Methods for Measuring the
Orientation and Rotation Rate of 3D-Printed Particles in Turbulence
AUTHORS:
Cole,
Brendan C.
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
bcole@wesleyan.edu
Marcus,
Guy G.
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
guygmarcus@jhu.edu
Parsa, Shima
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
sparsa@seas.harvard.edu
Kramel,
Stefan
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
skramel@wesleyan.edu
Ni, Rui
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
ruiniphy@gmail.com
Voth,
Greg A.
Department
of Physics
Wesleyan
University
Middletown,
CT, USA
gvoth@wesleyan.edu
CORRESPONDING AUTHOR:
Greg
Voth
KEYWORDS:
particles
in turbulence, anisotropic particles, turbulence, 3D printing, rotation, fluid
dynamics
SHORT ABSTRACT:
We use
3D printing to fabricate anisotropic particles in the shapes of jacks, crosses,
tetrads, and triads, whose alignments and rotations in turbulent fluid flow can
be measured from multiple simultaneous video images.
LONG ABSTRACT:
Experimental
methods are presented for measuring the rotational and translational motion of
anisotropic particles in turbulent fluid flows. 3D printing technology is used
to fabricate particles with slender arms connected at a common center. Shapes
explored are crosses (two perpendicular rods), jacks (three perpendicular rods),
triads (three rods in triangular planar symmetry), and tetrads (four arms in tetrahedral
symmetry). Methods for producing on the order of 10,000 fluorescently dyed
particles are described. Time-resolved measurements of their orientation and
solid-body rotation rate are obtained from four synchronized videos of their
motion in a turbulent flow between oscillating grids with
INTRODUCTION:
In a
recent publication, we introduced the use of particles made from multiple slender
arms for measuring rotational motion of particles in turbulence[1]. These particles can be fabricated
using 3D printers, and it is possible to accurately measure their position,
orientation, and rotation rate using multiple cameras. Using tools from slender
body theory, it can be shown that these particles have corresponding effective ellipsoids[2],
and the rotational motions of these particles are identical to those of their
respective effective ellipsoids. Particles with symmetric arms of equal length
rotate like spheres. One such particle is a jack, which has three mutually
perpendicular arms attached at its center. Adjusting the relative lengths of
the arms of a jack can form a particle equivalent to any tri-axial ellipsoid.
If the length of one arm is set equal to zero, this creates a cross, whose
equivalent ellipsoid is a disk. Particles made of slender arms take up a small
fraction of the solid volume of their solid ellipsoidal counterparts. As a
result, they sediment more slowly, making them easier to density match. This allows
the study of much larger particles than is convenient with solid ellipsoidal
particles. Additionally, imaging can be performed at much higher particle
concentrations because the particles block a smaller fraction of the light from
other particles.
In this
paper, methods for fabrication and tracking of 3D-printed particles are
documented. Tools for tracking the translational motion of spherical particles from
particle positions as seen by multiple cameras have been developed by several
groups[3],[4]. Parsa et al[5] extended this approach to track rods
using the position and orientation of the rods seen by multiple cameras. Here, we
present methods for fabricating particles of a wide variety of shapes and
reconstructing their 3D orientations. This offers the possibility to extend 3D
tracking of particles with complex shapes to a wide range of new applications.
This
technique has great potential for further development because of the wide range
of particle shapes that can be designed. Many of these shapes have direct
applications in environmental flows, where plankton, seeds, and ice crystals
come in a vast array of shapes. Connections between particle rotations and
fundamental small-scale properties of turbulent flows[6] suggest that study of rotations of
these particles provides new ways to look at the turbulent cascade process.
PROTOCOL:
1.
Fabrication of Particles
1.1.
Use
a 3D Computer Aided Drafting program to create particle models. Export one file
per model in a file format that can be processed by the 3D printer used.
1.1.1. Use the Circle
command to draw a circle with a diameter of .3 mm. Use the Extrude function to
make a cylinder with a length of 3 mm.
1.1.2. Make a cross
with two orthogonal cylinders with a common center; make a jack with three
mutually orthogonal cylinders with a common center; make a tetrad with four
cylinders sharing a common end at 109.5° angles to one another; make a triad
with three cylinders in a plane sharing a common end at 120° angles to one
another.
1.1.3. To tilt
cylinders (hereafter called “arms” of the particles) with respect to one
another, use the Rotate 3D command to draw a line across the diameter of the
circle at one of its ends and then enter the desired angle of rotation.
1.1.4. Use the Union
command to join the different arms together into a single watertight object.
