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.
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 Rλ = 91. In this relatively low-Reynolds number flow, the advected particles are small enough that they approximate ellipsoidal tracer particles. We present results of time-resolved 3D trajectories of position and orientation of the particles as well as measurements of their rotation rates.
In a recent publication, we introduced the use of particles made from multiple slender arms for measuring rotational motion of particles in turbulence1. 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 ellipsoids2, 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 groups3,4. 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 flows6 suggest that study of rotations of these particles provides new ways to look at the turbulent cascade process.
1. Fabrication of Particles
2. Preparation of Particles
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 and Rhodamine-B dye. Please click here to view a larger version of this figure.
3. Experimental and Optical Setup
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. Please click here to view a larger version of this figure.
4. Perform the Experiments
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. (See the file "Use_Instructions.txt" in the supplemental file "MATLAB_files.zip".)
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 direction12. 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, (φ, θ, ψ) , that specify its orientation matrix11 are saved. This is done for every frame where the particle is in view of all four cameras. These data enable the reconstruction of the complete trajectory of the particle across the viewing volume, as are shown in Figure 4 for a cross and a jack. Figure 4 was made using the Paraview open source visualization package and is based on measurements made with images from the experiments.
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. Please click here to view a larger version of this figure.
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 τη, and a cross is shown every 15 frames. (b) The blue, orange, and blue-green paths trace the paths of the three arms of the jack as the particle rotates and moves through the fluid. The dark green line denotes the path of the jack's center. The length of the particle track is 1,025 frames, or 17.5 τη, and a jack is shown every 50 frames. (Note: Neither the crosses nor the jacks above are drawn to scale.) Figure from 1, where it is Figure 3. Please click here to view a larger version of this figure.
Two different but related quantities based on particle orientations are calculated over the entire trajectory: tumbling rate and solid-body rotation rate. Tumbling rate, , is the rate of change of the unit vector defining the orientation of the particle. In previous measurements of rods, was defined as the axis of symmetry along the rod; for crosses and triads, is normal to the plane of the arms; for jacks and tetrads, is along one of the arms. Because rotation along the axis of rods cannot be directly measured, studies of the rotations of rods in turbulence have largely been limited to measuring the tumbling rate. This is not an issue for any of the particles in these experiments. All rotations of these particles can be measured and, with orientation measurements smoothed along a particle's trajectory, the full solid-body rotation rate vector, , can be found.
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 that relates an initial orientation to the measured orientations at a sequence of time steps:
where is the period between images and is the time of the initial frame. In Marcus et al.1, we used a nonlinear least-squares fit to determine the six Euler angles defining the initial orientation matrix, , and the rotation matrix over a single time step, , that best match the measured orientation matrices as a function of time. More recent work has shown that this algorithm sometimes has difficulty when the rotation rate is small because the nonlinear search is exploring the region where the Euler angles are approximately equal to zero and are degenerate. In the case where the rotation in a time step is sufficiently small, can be linearized using , where Ω is a rotation rate matrix. As described in the Discussion below, these experiments are in this low rotation limit, so Ω can be found from the measured using a linear least squares fit.
From the measured rotation matrix over a time step, , we can extract the solid-body rotation rate and the tumbling rate. By Euler's theorem11 can be decomposed as a rotation by an angle Φ about the solid-body rotation axis, . The magnitude of the solid-body rotation rate is . The tumbling rate is the component of the solid-body rotation rate perpendicular to the orientation of the particle, and so it can be calculated as . Figure 5 compares PDFs of the measured mean square tumbling rate for crosses and jacks to direct numerical simulations of spheres. Small jacks rotate just like spheres in fluid flows1, so the fact that the PDF for jacks agrees with the simulated PDF for spheres demonstrates that the experiments are able to capture the rare high rotation events that occur in turbulent flows.
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. Please click here to view a larger version of this figure.
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 probes14. 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 images15 and Lüthi, Tsinober, and Kinzelbach used 3D particle tracking velocimetry16. 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 probe17. 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 particles18,19 or fluorescent particles embedded in transparent hydrogel particles20. To track anisotropic particles, Bellani et al. have used custom-molded hydrogel particles21. 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 0.3 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 would not rotate by more than half of an angle 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 () is which for sec is 16.2 sec-2. This is a root mean square (RMS) tumbling rate of 4.0 sec-1. Since we record images at 450 frames per second, particles would then typically rotate 0.009 radians between frames. The smallest interior angle of any of the particles in these experiments was , so this smoothing method would fail if particles tumble more than radians between frames. Thus, we can accurately track particles with tumbling rates of more than 80 times the RMS, which is much faster than the times the RMS that we actually observe in Figure 5.
The authors have nothing to disclose.
We thank Susantha Wijesinghe who designed and constructed the image compression system we use. We acknowledge support from the NSF grant DMR-1208990.
Condor Nd:YAG 50W laser | Quantronics | 532-30-M | |
High speed camera | Basler | A504k | |
High speed camera | Mikrotron | EoSens Mc1362 | |
Rhodamine-B | ScienceLab.com | SLR1465 | |
Sodium Hydroxide | Macron | 7708 | Pellets. |
500 Connex 3D printer | Objet | Used to make smaller particles. Particles ordered from RP+M (rapid prototyping plus manufacturing). | |
VeroClear | Stratasys | RGD810 | Objet build material. |
Form 1+ 3D printer | Formlabs | Used to make larger particles. | |
Clear Form 1 Photopolymer Resin | Formlabs | ||
Cylindrical and spherical lenses | |||
200, 100, 50 mm macro camera lenses | F-mount. | ||
Ultrasonic bath | Sonicator | ||
Calcium Chloride | Spectrum Chemical Mfg. Corp. | CAS 10043-52-2 | Pellets. |
LabVIEW System Design Software | National Instruments | Used to trigger cameras, control grid, and trigger laser. | |
XCAP Software | EPIX | Used with LabVIEW to trigger cameras. | |
MATLAB | Mathworks | Used for all image and data analysis. Programs for extracting 3D orientations from multiple images are included with this publication. | |
OpenPTV: Open Source Particle Tracking Velocimetry | OpenPTV Consortium | ||
ParaView | Kitware | ||
AutoCAD | AutoDesk | Used to design all particles. Screenshots of particle designs are all of AutoCAD. | |
Mesh with 0.040 x 0.053 inch holes | Industrial Netting | XN5170–43.5 | |
Camera filters | Schneider Optics | B+W 040M |