1. Introduction
A large number of studies have attempted to obtain quantitative 3D information from flow fields. Several studies on stereo particle image velocimetry (PIV) have shown the capability of the PIV technique in visualizing complex flows quantitatively [1,2]. Hinsch [3] and Chan et al. [4] verified the possibility of the quantification of complex flows by holographic PIV (HPIV). The reconstructed intensity field was scanned by a charge-coupled device (CCD) sensor. However, the recording medium was a holographic film that requires wet processing. This process is time consuming and inaccurate because of the misalignment and distortion produced when re-positioning the hologram for object reconstruction. Hologram recording on a photographic plate can also be captured directly by a CCD sensor such as a digital HPIV (DHPIV) [5]. In this case, the light intensity distribution in the measurement volume is built numerically by solving the Fresnel diffraction formula for holographic near-field diffraction [6]. However, CCD sensors have very limited resolutions compared with photographic plates, returning about 2 to 3 orders less of particle images and velocity vectors. The large pixel pitch requires the recording to be obtained at a relatively small angle (a few degrees between the reference beam and scattered light) to resolve the interference pattern, thereby strongly limiting the numerical aperture and depth resolution [3]. Kim and Lee [7] constructed an in-line DHPIV by introducing a highdefinition camera (1 K × 1 K). The authors evaluated the maximum possible particle density of in-line DHPIVs. The results show that the recovery ratio (RR) of the whole particles in the measurement volume is approximately 65% when the particle density is 25 mm3. This result indicates that the identifiable number of whole particles is approximately 8500 among the 13,000 particles in the measurement volume.
Scarano et al. [8] developed a tomographic-PIV (TomoPIV). The Tomo-PIV was used for measuring a cylinder wake with a measurement volume of 80 mm × 80 mm × 15 mm. This algorithm was developed to increase the measurable number of vectors. Doh et al. [9,10] developed a genetic algorithm based on 3D particle-tracking velocimetry (3D-PTV) for measuring a cylinder wake with a volume of 50 mm × 50 mm × 50 mm. The authors obtained over 10,000 instantaneous vectors with a 1 K × 1 K camera. Lai et al. [11] developed a defocused 3DPTV with a measurable volume of 100 mm × 100 mm × 100 mm. Based on the number of vectors, the density of vectors per unit volume (1 mm3) obtained by the defocused 3D-PTV is more sparse than the one obtained by the DHPIV and Volume-PIV. The measured volume size of 3D-PTVs and Volume PIVs can be adjusted to that of HPIV by employing an optimized optical arrangement; thus, the identifiable particle numbers in the same particle density can be regarded as identical. The measureable thickness of the Volume-PIVs is restricted to a certain length because of the loss of image information, which is caused by an increase in the uncertainty of particle volume patterns with increasing camera-viewing depth. The drawbacks of 3D-PTVs are the numerous spurious vectors that exist among the calculated vectors and the long calculation time. No report has been made regarding the most appropriate method for various flow measurements. This study compared the efficiency of the Volume-PTV and Tomo-PIV algorithms in measuring complex flow fields.
2. Volume-PTV and Tomo-PIV
2.1. 3D Measurement Principle
A 3D measurement can be attained by matching the captured images of two or more cameras. To match the images, precise identification of the coordinate relations between the photographical coordinates of the cameras and the physical coordinates is required. Therefore, camera calibrations were conducted to determine these relations. The 10-parameter method [9] was used to obtain six exterior parameters (dis, α, β, γ, mx, my) and four interior parameters (cx, cy, k1, k2). The variables (α, β, γ) represent the tilting angles of the photographic coordinates for the absolute axes. The collinear equation for every point between the two coordinates is expressed in the following equation.
(1)
Variables cx and cy are the focal distances for the x and y components of the coordinate, respectively. Δx and Δy are the lens distortions and can be calculated by the following equation.
(2)
Equation (2) can be converted into the following Equation (3).
(3)
The study of Doh et al. [9,10] can be used as a reference for additional detailed calculation processes in obtaining camera parameters.
2.2. Volume-PTV
Figure 1 shows the developmental procedure for the Volume-PTV. After camera calibrations, 2D temporal vector trajectories for the whole particle pairs appeared on one camera image. The principle of finding the 2D vector trajectories is based on the 2D-PTV [12]. After obtaining the 2D vector candidates for each camera image, candidate trees were constructed by using the following hybrid fitness function (Figure 2).
(4)
where x is the scale sensitivity factor (x = δ/PF; Figure 2). δ is calculated using this equation.
(5)
where,
,
Figure 1. Overall procedure for Volume-PTV.
Figure 2. Fitness function against spatial distance.
where represents the velocity vectors comprising all candidate vectors, and represents the mean vector of nearby vectors in a certain area. The formula was calculated repeatedly until the maximum value of each particle was obtained.
Particle fitness (PF) indicates the data range, in which outliers are eliminated (Figure 2). Data from this PF value were not sorted into the correct candidate data group. Figure 3 shows the concept of the PF value. is the mean value of the velocity vectors in a certain searching region (Figure 4). A large PF value corresponds to large data variations in the correct data group. In other words, a larger PF value is recommended if the flow field is very complex and has a wide dynamic range.
Particle movement (PM) and particle neighborhood (PN) values are represented in [pixel] and [mm], respecttively. PM was used for finding the 2D particle trajectories of the same camera.
2D velocity vectors were installed within the PM [pixel] range, and all trajectories were stored in a candidate group. PM value was also used for other camera images. PN value was used for finding the same pairs in 3D space. As shown in Figure 4, particles in a sphere volume are regarded as one of the candidates and are subsequently sorted into the candidate group. The above-mentioned procedures indicate that two sets of 2D trajectories (i.e., 2D vectors) for the candidate data set were obtained from two camera images. The last data sets satisfied the PM and PN values.
2.3. Tomo-PIV
The principle of Tomo-PIV is based on the study of Elsinga et al. [13], except for the camera calibration process. Figure 5 shows the schematics of the Tomo-PIV. Tracer particles immersed in the flow were illuminated by a pulsed light source within a 3D region of space. The scattered light pattern was recorded simultaneously from several viewing directions using multiple cameras by applying the Scheimpflug condition on the image, lens, and mid-object plane. The particles within the entire volume have to be focused by setting a proper focal
Figure 4. Relations between particle searching region, particle neighborhood (PN), and particle movement (PM).
number. The 3D particle distribution (the object) was reconstructed as 3D light intensity distribution through the projections from the CCD arrays.
After reconstructing the whole images in virtual space, a voxel image was obtained. Thereafter, the cross-correlation of the particle intensity of the particles was calculated and the 3D vectors obtained on grids.
Figure 6 shows the relations between the projections