1.1.5. Use Rotate 3D
again to tilt the object so that no arms are along the vertical or horizontal
axes, because arms that lie along these axes tend to have defects, break off
more easily, or flatten out.
1.1.6.
Export
each object in a separate file in a format that can be used by 3D printers.
1.2.
Order
approximately 10,000 particles of each type from a commercial source that
specializes in additive manufacturing or print them at an available facility. Particles
should be printed on a polymer extrusion printer that uses a support matrix of
a different material that can be dissolved away.
1.2.1.
Order
the particles three weeks or more before experiments are planned because the
arrangement and printing of so many particles is a slow process. Ensure that
particles are printed on “high-resolution mode” because the particles are near
the minimum feature size of many 3D printers and the arms will not be as
symmetric and may break if printed at lower resolution.
2.
Preparation of Particles
2.1.
Prepare
a salt solution in which the particles are neutrally buoyant to minimize
particles’ arms bending while in storage and so that gravitational and buoyancy
forces do not have to be accounted for in the analysis.
2.1.1. Test average
particle densities by immersing particles in solutions of water mixed with calcium
chloride (CaCl2) at densities around 1.20 g/cm3.
2.1.1.1.
To
determine water density, first zero a scale while an empty 100 mL volumetric
flask is on top of it. Take the flask off and fill it with water mixed with
CaCl2. Place the flask back atop the scale and divide the given mass
by 100 mL.
Note:
Because 1 mL = 1 cm3, 1 g/mL = 1 g/cm3.
2.1.1.2.
Test
particles at many different solution densities, ranging from 1.16 g/cm3
to 1.25 g/cm3, in roughly 0.01 g/cm3 increments. Test
multiple particles at every density because not all particles will have the
same density: in the same solution, some will sink, some will be neutrally
buoyant, and some will float.
2.1.2.
Record
at which density particles are, on average, neutrally buoyant after several
hours.
Note:
The density found may be significantly different from the bulk density quoted
by the particle manufacturers.
2.1.3. Mix about 400
kg of CaCl2 into approximately 1600 L of water until the solution is
at the density recorded in 2.1.1 – 2.1.2.
2.1.4.
Remove
about 1 L of this mixed solution per particle type (jacks, tetrads, etc.) to be
used for storage of particles. Hold each liter in a different container at room
temperature. Store the remainder of the solution at room temperature in a large
storage tank.
[Place Figure 1 here]
2.2.
Manually loosen the support material in
which the particles come encased by gently breaking the large pieces (~ 5 x 320 mm, part of which is
shown in Figure 1a) into
small sections (~5 x 5 mm, Figure 1b), then manually massage each section until much of the
excess resin has come off (Figure 1c-e). Remove excess resin in this way
to reduce the amount of the NaOH solution that will need to be created for
steps 2.2.1 – 2.2.4.
2.2.1.
Place the remaining resin block in a
10% by mass sodium hydroxide (NaOH) solution immersed in an ultrasonic bath for
one hour. The resin is
a different material than the particles are, so the NaOH will remove the resin
without permanently affecting the particles.
CAREFUL:
The solution is corrosive and will get hot while in the ultrasonic bath.
2.2.2.
Filter
out particles.
2.2.2.1.
To
filter particles, create a funnel using netting with .1016 x .13462 cm plastic
holes. Hold the funnel over the container to be used for the disposal of the
NaOH solution and slowly pour the solution through. Dispose of NaOH solution in
accordance with environmental health and safety guidelines.
2.2.3.
Rinse
particles gently with water before immersing in a new 10% by mass NaOH solution
in an ultrasonic bath for another half hour.
2.2.4. Filter out
particles as in 2.2.2.1 and store in the density-matched solution separated in
2.1.4 while they harden. Handle the particles carefully because the NaOH
solution temporarily softens them.
Note: If particles are not stored in a density-matched solution, some
arms may bend. Keeping
them immersed in the density-matched solution for several hours also allows
some voids in the plastic to fill with fluid.
2.3.
Dye
particles with Rhodamine-B mixed with water so that they fluoresce under the
light emitted by a green laser.
2.3.1. Prepare a 1 L
solution of Rhodamine-B dye in water at a concentration of .5 g/L (afterward referred to as “dye”).
CAUTION:
Toxic.
2.3.2.
Heat
the dye to a temperature between 50 and 80 °C, depending on particle material.
Use higher temperatures for harder plastics; using too high of a temperature
will result in the arms bending.
2.3.3.
