Experimental determination of tomographic PIV accuracy by a 12-camera system

The experiment is performed in an open-circuit, open-test-section tunnel with test cross section of 40  ×  40 cm² (W-tunnel) at the Aerodynamics Laboratories of TU Delft Aerospace Engineering. A cylinder of 6 mm diameter spans the entire height of the test section and generates an unsteady wake, characterized by large-scale spanwise-coherent vortex shedding. The arrangement is schematically shown in figure 1. The tunnel is operated at an exit velocity of 12 m s^‒1, corresponding to a Reynolds number of 5000 based on cylinder diameter.

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A stage-smoke generator produces water-glycol droplets of approximately 1 micron diameter. To vary the particle seeding throughout the experiment, the laboratory is first filled to a high concentration of seeding particles. At the start of the experiment, an exhaust fan gradually vents the room, lowering the particle concentration.

The measurement volume size is 4 cm along the cylinder span, and 14 mm in depth. The volume is illuminated by a double-cavity Spectra-Physics Quanta-Ray PIV-400 Nd:YAG laser (400 mJ pulse^‒1). Light sheet optics shapes the 6 mm diameter beam into a collimated beam of rectangular cross section propagating parallel to the cylinder axis. An additional 90-degree high-power laser mirror is used to double-pass the laser slab through the measurement volume, which homogenizes the intensity across the cameras used in forward and backward scattering configuration. The time separation between laser pulses was set to 40 µs, yielding average displacements of approximately 0.5 mm (12 pixels) in the free stream region.

The illuminated volume has different thickness in two regions: close to the cylinder the thickness is 14 mm to measure the entire wake produced by the cylinder. Downstream, the thickness is reduced to 2.2 mm to enable an accurate estimate of the tracer's concentration. This configuration is realized by a knife-edge filter as illustrated in figure 1.

The 12-camera tomographic system is organized in a cross configuration to maximize the total angular aperture of the system (approximately 40 degrees total angle in both horizontal and vertical direction) as shown in figure 2. Twelve LaVision Imager LX 2MP interline CCD (1628  ×  1236 pixels, pixel pitch 4.4 µm) are used, equipped with 75 mm focal length objectives attached to custom-manufactured Scheimpflug adapters. The aperture is set to f/# = 8. The corresponding diffraction image size of particles is d_τ = 11 μm. Spatial autocorrelation analysis yields a particle image diameter of 2.05 px. The estimated depth of field is 18 mm, ensuring in-focus particle images through the entire depth of the measurement volume.

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A summary of illumination and imaging parameters is given in table 1.

The cameras are connected to a host computer equipped with three Gigabit Ethernet boards, and controlled using LaVision Davis 8.1 software and a PTU9 timing unit. Tomographic calibration is performed using a dual-plane target (LaVision Type 7). Initial fitting errors were in the range of 0.3 pixels. The volume self-calibration technique (Wieneke 2008) using a set of 50 images at low seeding density (ppp < 0.02) corrects the mapping function to within 0.01 pixel for all cameras. The self-calibration procedure was performed before and after each measurement as calibration errors are known to severely affect the reconstruction accuracy (Elsinga et al 2006).

Image pre-processing consists of two steps: first, the historical minimum is calculated at each pixel for each camera. Then the images are subtracted by the corresponding minimum. This has been suggested by Fukuchi (2012) as a method better suited for high source densities compared to local kernel methods such as sliding minimum subtraction. Following this, noisy fluctuations of the background intensity were estimated to be approximately 60 counts. This constant value was subtracted from the images. No spatial filtering (smoothing) or intensity normalization is applied. A sample preprocessed image for a value of N_S approximately 0.7 is shown in figure 3. Note the change in density across the image, corresponding to the knife-edge transition from 14 to 2.2 mm thickness.

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All measurements are performed simultaneously with the 12 cameras and subsequent operations consider subsets as illustrated in figure 2. The selection of cameras composing each subset maximizes the angular total of each subsystem.

