A three-dimensional particle tracking velocimetry (3D-PTV) system based on a high-speed camera with a four-view splitter is described here. The technique is applied to a jet flow from a circular pipe in the vicinity of ten diameters downstream at Reynolds number Re ≈ 7,000.
3D-PTV is a quantitative flow measurement technique that aims to track the Lagrangian paths of a set of particles in three dimensions using stereoscopic recording of image sequences. The basic components, features, constraints and optimization tips of a 3D-PTV topology consisting of a high-speed camera with a four-view splitter are described and discussed in this article. The technique is applied to the intermediate flow field (5 <x/d <25) of a circular jet at Re ≈ 7,000. Lagrangian flow features and turbulence quantities in an Eulerian frame are estimated around ten diameters downstream of the jet origin and at various radial distances from the jet core. Lagrangian properties include trajectory, velocity and acceleration of selected particles as well as curvature of the flow path, which are obtained from the Frenet-Serret equation. Estimation of the 3D velocity and turbulence fields around the jet core axis at a cross-plane located at ten diameters downstream of the jet is compared with literature, and the power spectrum of the large-scale streamwise velocity motions is obtained at various radial distances from the jet core.
Turbulent jet flows are ubiquitous in engineering applications. Detailed characterization of such flows is crucial in a wide spectrum of practical problems spanning from large-scale environmental discharge systems to electronic micro-scale devices. Because of its impact on a number of broad applications, jet flows have been studied in depth1–4. Several experimental techniques, including hotwire anemometry4–8, Laser Doppler Velocimetry (LDV)4,9–12, and Particle Image Velocimetry (PIV)12–16, have been used to characterize jet flows in a wide range of Reynolds numbers and boundary conditions. Recently, a few studies have been made using 3D-PTV to study the turbulent/non-turbulent interface of jet flows17,18. 3D-PTV is a technique especially suitable to describe complex turbulent fields from a different perspective. It allows the reconstruction of particle trajectories within a volume in a Lagrangian frame of reference using multi-view stereoscopy. The technique was first introduced by Chang19 and further developed by Racca and Dewey20. Since then, many improvements have been made on the 3D-PTV algorithm and experimental setup21–24. With these accomplishments and previous works, the system has been successfully used to study various fluid phenomena such as large-scale fluid motion in a domain of 4 m x 2 m x 2 m25, indoor airflow field26, pulsatile flows27 and aortic blood flow28.
The working principle of a 3D-PTV measurement consists of data acquisition system set-up, recording/pre-processing, calibration, 3D correspondences, temporal tracking and post-processing. An accurate calibration allows for a precise detection of particle positions. The correspondence of the particles detected in more than three image views allows for the reconstruction of a 3D particle position based on the epipolar geometry. A linkage from consecutive image frames result in a temporal tracking that defines the particle trajectories s(t). Optimization of the 3D-PTV system is essential to maximize the likelihood of multi-particle traceability.
First step of the optimization is to acquire an appropriate data acquisition system including high-speed cameras, illumination source and features of seeding particles. The camera resolution along with the size of the interrogation volume defines the pixel size and, therefore, the required seeding particle size, which should be larger than a single pixel. The centroids of detected particles are estimated with sub-pixel accuracy by taking the average position of particle pixels weighted by brightness21. The camera's frame rate is closely associated with Reynolds number and the ability to link detected particles. A higher frame rate allows for resolving faster flows or a larger number of particles since the tracking becomes more difficult when the mean displacement between images exceeds the mean separation of the particles.
Shutter speed, aperture and sensitivity are three factors to consider in the image capture. Shutter speed should be fast enough to minimize blurring around a particle, which reduces uncertainty of particle centroid position. Camera aperture should be adjusted to the depth of field of the interrogation volume to reduce the probability of detecting particles outside the volume. Since the maximum sensitivity of a camera is fixed, as the frame rate increases, the necessary light required to illuminate the particles should increase accordingly. Unlike PIV, complex optic settings and high-power lasers are not strictly required in 3D-PTV, as long as the light source is sufficiently scattered from the tracer particles to the camera. Continuous LED or halogen lights are good cost-effective options that bypass the need of synchronization21.