Put ~2,500 particles, enough to loosely
fill ~25 mL in the density-matched storage solution, in the dye and keep all at
80 °C for two to three hours to allow the dye to absorb into the polymer.
Remove particles once they are pink,
like the one in Figure 1f.
CAREFUL: The heat will soften the
particles temporarily.
2.3.4.
Filter
out particles and rinse them under a faucet before storage in the designated
solutions separated in 2.1.4. The particles will lose a small fraction of their
dye, making the solution pink, but rinsing under the faucet helps prevent
losing a detrimental amount of dye..
Note: Average particle density will
have changed due to dyeing, so test again as in 2.1.1–2.1.2 to find the new
solution density at which particles are, on average, neutrally buoyant.
2.4.
Change
bulk CaCl2 solution (from 2.1.3) density as needed. Repeat 2.1.4 and
remove new volumes of density-matched solution. Dispose of former storage
solutions, which will now have small amounts of Rhodamine-B dye in them, in
accordance with environmental health and safety regulations.
2.5.
Repeat
2.3.2–2.3.4 for successive sets of ~2500 particles, storing all particles of
the same shape in the same density-matched solutions created in 2.4, separated
from particles of different shapes.
Note:
After about 5 repetitions of 2.3.2–2.3.4, the Rhodamine-B solution will no
longer be a high enough concentration to effectively dye particles.
2.6.
Dispose
of the solution created in 2.3.1 in accordance with environmental health and
safety regulations, then repeat 2.3.1 and create a new .5g/L solution with
which to dye particles.
2.7.
Repeat
2.6 every 5 repetitions of 2.3.2–2.3.4.
3.
Experimental and Optical Setup
[Place
Figure 2 Here]
3.1.
Prepare
the cameras.
3.1.1.
Use
cameras capable of at least 1 megapixel resolution at 450 frames per second.
3.1.2.
Arrange
the cameras such that each camera is pointing at, and is focused on, the center
of the viewing volume. Fewer cameras can be used, however shadowing of an arm
of a particle by another arm limits the orientation measurement accuracy, and
having fewer cameras makes experiments more susceptible to this effect. Using
more than four cameras could likewise increase orientation measurement precision
because it will reduce the chance of arms being shadowed on all cameras, which
is a primary source of uncertainty.
3.1.3. Position the
cameras with large (~90°) angles between each pair subject to constraints of
the apparatus. Place
cameras as shown in Figure 2 to balance experimental access and the size of the
angle between individual cameras. Minimize optical distortions by building viewing ports into the
apparatus perpendicular to each camera viewing direction.
3.1.4.
Use
200 mm macro lenses on each camera to obtain the desired measurement volume
from a working distance of half a meter. The volume viewed by all four cameras
determines the detection volume, which is about 3 x 3 x 3 cm3.
3.1.5.1.
Set
the apertures to f/11 and mount 532 nm notch filters to remove laser light while
allowing through longer-wavelength fluorescence onto the cameras
3.1.5.2. Place an image calibration mask in the tank, fill the tank with the
bulk solution from 2.4, and illuminate the mask.
3.1.5.3. Adjust the cameras so they each have the mask in view and they all are
focused on the same point on the mask. Carefully align the cameras to optimize
the shape of the detection volume.
3.1.5.4.
Be
careful to change as little as possible about the optical setup from this point
forward.
3.1.5.5.
Acquire and store images of the mask from each
camera.
3.1.5.6.
Drain
the solution out of the tank and pump it back where it had previously been
stored.
3.1.5.7.
Extract
the parameters specifying the position, viewing direction, magnification, and
optical distortions of each camera from the calibration images. Do this by
identifying places on the calibration mask visible on all four cameras and
defining the distance between these points. With this information, use standard
calibration methods to extract relevant parameters.
Note: The basic calibration method
is described in Tsai, 1987[7].
The implementation used in these experiments is described in Oullette et al3. Researchers wishing to develop camera calibration
software may also want to consider OpenPTV4.
3.1.5.8.
Create
a final calibration file using a dynamic calibration process. This is done
after tracer particle data has been acquired. Use a nonlinear least squares
search to optimize the camera calibration parameters and obtain the smallest
mismatch between the positions of particles seen on multiple cameras. These
methods are described in Ref. [8]
and [9].
3.2.
With
a Q-switched green Nd:YAG laser capable of 50 W average power (hereafter called
the “laser”), illuminate a cylinder in the center of the tank with roughly a 3
cm cross-sectional diameter, where the flow is homogeneous.8
Note:
The laser power is specified at a pulse frequency of 5 kHz. The pulse frequency
in these experiments is 900 Hz, where the output power is significantly lower.
