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Settling behaviour of thin curved particles in quiescent fluid and turbulence

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  • 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
Email address for correspondence: g.bharath@soton.ac.uk
© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
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
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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
natural gas industry (Bahadori 2014), to improve clinker burning processes (Wasilewski
& 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
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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)
Steady
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
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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.
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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.
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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
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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.
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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
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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).
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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.
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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)
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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
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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
constant vertical descent velocity reads
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
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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
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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
the added-mass coefficient m
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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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... 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. ...
Preprint
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We study the orientation dynamics of two-dimensional concavo-convex solid bodies more dense than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle $\phi$ relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number $Re_{c}^{(1)}$, and a subcritical pitchfork bifurcation at a Reynolds number $Re_{c}^{(2)}$. For $Re<Re_{c}^{(1)}$, the concave-downwards orientation of $\phi=0$ is unstable and bodies overturn into the $\phi=\pi$ orientation. For $Re_{c}^{(1)}<Re<Re_{c}^{(2)}$, the falling body has two stable equilibria at $\phi=0\text{ and }\phi=\pi$ for steady descent. For $Re>Re_{c}^{(2)}$, the concave-downwards orientation of $\phi=0$ is again unstable, and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable $\phi=\pi$ orientation. The $Re_{c}^{(2)}\approx15$ at which the subcritical pitchfork bifurcation occurs is distinct from the $Re$ for the onset of vortex shedding, which causes the $\phi=\pi$ equilibrium to also become unstable, with bodies fluttering about $\phi=\pi$. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to settling of mollusk shells.
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This paper presents a study of the terminal fall velocity, drag coefficient and descent style of ‘wavy-edge’ flat particles. Being highly non-spherical and with a size of up to a few centimeters, these particles show strong self-induced motions that lead to various falling styles that result in distinct drag coefficients. This study is based on experimental measurements of the instantaneous 3D velocity and particle trajectory settling in water. A disk of diameter 30 mm, thickness 1.5 mm and density 1.38 g/cm3 is manufactured as a reference particle. The disk was initially designed to lie within the Galileo number - dimensionless moment of inertia (G-I*) domain corresponding to the fluttering regime. A total of 35 other particles with the same frontal area and material properties were manufactured. These are manufactured to have different amplitudes (a) of the sinusoidal wave on the edge and number of cycles (N) around the entire perimeter. Thus, 5 sets of particles are manufactured with different relative wave amplitudes; i.e. a/D= 0.03, 0.05, 0.1, 0.15, 0.2. Each set consisting of 7 particles from N=4 to N=10. The isoperimetric quotient is used as a measure of the particle circularity and is also linked with different characteristic settling behaviours. Disks and other planar particles with small a/D ratio were found to descent preferably with ‘Planar zig-zag’ behaviour with events of high tilted angle. In contrast, particles with high a/D ratio were found to follow a more uniform descent with low tilted angle to the vertical motion. The differences in projected frontal area were shown not to be sufficient to compensate the differences in descent velocity, leading to unequal drag coefficients. Therefore, we believe that the falling styles of these irregular particles go hand by hand with characteristic wake structures, as shown for disks with various dimensionless moment of inertia, that enhance the descent of particles with low circularity.