In 3D-PTV, like other optical flow measurement techniques, tracer particle velocity is assumed to be the local instantaneous fluid velocity29. However, this is only the case for ideal tracers of null diameter and inertia; tracer particles should be large enough to be captured by a camera. The fidelity of a finite particle can be determined by the Stokes number St, i.e. the ratio of the relaxation time scale of particles and the time scale of turbulent structures of interest. In general, St should be substantially smaller than 1. For St ≤0.1 flow tracking errors are below 1%30. In-depth discussion can be found in Mei et al.29–31. Recommended particle size for a 3D-PTV experiment varies depending on the light source and camera sensitivity. With halogen or LED lights as illumination sources, relatively larger particles are used (e.g. 50-200 µm)32, whereas smaller particles (e.g. 1-50 µm)33,34 can be used with a high power laser (e.g. 80-100 Watts CW laser). Particles with high reflectivity for a given wavelength light, like silver coated under halogen light, can amplify their mark into an image. The seeding density is another important parameter for a successful 3D-PTV measurement. Few particles result in low number of trajectories, while an excessive number of particles cause ambiguities in establishing correspondences and tracking. Ambiguities in establishing correspondences include overlapping particles and detecting multiple candidates along the defined epipolar line. In the tracking process, the ambiguity due to a high seeding density is occurred because of the relatively short mean separation of particles.
Second step is optimal settings in recording/pre-processing to enhance the image quality. Photographic settings, such as gain & black level (G&B), play an important role in optimizing the image quality. Black level defines the brightness level at the darkest part of an image, whereas gain amplifies the brightness of an image. Slight variations of the G&B levels can significantly impact the likelihood of traceability. In fact, high G&B may over-brighten an image and eventually damage the camera sensor. To illustrate this, the impact of G&B levels on the flow reconstruction is also examined in this article. In the pre-processing step, the images are filtered with a high-pass filter to emphasize light scatter from particles. The pixel size and gray scale are adjusted to maximize the particle detection within the interrogation volume.
Third step of the optimization is accurate calibration of the stereoscopic imaging, which is based on epipolar geometry, camera parameters (focal length, principle point, and distortion coefficients), and refractive index changes. This process is essential to minimize the 3D reconstruction error of the fiducial target points. Epipolar geometry uses relative distances (between camera and interrogation volume) and tilted angle from the target image. Refractive index changes along the camera view through the interrogation volume can be taken into account based on the procedure of Mass et al.21. In this experiment, a 3D stair-like structure with regularly distributed target points is used as a target.
In a 3D-PTV experiment, although only two images are needed to determine a 3D particle position, typically more cameras are used to reduce ambiguities21. An alternative to expensive setups with multiple high-speed cameras is the view splitter, proposed by Hoyer et al.35 for the use of 3D-PTV and recently applied by Gulean et al.28 for the biomedical applications. The view splitter consists of a pyramid-shaped mirror (hereon primary mirror) and four adjustable mirrors (hereon secondary mirror). In this work, a four-view splitter and a single camera were used to mimic the stereoscopic imaging from four cameras. The system is used to characterize the intermediate flow field of a pipe jet with a diameter, dh = 1 cm and Re ≈ 7,000 from a Lagrangian and Eulerian frames at around 14.5-18.5 diameters downstream from the jet origin.
1. Lab Safety
2. Experimental Set-up
3. Set-up Optimization
4. Calibration
5. Flow Setting/Data Collection
6. Data Processing (Via OpenPTV Software)
7. Post Processing (Optative)
Note: the reach and type of post-processing depends on individual needs and it is, therefore, customizable. Here, point base calculations are briefly described as an example.
A photograph and a schematic of the setup are shown in Figures 1 and 2. The calibration target, the fiducial marks reflected on the view-splitter and 3D calibration reconstruction are illustrated in Figure 3. The RMS of the recognized calibration targets is 7.3 µm, 5.7 µm and 141.7 µm in the streamwise x, spanwise y, and depth z directions. The relative higher RMS in the z-coordinate is due to the reduced targets points with respect to those in the other directions and relatively small angles of four views with the z-axis compared to x and y coordinates. The detected particles in each of the four views at any given instant were on the order of 103. Among the detected particles, the number of successful 3D reconstructions is reduced to roughly half due to the fact that only particles in the intersection region are captured. Video 1 shows a high-speed video sample of the jet flow from the four-view splitter.