3.2.1. Split the light
from the laser using a beam splitter and use mirrors to guide one beam into the
front of the tank and the other, orthogonal to the first, into the side of the
tank.
3.2.2. Place two
additional mirrors outside of the tank, opposite where the beams are entering,
in order to reflect light back into the tank and create more uniform
illumination, dramatically decreasing shadowing effects.
Note:
The length scale of interference effects from the counter-propagating beams is
too small to significantly affect these experiments.
4.
Perform the Experiments
4.1.
Prepare
to record video from each camera.
4.1.1.
Program
an image compression system that removes unwanted image data in real time.[10],13
4.1.1.1.
If the camera does not have a particle in
view, do not save the image.
4.1.1.2.
Where there are bright pixels, save only the
location and brightness of bright pixels instead of the whole image.
Note:
Because each particle typically covers approximately 5,000 bright pixels and
there is rarely more than one particle in view at a time, the image compression
system dramatically reduces the amount of storage required to record with
high-speed cameras for many hours.
4.1.2.
Prepare
the data acquisitioning software.
4.2.
Prepare the turbulent flow in a 1 x 1 x
1 m3 octagonal tank using two parallel 8 cm mesh grids oscillating
in phase.8
4.2.1.
Pump
the CaCl2 solution from 2.4 into a vacuum chamber and keep it in the
chamber overnight to degas the solution, which minimizes air bubbles in the
experiments.
4.4.
Choose one particle type (tracer
particles, jacks, crosses, tetrads, or triads) to be used for the first round
of experiments and add all 10,000 of those particles into the water through a
port at the top of the apparatus. Close this port after adding particles.
4.4.1. Turn the laser
on.
4.4.2.
Set
cameras and laser to respond to an external trigger and set the frequency of
the trigger to 450 Hz for the cameras and 900 Hz for the laser. Use the
external trigger to ensure all cameras start acquisition simultaneously and
remain synchronized throughout the recording
4.4.3. Open the laser
aperture.
4.4.4. Set the grid to
the chosen frequency (1 or 3 Hz) and start it running. Before starting data
acquisition, run the grid for about 1 minute to allow turbulence to fully
develop.
4.4.5. Record 106
frames in order to keep the file size manageable and to keep any errors that
may occur in the image compression systems from compromising too much data.
4.4.6. Close the laser
aperture and stop the camera trigger. Reset the image compression systems and
the cameras.
4.4.6.1.
Check
that the video files are not corrupted by viewing portions of each file.
4.4.7.
Repeat
4.4.1 – 4.4.6 until 107 images have been recorded at the chosen grid
frequency for the chosen particle.
4.5.
Change
the grid frequency to the one not chosen in 4.4.4 and repeat 4.4.4 – 4.4.7
4.6.
Empty the tank and filter the water to
remove all particles. Save particles in the storage water
from 2.4 if desired.
4.7.
Repeat
4.4 – 4.6 for all particle types.
4.8.
After
all experiments have been finished, calibrate cameras once more, as in 3.1.5–3.1.5.7.
5.
Data Analysis
Note: This section of the Protocol
presents an overview of the process used to obtain particle orientations and
rotation rates. The specific programs used, along with test images and
calibration files, are included as a supplement to this publication, and are
open to use by any interested readers.
5.1.
Using
the camera calibration parameters, obtain the 3D position and orientation from
images of particles on multiple cameras.
5.1.1.
At
every frame, find the center of the particle on each of the four images. All
particles in these experiments are sufficiently symmetric that the center of
the object is at the geometric center of the bright pixels on the image when
viewed from any perspective.
5.1.2.
Find
the 3D position of the particle by stereomatching its simultaneous 2D positions
on all four cameras3,8.
5.1.3. Create a
numerical model of the particle that can be projected onto each camera to model
the intensity in the image from that camera.
5.1.3.1. Model the particle as a composite of rods. Using the camera calibration
parameters from 3.1.5.7 and 3.1.5.8, project the two end points of each rod
onto the cameras and then model the distribution of light intensity in two
dimensions, with a Gaussian function across the width of the rod and a
Fermi-Dirac function across its length according to software protocol.
5.1.3.2.
Model
light intensity in two dimensions in this way to minimize the computational
cost of the data analysis. Projection of a full three-dimensional model of the
fluorescent particle could improve on this approach, but would be much more
computationally intensive.
5.1.3.3.
