled is shorter compared with zigzag, and downdraughts force the particle
down. Although the limited depth of our tank means the vertical descent in ﬁgure 15(d)
is incomplete, we are conﬁdent that the complete event still increases Vzfor the reasons
above. In summary, as long gliding motions and vertical descents have small fevent/fq,they
correspond to points with small fevent and large Vevent in ﬁgure 14(a).
So far, it has been shown that low-frequency events such as long gliding motions and
vertical descents could enhance settling. However, Vzis smaller than the quiescent value
on the whole. This result is captured when plotting the p.d.f. of Vevent (ﬁgure 16). Before
proceeding, note that the deﬁnition of events used may overcount the slow ones. This is
mitigated by combining successive events with Vevent <0.4Vq. Though the threshold is
922 A30-19
T.T.K. Chan and others
Vz /VqVz /Vq
fevent/fq
= 1.14
fevent/fq
= 1.27
(a)
(d)
(c)
(b)2.5
2.9
3.3 0123
0123
5.5
5.9
6.3
z/Dz/D
2.5
3.0
3.5
4.0
4.5
5.0
5.5
fevent/fq
= 0.63
fevent/fq
= 0.40
0123 0123
6.0
6.5
7.0
7.5
8.0
8.5
9.0
Figure 15. The instantaneous descent velocities Vz(solid lines) and the mean values Vevent (dotted lines) of
the four types of events identiﬁed: (a) slow descent, (b) rapid rotation, (c) long gliding motion and (d) vertical
descent. The green lines show the locations of the events.
0.5 1.0 1.5 2.0
0
0.1
0.2
0.3
0.4
p.d.f.
0.5
Vevent /Vq
Figure 16. The p.d.f. of Vevent relative to Vq. To avoid overcounting slow events, the successive ones with
Vevent <0.4Vqhave been merged.
somewhat arbitrary, it does not affect the following discussion. The reduction in Vzis
manifested as a slight leftward shift of the entire p.d.f.
Among the four types of events identiﬁed, only slow descents reduce the settling speed.
However, we recognise that the events described are the most readily detected ones and
do not constitute an exhaustive list. Particle settling in turbulence is a highly complex
and multiscale phenomenon (Tom & Bragg 2019) that exhibits a number of more subtle
922 A30-20
Settling behaviour of thin curved particles
unclassiﬁed events. We therefore believe the attenuation in settling may be caused by the
less discernible events. As the falling particle resembles a swept-back wing in the direction
of motion and larger particles glide more in quiescent ﬂuid, it is possible that under most
conditions, the turbulence provides slightly more lift without considerably changing the
basic zigzag motion.
Admittedly, such a result is unexpected. Since the particle sizes are of the same order
as the integral length scale Lturb, we anticipated the solid to exhibit downward sweeping
motions triggered by interactions with large vortices. However, the object’s inherent
stability likely suppresses these motions.
4. Concluding remarks
Motivated by the numerous applications of particle settling, such as differentiating plastic
from glass in hydrodynamic separators, 3D-printed rigid thin curved solids resembling
bottle fragments were dropped in a water tank in quiescent ﬂuid and in homogeneous
anisotropic turbulence.
In quiescent liquid, the particles underwent planar zigzag descent and their trajectories
were divided into gliding and turning sections. While one might expect the average
vertical velocity Vzto vary monotonically with particle size, a minimum was found at
D38 mm (Ar 1.8×104). Closer examination of the settling behaviour showed that
the horizontal oscillation amplitude Aand radius of curvature Lpend normalised by particle
size were generally enhanced for larger particles. On the contrary, the oscillation frequency
fand the maximum pitch angle αmax, which was directly proportional to az,max, decreased
monotonically. These suggested enhanced lift generation as the particle size grew, which
was supported by a closer alignment between αand the direction of motion. This led to
the initial reduction in Vz. The subsequent settling enhancement for the larger objects
was due to more rapid horizontal motion at midswing locations coupled with their initial
pitch down attitude at the beginning of each swing. All the trajectories observed could be
modelled reasonably well by undamped underwater pendulums descending at a constant
velocity.