The reconstruction is conducted using the fast MART algorithm within the LaVision 8.1 software, which corresponds to a more efficient implementation of the sequential MART algorithm, i.e. Elsinga et al (2006). The computations are optimized taking into account the sparsity of the reconstructed volume and voxels with intensity below a specified threshold level are not updated (Atkinson and Soria 2009). The threshold level of 0.001 counts was determined by comparison to the conventional MART algorithm taken for reference. The settings for the processing are outlined in table 2. The reconstruction volume depth is increased by approximately 25% in the front and rear to be able to estimate the SNR_R. The reconstructed volume size is thus 47  ×  52  ×  19 mm. The calculations were performed on a 24-processor PC and the reconstruction time was approximately 5 min for a four-camera system and 15 min for the 12 cameras system.

The particle density varies in time throughout the run. At the beginning of the experiment, the highest concentration of particle tracers is approx. 40 part mm^‒3 (as determined by the following analysis). The concentration decays by venting the room during the experiment. As a result, a continuous range of particle concentrations is experienced; however, an in-situ method is required for an accurate determination of the concentration, ppp, and source density at each snapshot. In a tomographic PIV experiment, the particle image density on the cameras is typically too high for an unbiased estimate based on 2D peak detection. As the particle image density exceeds approximately 0.04 ppp (Fukuchi, 2012), the number of detected particles no longer follows a linear relationship with the number of true particles due to high overlap probability. From previous studies dealing with this problem, the most practiced technique for concentration measurement is the slit-test (e.g. Novara and Scarano 2012, Schanz et al 2013, Fukuchi 2012, among others). A removable slit acts as knife-edge filter placed into the laser beam before it reaches the laser volume. The reduced thickness of the laser illumination results in a lower particle image density, allowing an accurate estimate of particle concentration by 2D peak detection. The actual particle image density is estimated by multiplying the obtained particle density by the ratio of the slit to full volume thickness.

This technique can be applied provided that two caveats are considered. First, the method requires the seeding density to be held constant during the time between the recordings with and without the slit. Second, because of differences in optical magnification and viewing angle, the seeding density varies from camera to camera. The present approach addresses both issues. First, the experiment is designed such that the slit measurement is simultaneous to the thick volume measurement (figure 3), which enables a particle density measurement while the concentration is varying in time.

Because the volumetric particle density (ppv, particles per voxel) does not vary depending on the viewing angle like the particle image density, the former is considered for estimating the particles' concentration. Tracer particles are counted by a 3D peak-finding algorithm within the reconstructed volume corresponding to the slit region, where the 12-camera system is essentially free from the phenomenon of the ghost particles. This was further verified using fewer cameras, and the estimated ppv did not change when at least 6 cameras were used.

The 3D peak-finder operates on a 6-connected neighborhood, which is considered to be a robust approach in view of the sparsity of the reconstructed object. The corresponding ppp is obtained by multiplying the ppv by the volume thickness and it represents the particle image density that would be detected by a camera with a viewing axis normal to the illuminated volume median plane.

The time history of measured C, ppv, ppp, and N_S within the experiment are given in figure 4. At the beginning of the experiment, the particle concentration exceeds 40 particles mm^‒3, which corresponds to a source density of approximately 3.5. During a period of 300 s, the seeding density decays to values reported in many tomographic PIV experiments for air flows (Scarano 2013), with particle concentration around 3–5 particles mm^‒3 and a source density of 0.25–0.5. The scatter in particle concentration is largely due to small inhomogenetities in the seeding of the room, leading to slightly higher or lower concentration. This was verified by inspection of the particle images that exhibited the same trend as figure 4.

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As reported in previous studies, the most prominent artifact of MART reconstruction for a particle field is the presence of ghost particles. In the illuminated region, both real particles and ghost particles are formed, whereas outside the reconstructed intensity only ghost particles are expected to form.