Click
Run to begin analysis.
5.1.4.
Choose
an initial guess of the particle orientation.
5.1.4.1.
If analyzing
the first frame in which this particle is visible, the first guess can be a
random set of Euler angles.
5.1.4.2.
If
this particle was in at least one previous frame, use the orientation found
using the previous frame as the initial guess.
5.1.5. Perform a nonlinear
least-squares fit to determine the particle orientation.
Note:
There are multiple conventions for defining Euler angles. Define the angles,
Note:
The method in 5.1.6 assumes that the particle will not rotate more than half of
one of its interior angles between frames. Justification for this assumption is
given in the Discussion.
5.2.
Save
the position and Euler angles as a function of time.
5.3.
Use
these data to extract solid-body rotation rate and other quantities.
REPRESENTATIVE RESULTS:
Figure
3a shows an image of a tetrad from one of our cameras above a plot of the Euler
angles obtained from a section of its trajectory (Figure 3c). In Figure 3b, the
results of the orientation-finding algorithm, described in Protocol 5 – 5.3,
are superimposed on the tetrad image. The arms of the tetrad in Figure 3a do
not follow the simple intensity distributions that are used to create the model
(Protocol 5.1.3.1). This is true for all of the particles. The observed
intensity furthermore has a non-trivial dependence on the angles between the
arms, the illumination, and the viewing direction[12]. The models do not include any of
these factors but nonetheless produce very accurate measurements of particle
orientations.
Once
an orientation is found with a least-squares fit, the 3D coordinates of the
particle center and the three Euler angles,
[Place
Figure 3 Here]
[Place
Figure 4 Here]
Two
different but related quantities based on particle orientations are calculated
over the entire trajectory: tumbling rate and solid-body rotation rate.
Tumbling rate,
To
extract the solid-body rotation rate from measured particles orientations, smoothing
needs to be done over several time steps. The problem is to find the rotation
matrix
where
From
the measured rotation matrix over a time step,
[Place Figure 5 Here]
Figure
1: A jack at various stages of resin removal.
a) The blocks of support resin that the particles arrive in. b) A single block
separated from the rest. c-e) Multiple stages of resin removal done by hand. f)
A single jack after the NaOH bath.
Figure
2: Experimental setup. In the
octagonal flow between oscillating grids, a central viewing volume in the focus
of the four video cameras is illuminated by a green Nd:YAG laser. a) Side view
showing how the four cameras are arranged and connected to computers. Figure
from [13]. b) Top view showing laser, mirror,
and lens configuration to achieve uniform illumination in the central volume.
Figure
3: Reconstructed particle orientations
from measured images. a) A sample image from one of the four cameras. The
object shown is a tetrad, which has four arms at 109.5° interior angles to one
another. b) The same tetrad shown with the results of our orientation-finding
algorithm. c) Measured Euler angles plotted as a function of time for a single
trajectory.
Figure
4: Reconstructed trajectories of a cross
(a) and a jack (b) in three-dimensional turbulence. (a) The two different
color sheets trace the path of the two arms of the particle through space over
time. The length of the track is 336 frames, or 5.7
Figure
5: PDF of mean-square tumbling rate.
The probability density function of the measured mean-square tumbling rate for
our crosses (red squares) and jacks (blue circles) as well as direct numerical
simulations of spheres (solid line). Error bars include the random error due to
limited statistical sampling estimated by dividing the data set into subsets,
as well as the systematic error that results from the fit length dependence of
the tumbling rate, which is estimated by performing the analysis at a range of
fit lengths. Figure from 1 where it is Figure 5.
DISCUSSION:
Measurements
of the vorticity and rotation of particles in turbulent fluid flow have long
been recognized as important goals in experimental fluid mechanics. The
solid-body rotation of small spheres in turbulence is equal to half the fluid
vorticity, but the rotational symmetry of spheres has made direct measurement
of their solid-body rotation difficult. Traditionally, the fluid vorticity has
been measured using complex, multi-sensor, hot-wire probes[14].
But these sensors only get single-point vorticity measurements in airflows that
have large mean velocity. Other vorticity measurement methods have been
developed. For example, Su and Dahm used flow field velocimetry based on scalar
images[15]
and Lüthi, Tsinober, and Kinzelbach used 3D particle tracking velocimetry[16].
Measurements of vorticity in turbulence by tracking rotations of single
particles were pioneered by Frish and Webb, who measured the rotations of solid
spherical particles using a vorticity optical probe[17].