A sample of four representative particle trajectories in the intermediate-field region around and crossing the x/dh=16 plane at radial distances r/dh=v0, 1.5, 3 from the jet core is illustrated in Figure 4. As expected, longer trajectories in the given time interval (Δt ≈ 1 sec) are observed around the jet core. At the edge of the jet (r/dh ≥2), tracer particles exhibit short and more complex trajectories. Figure 5 shows all the successfully reconstructed particle trajectories crossing the x/dh = 16 plane. The particle velocities in the selected domain exhibit a wide distribution ranging from nearly 0 to 0.6Uj, where Uj ≈ 0.6 m/sec is the exit velocity of the jet, and acceleration/deceleration, ∂U⁄∂s, where is the Cartesian coordinate. It also reveals that an substantial portion of the particles within the vicinity of the jet core exhibit smoother trajectories. The particle trajectories, s(t), allows for the estimation of the position, velocity and acceleration. Figure 6a shows the case of a particle crossing the x/dh = 16 plane around the jet core. Figure 6b, 6c, and 6d show the 3 components of the particle trajectory, velocity and acceleration as a function of normalized time. It is worth highlighting that the local particle acceleration can be several times the standard gravity. The particle trajectories allow for obtaining specific features of the particle trajectories via the so-called Frenet-Serret frame. It describes the changes of the orthonormal vectors (tangential, normal, binormal) along s(t). Of particular relevance is the curvature, κ, which is the inverse of the radius of curvature, ρ, and defined as:
where = dr/ds is the tangent unit vector of the trajectory and r is the position vector (Euclidean space) of the particle as a function of time, which can be written as a function of , i.e., r(s) = r(t(s)). The curvature, κ, is computed for all the particles crossing the x⁄dh = 16 and x⁄dh = 17 planes. The mean curvature, , as a function of the distance from the jet core r is calculated as:
where Δr = 0.2dh is used in here. Figure 7 illustrates = f(r) normalized by dh. It shows a relatively low and nearly constant within the area defined by the circular cross section of the pipe, r/dh ≤0.5. At a larger distance from the jet core in the x/dh = 16 plane, increases monotonically. A similar trend is obtained at the x/dh = 17 plane, but with a reduced outside the jet core (r/dh ≥0.5). It is worth highlighting that this flow feature can be inferred only with the 3D-PTV technique. The data quality based on various levels of G&B settings is assessed in terms of the ratio of linked particles to the rest of 3D-reconstructed particles shown in Table 1. The highest link ratio is observed at the G&B setting of 300&500.
Eulerian flow characteristics can be achieved by grid-interpolation, which mimics 3D particle image velocimetry (3D-PIV). It is important to note that due to the comparatively low particles tracked at each time, a significantly higher number of frames are needed to truly mimic PIV quality for an Eulerian description. This is more critical in the estimation of high-order statistics (e.g., turbulence intensity and Reynolds stresses). The streamwise velocity at the jet core for various G&B levels is illustrated in Figure 8. The measurements are compared with the theoretical behavior:
where U0(x) is the streamwise velocity at the jet core, B ≈ 6 is a constant, and x0 is the virtual origin38. The figure shows the relevance of setting the G&B levels. Figure 9 illustrates the mean velocity distribution of the jets in the x/dh = 16 plane.
Finally, the spectral distribution ϕ(f) of the large-scale motions of the streamwise velocity at locations r/dh = 0, 0.6, and 1 in the x/dh = 10 plane is illustrated in Figure 10. A Butterworth low-pass filter was applied to the velocity time series with cut-off frequency, fc = 200 Hz.
Figure 1: Schematic of the experimental set-up. Please click here to view a larger version of this figure.
Figure 2: Experimental set-up. This illustrates various views of the camera and the four-image view splitter, flume and interrogation volume: (top left) top view, (bottom left) back view of the camera and view splitter system, (top middle, bottom middle) side views of the overall experimental set-up, (right) zoom-in view of seeding particles in the jet flows. Please click here to view a larger version of this figure.
Figure 3: Calibration: (a) Calibration target, (b) Image-set of the calibration target from the view splitter, (c) 3D recognition of the fiducial marks from the calibration target. Please click here to view a larger version of this figure.
Figure 4: Selected particle trajectories at r⁄dh = 0, 1.5, 3. Please click here to view a larger version of this figure.
Figure 5: Particle trajectories crossing the x/dh = 16 plane, where velocity is shown as a color level. The interrogation volume shown in the figure is contained between (x)⁄dh (14.5,18.5), y⁄dh (-2,2), and z⁄dh (-2,2), where (x, y, z) = (0, 0, 0) is located at the center of the jet origin. The velocity along the individual trajectories, normalized by the bulk velocity U0, is illustrated as a color level. Please click here to view a larger version of this figure.