The zigzag motion was also observed for settling in turbulence, but ﬂuctuations in the
ﬂow modiﬁed it so the radial dispersion increased considerably. Notably, the particles
never ﬂipped over although their sizes were comparable to Lturb. In agreement with Good
et al. (2014), Vzwas slightly lower than in quiescent ﬂuid for Vq/u
rms >1. Four special
types of events comprising slow descents, vertical descents, long gliding motions which
were sometimes preceded by large pitch angles, and rapid rotations, were identiﬁed. Also,
each type of motion was related to the particle kinematics and to the descent velocity.
In general, vertical descents and long gliding sections sped up settling. By dividing
each trajectory into a collection of events, those with a low frequency were found to be
capable of enhancing the descent, while the opposite occurred for high-frequency events.
Nevertheless, the p.d.f. of Vevent was unimodal and the reduction of Vzwas reﬂected
by a leftward shift. This may suggest the background ﬂow slightly modulated each event
by enhancing lift production, so the change in Vzcould not be simply connected to the
special events. The above also underlines the difﬁculty of studying descent behaviour with
background turbulence.
Future research may therefore focus on wake visualisation of these particles in both
turbulence and quiescent ﬂuid. As transitions in settling behaviour are usually correlated
to a change in wake structure (see e.g. Ern et al. 2012; Auguste et al. 2013;Leeet al.
2013; Esteban et al. 2019c; Toupoint et al. 2019)andαmax az,max found here indirectly
supports this argument, observing the wake may reveal other types of events and the effects
922 A30-21
T.T.K. Chan and others
of anisotropic geometries. This may further explain the change of Vzin turbulence
and the lift enhancement as the particle size increased in quiescent ﬂuid. Moreover, it
may uncover why certain trends reversed for particle no. 4, where θ>90. Theoretical
development may concentrate on ﬁnding a suitable scale-dependent metric for anisotropic
particles to distinguish between enhanced and hindered settling in turbulence. Finally,
additional development of the pendulum model is desirable. An emphasis should be placed
on interpreting Cas it may complement the current experimental observations and improve
the predictive power of the model.
Supplementary movies. Supplementary movies are available at https://doi.org/10.1017/jfm.2021.520.
Acknowledgements. We thank J.B. Will for fruitful discussions and D. Krug for drawing our attention to
literature that modelled settling behaviour with pendulums.
Funding. T.T.K.C. thanks the Internship Ofﬁce at the University of Twente and the Faculty Ofﬁce of the
Faculty of Engineering and Physical Sciences at the University of Southampton. He is partially funded by the
University of Twente Scholarhip and the Erasmus+Traineeship Scholarship. S.G.H. acknowledges MCEC for
ﬁnancial support.
Declaration of interests. The authors report no conﬂict of interest.
Data availability statement. All data supporting this study are openly available from the University of
Southampton repository at https://doi.org/10.5258/SOTON/D1859.
Author ORCIDs.
Timothy T.K. Chan https://orcid.org/0000-0003-2363-0403;
Luis Blay Esteban https://orcid.org/0000-0003-4675-6957;
Sander G. Huisman https://orcid.org/0000-0003-3790-9886;
John S. Shrimpton https://orcid.org/0000-0003-2510-6373;
Bharathram Ganapathisubramani https://orcid.org/0000-0001-9817-0486.
Appendix A. Turbulence generation and characteristics
As explained in § 3, the experiments are conducted in a random jet array facility (ﬁgure 2e),
where turbulence is generated by the continuous action of submerged water pumps. These
pumps, arranged in two 8 ×6 arrays with vertical and horizontal mesh lengths of 10 cm
on either side of the tank, ﬁre independently according to the ‘sunbathing algorithm’ to
generate statistically stationary homogeneous anisotropic turbulence with negligible mean
ﬂow (Variano & Cowen 2008; Esteban et al. 2019a). The durations of the ‘on’ and ‘off’
signals are randomly selected from two separate Gaussian distributions with their mean
values and standard deviations denoted by μon/off and σon/off , respectively. In this case,
μon ±σon =(3±1)sandμoff ±σoff =(21 ±7)s. When the pumps are active, water is
drawn radially at their bases and expelled horizontally out of their cylindrical nozzles with
a diameter of 18 mm. To improve isotropy and protect the particles from collisions with the
pumps, a 13 mm square mesh is placed 3 cm downstream of the jets. Turbulence intensity
is controlled through modulating the power supplied by pulse-width modulation. For more
information on the turbulence facility, the reader is referred to Esteban et al. (2019a). The
equipment is identical apart from the addition of the mesh and the power control system.