As more cameras are added, fewer ghost particles are formed with the double effect of increasing the signal level within the laser sheet and decreasing it outside. Figure 5 illustrates a reconstructed intensity profile along the volume depth making use of 4, 6, and 12 cameras. The data is taken from an average of five snapshots at approximately N_S = 1.4. The intensity is normalized by the maximum of each profile. As more cameras are added, the intensity outside of the laser sheet is dramatically reduced. The average value inside and outside the laser sheet is estimated by the solid and dashed lines, and represents E_O and E_I in equation (3), respectively. The reconstruction signal-to-noise ratio SNR_R is reported in figure 5. The four-camera system yields SNR_R < 2, while for 12 cameras SNR_R approaches 5.

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Extending the procedure to the entire experiment transient yields the relation between the SNR_R and the source density for different numbers of cameras. Results are plotted in figure 6. Note that the condition SNR_R > 2 is met at a source density up to 0.9 for the four-camera system, 1.9 for six cameras, and beyond 3.0 for the 12-camera system. As a result, one may conclude that experiments conducted with more than six cameras satisfy the reconstruction quality criterion even at particle image density up to 0.5 ppp, which significantly moves forward the value currently practiced (ppp = 0.05, Elsinga et al 2006) by about one order of magnitude. The latter result agrees with the experimental observations made by Fukuchi (2012).

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The same procedure is followed to generate the normalized intensity variance (equation (4)). Only the regions of the reconstruction within the illuminated volume are considered. Figure 7 (left) shows the value of $\sigma _{\text{E}}^{\text{*}}.$ The trend is consistent with that of SNR_R, with higher values obtained at low particle concentration indicating a high signal sparsity and low occurrence of ghost particles. The variance of the reconstructed object increases when more cameras are added. However, the difference in variance between systems of different numbers of cameras is not as pronounced as for the SNR_R. For example, if a criterion for acceptable reconstruction quality is taken with a variance above 20, then the improvement in source density moving from four to six cameras is only of 0.6, compared to 1.0 suggested by the SNR_R criterion.

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The value of the variance shown in figure 7 is comparable to the results shown by Novara and Scarano (2012), who reported typical values of the variance for converged MART with different cameras and the application of MTE in the range of 20–30. The maximum reported value of $\sigma _{\text{E}}^{\text{*}}$ for a four-camera system in their work is approximately 28, which compares well to the four-camera system operating at a source density of 0.9 in the present study. In conclusion, the agreement suggests a design note for tomographic PIV reconstructions: the normalized intensity variance should exceed 20 to compare with reconstruction SNR beyond 2 and ensure a reliable analysis by cross-correlation.

The use of a relatively large number of cameras enables the evaluation of reconstruction accuracy by the quality factor Q. The latter is defined as the normalized correlation coefficient between the reconstructed volume and the exact intensity distribution (Elsinga et al 2006). In contrast to the SNR_R and $\sigma _{\text{E}}^{\text{*}},$ which are properties of a reconstructed object, the quality factor specifies the degree of matching to a reference reconstructed object and is therefore a more rigorous check of the reconstruction quality. This criterion has been modified by Novara et al (2010) for experimental data, considering the normalized cross-correlation between the reconstructed object E₁ and a reference object E₂,

$$\begin{eqnarray}&&{Q^*} = \frac{{\sum {{E_1} \cdot {E_2}} }}{{\sqrt {\sum {{{({E_1})}^2}} \cdot \sum {{{({E_2})}^2}} } }}.\end{eqnarray} \tag{ 7 }$$

Note that the mean value of E₁ and E₂ are subtracted. E₁ and E₂ are chosen such that E₁ is a reconstruction with a smaller number of cameras (i.e. E₁ = E_Nc=4) and E₂ is a reconstruction with the full number of cameras (E₂ = E_Nc=12). These values are denoted as Q* and shown in figure 8.