Figure 6: (a) Particle trajectory, (b) displacement, (c) velocity, and (d) acceleration of an arbitrary particle. Please click here to view a larger version of this figure.
Figure 7: Curvature of the particles: Graph showing mean curvature of the particles as a function of the radial distance from the jet core at the planes x⁄dh = 16 and x⁄dh = 17. Please click here to view a larger version of this figure.
Figure 8: Streamwise velocity at the jet core within (x)⁄dh (15,18) for various G&B levels. Three G&B levels are included (300&500 (optimum), 300&250, 100&250). Please click here to view a larger version of this figure.
Figure 9: Non-dimensional distribution of the streamwise velocity component at x/dh = 16. Please click here to view a larger version of this figure.
Figure 10: Power spectrum ϕ(f) of the streamwise velocity component at a point located at r/dh = 0 (jet core), 0.6, and 1 in the x/dh = 16 plane. Please click here to view a larger version of this figure.
Video 1: Video sample of the jet flow from the four-view splitter, 10 times slower than actual speed obtained at 550 fps (right click to download).
Table 1: The ratio of linked particles to the rest of 3D-reconstructed particles at various G&B levels. Three G&B levels are included (100&250, 300&250, 300&500).
3D-PTV has great potential to unravel the complex physics of a variety of turbulent flows such as large-scale turbulent motions in the lower atmosphere25, indoor air distribution26, or pulsatile flows in aortic topology28 among many others. However, an understanding of its advantages and limitations as well as experience is essential to maximize its potential. Trial and error preliminary testing and exhaustive iterations for optimum settings, including frame rate, illumination source, G&B level and image-filtering method, are directly correlated with the ability of reconstructing the Lagrangian paths of a set of (e.g., tracer) particles. It is important to note that the critical protocol steps, as demonstrated here, are the adjustments of the G&B levels and the illumination of the FOV (combination of halogen spots lights, magnifying lens and reflecting mirror from the bottom of the flume).
These adjustments help to optimize the light scatters within the investigation to the four views. After identifying the experimental settings for high-fidelity measurements, thorough modification and troubleshooting should be made to compute the maximum number of accurate trajectories based on the frame rate, camera resolution and the size of investigation volume. Although the number of captured particles can be increased with higher frame rates, it is worth noticing that the number of tracked particles in 3D-PTV is much lower compared to PIV. The biggest potential of 3D-PTV is in its unique ability of describing the Lagrangian paths of multiple particles. In this demonstration, the view splitter set-up was implemented to avoid using multiple expansive cameras, however, it is important to note that this set-up requires higher camera resolution and limits the size of sample volume.
In this study, the intermediate-field features of a circular jet are analyzed with the 3D-PTV technique. The approach allowed obtaining important features of the flow from Eulerian and Lagrangian frames. In particular, the average curvature of the particles as a function of the radial distance is characterized for first time, at two cross-sectional planes using the Lagrangian features of the particle trajectories. The RMS of the recognized calibration targets ranges between 7.3 µm, to 141.7 µm in the streamwise and spanwise directions. Although this high relative error in the spanwise direction due to small angles of the views in z-direction may not be completely overcome, it can be further reduced by adding more target points in z-direction such as using a 2D calibration target at various locations (multiplane calibration).
Overall, 3D-PTV is a useful technique that can be applied in a number of other problems including time-dependent flows or the dynamics of active scalars. For instance, it can be highly useful to study the interplay between turbulence and species in aquatic environments.
The authors have nothing to disclose.
This work was supported by the Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, as part of the start-up package of Leonardo P. Chamorro.
ImageOps | CAMMC4082 | High-speed camera |
ImageOps | FBD-4XCXP6 | Frame Grabber |
Potters Industries LLC | AG-SL150-30-TRD | Seeding Paritcles |
Upstate Technical Equipment CO.,INC | MISNOR-STP-6-S-CL | Camera appliation |
Photrack AG | Customized part and necessary if performing 3D-PTV with one camera | |
General Electrics | 23719 | Light source |
OpenPTV(http://www.openptv.net) | Open source particle tracking software (Note: available as a service for anyone who wants to use it without all the installation mess or computer power availability problems). |