Prior to releasing particles, the turbulence generated is characterised with particle image
velocimetry (PIV). The ﬂow was seeded with 56 µm polyamide particles (Vestosint 2157).
A laser sheet passing through the centre of the tank contained in the xz-plane is generated
(Litron BERN 200-15PIV), and 3000 image pairs are taken at 0.8 Hz (VC-Imager Pro
LX 16M). The interpulse time is set to 4000 µs to limit the tracer displacements to
approximately 6 pixels and reduce the out-of-plane displacements between image pairs.
922 A30-22
Settling behaviour of thin curved particles
(|Ux|+2|Uz|)/3 (mm s–1)
6
4
2
0
x
16
15
14
z
(|ux|+2|uz|)/3 (mm s–1)
17
18
(a)(b)
25 mm 25 mm
x
z
Figure 17. The PIV measurements of the ﬂow in the turbulence box. The time-averaged (a) r.m.s. ﬂow velocity
ﬂuctuation ﬁeld and (b) mean ﬂow ﬁeld at the middle of the tank. The subscripts (x,z)denote the corresponding
velocity components.
To characterise the turbulence generated, we decompose the ﬂow velocity into mean
and ﬂuctuating components Uf(x,t)=Umean(x,t)+uﬂuc (x,t),wherexis the position
vector. Figure 17 shows the two ﬁelds, where (Ux(x), Uz(x)) and (ux(x), uz(x)) are
the time-averaged (x,z)components of Umean and of the r.m.s. of uﬂuc, respectively.
The ﬂuctuations appear homogeneous although there is some horizontal mean ﬂow caused
by the synthetic jets emitted by the pumps. These are quantitatively expressed by the
homogeneity deviation HD and the mean ﬂow factor MFF. Assuming symmetry about
the x-axis (Carter & Coletti 2017), u
rms =(ux+2uz)/3, where the line above denotes
spatial averaging. Then HD =2σu/u
rms =0.07 1 (Esteban 2019), where σuis the
standard deviation of u
rms in space. Thus the turbulence is indeed homogeneous. Denoting
the mean ﬂow speed by U, the relative magnitude of the mean ﬂow is assessed by
MFF =U/u
rms =0.48. While a small mean ﬂow is present, velocity ﬂuctuations still
dominate so we believe it has no signiﬁcant effect on the settling characteristics of the
particles tested. Nonetheless, the global isotropy ux/uz=1.34 >1 shows the turbulence
is mildly anisotropic. This implies the integral length scales and Taylor microscales depend
on the direction of the velocity component and of the spatial separation.
Taking this into account, ﬁgure 18 gives the various autocorrelation functions along the
vertical and horizontal directions, ρij . They decay as rincreases and approach 0 at r
+∞. Thus, we deﬁne the upper integration limit r0for the integral length scale Lij such that
ρij(r0)ﬁrst reaches 0.01. This is in line with the suggestion in O’Neill et al. (2004): taking
r0as the ﬁrst zero-crossing of ρij balances accuracy with ease of calculation. Furthermore,
if the directly measured autocorrelation does not reach ρij 0.01, an exponential tail is
ﬁtted for ρij 0.35. Table 3 includes the various Lij found.
Integral length scales involving velocity ﬂuctuations along the x-direction are larger than
those along z. This was also found by Carter & Coletti (2017) in a similar facility despite
a different Reλ, suggesting eddies were elongated by the larger ﬂuctuations. Following
their suggestion, the geometric mean of integral length scales involving ﬂuctuations along
one direction is taken to represent the size of the largest vortices in that orientation,
i.e. Lx=(LxxLzx )1/2, for instance. To facilitate comparison with previous experiments,
a conventional integral length scale assuming axisymmetry
Lturb =Lx+2Lz
3(A1)
is evaluated too.