This probe uses small particles with planar crystals embedded that act as
mirrors to create a beam whose direction changes as the particle rotates. Recently,
methods have been developed for measuring the rotational motion of large
spherical particles using imaging of patterns painted on the particles[18],[19]
or fluorescent particles embedded in transparent hydrogel particles[20].
To track anisotropic particles, Bellani et
al. have used custom-molded hydrogel particles[21]. Parsa et al. have tracked the rotations of segments of nylon threads5,6,12. The methods for measuring vorticity and particle
rotations presented in this paper have advantages over these alternative methods.
3D-printed anisotropic particles can be small, with arm thicknesses down to .3 mm
in diameter, and their rotations can still be resolved very accurately. Other
methods traditionally require larger particles because they involve the
resolution of structures on or within the particles themselves. In addition,
the use of image compression systems allows for many more particle trajectories
to be recorded and measured than would otherwise be reasonable. Having more
measurements makes it possible to study rare events like those with very high
rotation rates in Figure 5, which reveal intermittency phenomena of great
interest to researchers.
Particle
concentrations in these experiments were about 5 x 10-3 cm-3,
which meant that typically only about 20% of images from the cameras had a
particle. To study rare events, thousands of particle trajectories are typically
required, which meant that hundreds of thousands of images of particles were
needed. With these low concentrations, therefore, millions of images needed to
be recorded to obtain an adequate volume of data. If real-time image
compression systems were not used to facilitate data acquisition, this would
require hundreds of TB of data storage and the analysis would be much more
computationally intensive. Image compression systems decrease this load by
factors of several hundred10. However, standard video recording would be adequate
for higher particle densities and if data storage space is not an issue. If
100,000 particles of each type were ordered instead of 10,000, fewer images
would, in principle, be needed to capture the same statistics. However, at
higher particle densities particles begin to shadow one another more often.
That is, there will be more times when there are particles between the laser
and the particle in view, or between the particle in view and the camera. These
shadowing events make measuring orientations throughout a track across the
viewing volume more difficult and less reliable. For these reasons, lower
particle concentrations were chosen for these experiments and image compression
systems were therefore necessary.
There
may be times when arm shadowing will affect the results of the nonlinear search
algorithm. For certain orientations of the jack, arm shadowing causes there to
be multiple minima in Euler angle space, which lead to indeterminacies in the
measured orientations. This reduces the accuracy of orientation measurements
for these particular orientations and occasionally leads to erroneously high
measurements of the solid-body rotation rate, which pushes additional
probability density towards the tail of the PDF in Figure 5. For jacks, whose
arms are perpendicular to each other, this issue could be decreased by changing
the angles of the cameras with respect to one another to be farther away from 90°.
If the configuration of the apparatus makes this change difficult to implement,
one alternative is to change the geometry of the particles to decrease
shadowing. This was the reason tetrads were chosen for experiments after those
with jacks had been completed, and recent tetrad measurements have shown
significantly improved orientation accuracy when compared to jacks.
The
methods of 3D particle tracking presented here are not confined to this
particular flow or the particle sizes and shapes we use. We have already begun
experiments tracking tetrads and triads with much larger sizes using similar
techniques. The use of high-speed cameras to measure particle orientations and
rotations can be extended to a wide array of shapes and can be used for
inertial particles as well as in the neutrally buoyant case presented here.
Using more cameras would allow for an even wider array of potential particle
shapes, as the primary limitations to this method are the resolution of the
cameras and particles’ self-shadowing, as discussed in the previous paragraph.
In
step 5.1.6 of the Protocol, we smooth Euler angles measurements by assuming
that a particle has not rotated by more than half of one of the interior angles
between arms over the course of two frames – that is, we assume that the
accurate orientation measurement at frame i+1
retains the chosen symmetric orientation found for frame i. If the particle had rotated by more than half of one of these
interior angles, then smoothing in this way would result in a sudden and
incorrect reversal of the direction of rotation. In Ref. 5 we show that an upper limit on particle tumbling rate
is
So
the largest tumbling rate (
ACKNOWLEDGEMENTS:
We thank
Susantha Wijesinghe who designed and constructed the image compression system
we use. We acknowledge support from the NSF grant DMR-1208990.
DISCLOSURES:
The
authors have no competing financial interests to disclose.
[1] Marcus,
G., Parsa S., Kramel, S., Ni, R., and Voth, G. Measurements of the Solid-body
Rotation of Anisotropic Particles in 3D Turbulence. New J. Phys. 16, 102001, doi:10.1088/1367-2630/16/10/102001
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