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A solid horizontal line provides the reconstruction quality criterion (Q > 0.75) proposed by Elsinga et al (2006) on the basis of numerical simulations. Furthermore, part of the diagrams obtained with synthetic data in Scarano (2013) are shown as overlays. At low seeding density (N_S < 0.3), only the four- and six-camera systems are able to achieve reconstructions with a quality factor around 0.75. The reduction of Q* is significant as the number of cameras is reduced. For the systems with fewer cameras, the downward trend in reconstruction quality comes back to an agreement with available data from Scarano (2013) until approximately N_S of 0.4.

A discrepancy is noted between the data from the six-camera system and synthetic data, attributed to the use of the 12-camera data as the reference volume. Because the 12 camera system also contains reconstruction artifacts, the value of Q* will not asymptote to a value of 1.0. However, a clear trend can be noted in the data, which allows a connection to be made between the quality factor and the reconstruction SNR. For Q*, the 0.75 threshold is crossed at an N_S of 0.9 and 0.3, respectively. The corresponding SNR_R for these cameras at this source density is 4.5 and 5, respectively.

Although the seeding concentration varies throughout the experiment, the analysis of the velocity field measurement is performed maintaining constant the size of the interrogation box to 32  ×  32  ×  32 voxels. The overlap factor is kept at 75%. With a particle concentration of 5 particles mm^‒3 (average concentration from 400 to 600 s), the number of particles inside the window is approximately N_I = 24, ensuring a robust analysis by cross-correlation.

An in-house algorithm (Fluere) performs 3D cross-correlation by the multi-pass, multi-grid volume deformation (VOLDIM) method similar to that used in Scarano and Poelma (2009). Correlation is performed using symmetric block direct correlation and Gaussian window weighting. Spurious vectors are detected by universal outlier detection (Westerweel and Scarano 2005) and replaced with a distance-weighted neighborhood average. No spatial filtering (smoothing) is applied to the final vector field.

A visual inspection of the measured velocity field by means of Q-criterion visualization (Hunt et al 1988) of vortices is given in figure 9. The conditions are taken as the same shown earlier for the reconstruction (N_S = 1.2, N_C = 3, 6, and 12). The calculation of the velocity gradient tensor components is based on second-order central finite differences.

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The largest flow features are the spanwise coherent rollers, which are captured in all cases. At the present Reynolds number (Re = 5000), the turbulent regime results in significantly smaller flow scales compared to, e.g., Scarano and Poelma (2009). Although the general topology of vortices is still reproduced at such a high seeding density for the three-camera system, the secondary rollers mostly inclined along the streamwise direction are not captured especially by the three-camera system, whereas they become visible in the snapshot reconstructed by 12 cameras. Elsinga et al (2011) found that the flow topology remains unaffected by increased levels of ghost particles. However, a gradient modulation effect is visible for the three-camera analysis, with the large-scale coherent structures along the span of the model diminished in intensity and appearing intermittently broken apart.

A quantitative comparison is not straightforward since the actual velocity field is unknown. A method is followed here that directly estimates the velocity error by comparing the measurements from two independent systems. Under the assumption that the error from such two systems is uncorrelated, the error estimate is taken as the norm of the difference between the two velocity fields. Additionally, the objects composing the pair considered for cross-correlation are taken by different sets of cameras. This is done in order to minimize any possible correlation/bias from ghost particles. For example, to calculate the first displacement (viz. velocity) field $\Delta {{x}_{1,}}$ frame 1 from system 1 would be correlated with frame 2 from system 2. The opposite is used to calculate $\Delta {{x}_{2}}.$ The error ${{\varepsilon}_{\Delta x,1}}$ is computed as the average norm of the difference of these fields following Christensen and Adrian (2002),

$$\begin{eqnarray}&&{{\varepsilon}_{\Delta x,1}}=\frac{1}{N}\underset{i=0}{\overset{N}{\sum}} \left(\frac{1}{\sqrt{2}}||\Delta{{x}_{2}}\left(i\right)-\Delta{{x}_{1}}\left(i\right)||\right).\end{eqnarray} \tag{ 8 }$$