922 A30-23
T.T.K. Chan and others
r (mm)
1.0
ρxx
ρij
ρzx
ρxz
ρzz
0.8
0.6
0.4
0.2
0
–0.2 0 40 80 120 160 200
Figure 18. Autocorrelation functions ρij of the j-component velocity ﬂuctuations along the i-direction. The
solid lines give the measured data while the dashed line shows the exponential ﬁt. The spacing between markers
is not indicative of the resolution.
The Taylor microscale along the i-direction of j-component velocity ﬂuctuations, on
the other hand, is evaluated according to its deﬁnition λij =(1
2(d2ρij/dr2)|r=0)0.5.To
minimise PIV error, we only consider the ﬁrst two values of ρij with a positive separation
whose interrogation windows do not overlap (Adrian & Westerweel 2011). The horizontal
intercept of the ﬁtted parabola then equals λij. The conventional longitudinal and
transverse Taylor microscales λfand λgare found assuming axisymmetry in analogy to
(A1).
The related Reynolds number Reλis also determined using the measured water
temperature of 17 C. The direction-dependent values Reλ,i=λg,iui ,whereλg,iis
the transverse Taylor microscale involving i-component velocity ﬂuctuations ui.The
conventional axisymmetric Reλand all the quantities discussed above are displayed in
table 3.
All in all, these measurements show the background turbulence is homogeneous but
mildly anisotropic.
REFERENCES
ADRIAN,R.J.&WESTERWEEL,J.2011Particle Image Velocimetry. Cambridge University Press.
ARDEKANI, M.N., COSTA,P.,BREUGEM,W.P.&BRANDT, L. 2016 Numerical study of the sedimentation
of spheroidal particles. Intl J. Multiphase Flow 87, 16–34.
AUGUSTE,F.,MAGNAUDET,J.&FABRE, D. 2013 Falling styles of disks. J. Fluid Mech. 719, 388–405.
BAHADORI,A.2014Natural Gas Processing: Technology and Engineering Design. Gulf Professional
Publishing.
BEC,J.,CENCINI,M.,HILLERBRAND,R.&TURITSYN, K. 2008 Stochastic suspensions of heavy particles.
Physica D237 (14–17), 2037–2050.
BELMONTE,A.,EISENBERG,H.&MOSES, E. 1998 From ﬂutter to tumble: inertial drag and Froude
similarity in falling paper. Phys. Rev. Lett. 81 (2), 4.
BYRON, M.L., TAO,Y.,HOUGHTON,I.A.&VARIANO, E.A. 2019 Slip velocity of large low-aspect-ratio
cylinders in homogeneous isotropic turbulence. Intl J. Multiphase Flow 121, 103120.
CARTER,D.W.&COLETTI, F. 2017 Scale-to-scale anisotropy in homogeneous turbulence. J. Fluid Mech.
827, 250–284.
CHRUST,M.,BOUCHET,G.&DUŠEK, J. 2013 Numerical simulation of the dynamics of freely falling discs.
Phys. Fluids 25 (4), 044102.
922 A30-24
Settling behaviour of thin curved particles
ERN,P.,RISSO,F.,FABRE,D.&MAGNAUDET, J. 2012 Wake-induced oscillatory paths of bodies freely
rising or falling in ﬂuids. Annu. Rev. Fluid Mech. 44 (1), 97–121.
ESTEBAN, L.B. 2019 Dynamics of non-spherical particles in turbulence. PhD thesis, University of
Southampton.
ESTEBAN, L.B., SHRIMPTON,J.&GANAPATHISUBRAMANI, B. 2018 Edge effects on the ﬂuttering
characteristics of freely falling planar particles. Phys. Rev. Fluids 3(6), 064302.
ESTEBAN, L.B., SHRIMPTON,J.&GANAPATHISUBRAMANI,B.2019aLaboratory experiments on the
temporal decay of homogeneous anisotropic turbulence. J. Fluid Mech. 862, 99–127.
ESTEBAN, L.B., SHRIMPTON,J.&GANAPATHISUBRAMANI,B.2019bStudy of the circularity effect on
drag of disk-like particles. Intl J. Multiphase Flow 110, 189–197.
ESTEBAN, L.B., SHRIMPTON,J.&GANAPATHISUBRAMANI,B.2019cThree dimensional wakes of freely
falling planar polygons. Exp. Fluids 60 (7), 114.
ESTEBAN, L.B., SHRIMPTON,J.&GANAPATHISUBRAMANI, B. 2020 Disks settling in turbulence. J. Fluid
Mech. 883, A58.
ESTEBAN, L.B., SHRIMPTON,J.,ROGERS,P.&INGRAM, R. 2016 Three clean products from co-mingled
waste using a novel hydrodynamic separator. Intl J. Sustain. Dev. Plan. 11 (5), 792–803.
FERNANDES, P.C., ERN,P.,RISSO,F.&MAGNAUDET, J. 2005 On the zigzag dynamics of freely moving
axisymmetric bodies. Phys. Fluids 17 (9), 098107.
FIELD, S.B., KLAUS,M.,MOORE,M.G.&NORI, F. 1997 Chaotic dynamics of falling disks. Nature 388
(6639), 252–254.
FORNARI,W.,PICANO,F.&BRANDT,L.2016aSedimentation of ﬁnite-size spheres in quiescent and
turbulent environments. J. Fluid Mech. 788, 640–669.
FORNARI,W.,PICANO,F.,SARDINA,G.&BRANDT,L.2016bReduced particle settling speed in turbulence.
J. Fluid Mech. 808, 153–167.
GOOD, G.H., IRELAND, P.J., BEWLEY, G.P., BODENSCHATZ, E., COLLINS, L.R. & WARHAFT,Z.2014
Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech. 759,R3.
HEISINGER, L., NEWTON,P.&KANSO, E. 2014 Coins falling in water. J. Fluid Mech. 742, 243–253.
HO, H.W. 1964 Fall velocity of a sphere in an oscillating ﬂuid. PhD thesis, University of Iowa.
HOROWITZ,M.&WILLIAMSON, C.H.K. 2006 Dynamics of a rising and falling cylinder. J. Fluid Struct. 22,
837–843.
HOROWITZ,M.&WILLIAMSON, C.H.K. 2010aThe effect of Reynolds number on the dynamics and wakes
of freely rising and falling spheres. J. Fluid Mech. 651, 251–294.
HOROWITZ,M.&WILLIAMSON, C.H.K. 2010bVortex-induced vibration of a rising and falling cylinder.
J. Fluid Mech. 662, 352–383.
JENNY,M.,DUŠEK,J.&BOUCHET, G. 2004 Instabilities and transition of a sphere falling or ascending
freely in a Newtonian ﬂuid. J. Fluid Mech. 508, 201–239.
LAU, E.M., HUANG,W.X.&XU, C.X. 2018 Progression of heavy plates from stable falling to tumbling
ﬂight. J. Fluid Mech. 850, 1009–1031.
LEE,C.,SU, Z., ZHONG,H.,CHEN,S.,WU,J.&ZHOU, M. 2013 Experimental investigation of freely
falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral. J. Fluid Mech. 732,
77–104.
MAHADEVAN, L., RYU,W.S.&SAMUEL, A.D.T. 1999 Tumbling cards. Phys. Fluids 11 (1), 1–3.
MATHAI,V.,LOEFFEN, L.A.W.M., CHAN, T.T.K. & SANDER, W. 2019 Dynamics of heavy and buoyant
underwater pendulums. J. Fluid Mech. 862, 348–363.
MATHAI,V.,ZHU,X.,SUN,C.&LOHSE, D. 2018 Flutter to tumble transition of buoyant spheres triggered
by rotational inertia changes. Nat. Commun. 9(1), 1792.
MAXEY, M.R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random
ﬂow ﬁelds. J. Fluid Mech. 174, 441–465.
MAXEY,M.R.&CORRSIN, S. 1986 Gravitational settling of aerosol particles in randomly oriented cellular
ﬂow ﬁelds. J. Atmos. Sci. 43 (11), 1112–1134.
NIELSEN, P. 1984 On the motion of suspended sand particles. J. Geophys. Res. 89 (C1), 616–626.
NIELSEN, P. 1992 Coastal Bottom Boundary Layers and Sediment Transport. World Scientiﬁc.
NIELSEN, P. 1993 Turbulence effects on the settling of suspended particles. J. Sedim. Res. 63 (5), 835–838.