In the above equation, i is the index of each gridpoint, and N the total number of vectors in the measurement domain. This approach not only accounts for the random error, but it also considers the effect of calibration errors. Although useful in the present case where two tomographic PIV systems are used, the present method is generally not viable for studies where a single tomographic system is used. Therefore, the results are also compared with the error estimator based on the velocity divergence operator. For an incompressible flow, any nonzero measured value of the divergence is ascribed to measurement errors. This error ${{\varepsilon}_{\Delta x,2}}$ has been related to the standard deviation of the measured divergence via linear error propagation, under the assumption that the errors at neighboring grid points are uncorrelated (Atkinson et al 2011),

$$\begin{eqnarray}&&{{\varepsilon}_{\Delta x,2}}=\frac{{{\sigma}_{\nabla \cdot \left(\Delta x\right)}}}{\sqrt{\frac{3}{2{{h}^{2}}}}}.\end{eqnarray} \tag{ 9 }$$

In the preceding expression, ${{\sigma}_{\nabla \cdot \left(\Delta x\right)}}$ is the standard deviation of the divergence field and h is the grid spacing (8 voxels presently). Figure 10 presents the ${{\varepsilon}_{\Delta x,1}}$ (left) and ${{\varepsilon}_{\Delta x,2}}$ (right) for three-, four-, and six-camera systems. At relatively low source density (N_S < 0.5) the measurement error is dominated by the limited number of particles within a window in the cross-correlation analysis. In particular, the values of ${{\varepsilon}_{\Delta x,2}}$ at low seeding density is little dependent upon the number of cameras used. As the particle density is increased, both ${{\varepsilon}_{\Delta x,1}}$ and ${{\varepsilon}_{\Delta x,2}}$ decrease, which is ascribed to the higher number of particles considered for the cross-correlation. Further increases of seeding density show a greater disparity between the systems of different numbers of cameras. This indicates that the error here becomes dominated by the reconstruction accuracy, which prevails over the increase of cross-correlation accuracy and robustness.

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The minimum error for both estimators (${{\varepsilon}_{\Delta x,1}}$ and ${{\varepsilon}_{\Delta x,2}}$) results from a minimization of the combined effect of limited particles within the interrogation volumes and reconstruction artifacts. In the three-camera system, a minimum error is ${{\varepsilon}_{\Delta x,1}}$ approximately 0.6 vox at a source density N_S = 0.5. In comparison, the six-camera system attains a minimum error ${{\varepsilon}_{\Delta x,1}}$ of approximately 0.35 vox across a broad range of source densities ranging from N_S = 1.0–2.5.

It is worth noting the discrepancy between ${{\varepsilon}_{\Delta x,1}}$ and ${{\varepsilon}_{\Delta x,2}},$ in general, the values of ${{\varepsilon}_{\Delta x,2}}$ are found to underestimate ${{\varepsilon}_{\Delta x,1}}$ by approximately 30%, which is ascribed to the fact that for the grid points used for the finite differences estimate of the divergence, the random error is not uncorrelated. Moreover, any error arising from discrepancies in the geometrical calibration of the two separate tomographic systems will not be accounted for by the divergence estimator.

The error magnitude reported here is comparable to numerous studies in the literature. In Scarano and Poelma (2009), a cylinder wake at Reynolds number of 1000 was investigated using a four-camera system, with a standard deviation in the divergence of 0.02. This is lower than the values in figure 10, which is ascribed to the greater spatial resolution and lower Reynolds number of the experiment. Atkinson et al (2011) performed an experiment in a turbulent boundary layer with discrete volume offset methods and reported a divergence of 0.05, which corresponded to a velocity error of 0.5 px; these values compare favorably with the four-camera case. Wieneke and Taylor (2006) reported errors of around 0.2 vox for a laminar flow with low velocity gradients. This wide variability in reported error estimates is in part due to the variation in complexity of the underlying flow field. It is well established that the different flow regimes and properties of the turbulent flow can cause large discrepancies in the measured random error.

In conclusion, the trends shown in figure 10 provide guidance for choosing the optimal source density that maximizes the accuracy of the velocity estimation for a given number of cameras.