O’NEILL, P.L., NICOLAIDES,D.,HONNERY,D.&SORIA, J. 2004 Autocorrelation functions and the
determination of integral length with reference to experimental and numerical data. In Proceedings of
the Fifteenth Australasian Fluid Mechanics Conference (ed. M. Behnia, W. Lin & G.D. McBain), pp. 1–4.
The University of Sydney.
SMITH, E.H. 1971 Autorotating wings: an experimental investigation. J. Fluid Mech. 50 (3), 513–534.
922 A30-25
T.T.K. Chan and others
STRINGHAM, G.E., SIMONS,D.B.&GUY, H.P. 1969 The behaviour of large particles falling in quiescent
liquids. USGS Paper 562-C. US Department of the Interior.
TANABE,Y.&KANEKO, K. 1994 Behavior of a falling paper. Phys. Rev. Lett. 73 (10), 1372–1375.
TOM,J.&BRAGG, A.D. 2019 Multiscale preferential sweeping of particles settling in turbulence. J. Fluid
Mech. 871, 244–270.
TOUPOINT,C.,ERN,P.&ROIG, V. 2019 Kinematics and wake of freely falling cylinders at moderate
Reynolds numbers. J. Fluid Mech. 866, 82–111.
TRIPATHY, S.K., BHOJA, S.K., KUMAR,C.R.&SURESH, N. 2015 A short review on hydraulic classiﬁcation
and its development in mineral industry. Powder Technol. 270, 205–220.
VARIANO, E.A. & COWEN, E.A. 2008 A random-jet-stirred turbulence tank. J. Fluid Mech. 604, 1–32.
VELDHUIS, C.H.J. & BIESHEUVEL, A. 2007 An experimental study of the regimes of motion of spheres
falling or ascending freely in a Newtonian ﬂuid. Intl J. Multiphase Flow 33 (10), 1074–1087.
VOTH,G.A.&SOLDATI, A. 2017 Anisotropic particles in turbulence. Annu. Rev. Fluid Mech. 49, 249–276.
WASILEWSKI,M.&SINGH BRAR, L . 2017 Optimization of the geometry of cyclone separators used in clinker
burning process: a case study. Powder Technol. 313, 293–302.
WILLMARTH, W.W., HAWK, N.E. & HARVEY, R.L. 1964 Steady and unsteady motions and wakes of freely
falling disks. Phys. Fluids 7(2), 197.
ZHONG,H.,CHEN,S.&LEE, C. 2011 Experimental study of freely falling thin disks: transition from planar
zigzag to spiral. Phys. Fluids 23 (1), 011702.
ZHONG,H.,LEE,C.,SU, Z., CHEN,S.,ZHOU,M&WU, J. 2013 Experimental investigation of freely falling
thin disks. Part 1. The ﬂow structures and Reynolds number effects on the zigzag motion. J. Fluid Mech.
716, 228–250.
ZHOU,W.,CHRUST,M&DUŠEK, J. 2017 Path instabilities of oblate spheroids. J. Fluid Mech. 833, 445–468.
ZHOU,W.&DUŠEK, J. 2015 Chaotic states and order in the chaos of the paths of freely falling and ascending
spheres. Intl J. Multiphase Flow 75, 205–223.
922 A30-26
... At low Re, in analogy with [7], we expect that a body will settle concave-upwards. Whereas experiments [4,11] suggest that only the concave-upward orientation is stable at moderate to large Re, we find that for a finite range of Re, the φ = 0 orientation is stable, indicating a stability boundary. In particular, we find a transcritical bifurcation at Re (1) c ≈ 2.5, where the unstable φ = 0 orientation becomes a stable spiral, and a subcritical pitchfork bifurcation at Re (2) c ≈ 15, where the stable spiral at φ = 0 becomes an unstable spiral. ...
... The point φ = π is weakly unstable, due to the periodic shedding of vortices behind the body as shown in Fig. 2(d). The trajectories in φ,φ space reach limit cycles whose amplitudes increase with increasing Re, as seen in Fig. 4. The periodic oscillations about φ = π are consistent with the experimental observations of curved bodies by Chan et al. [11]. The weak instability at φ = π is also reminiscent of the behavior of passively hovering bodies in oscillatory flow [e.g., 8] where the concave-downwards c , the φ = 0 orientation is unstable and all initial orientations φ 0 lead to steady descent at φ = π. ...
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