On the velocity of ghost particles and the bias errors in Tomographic-PIV
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- Elsinga, G.E., Westerweel, J., Scarano, F. et al. Exp Fluids (2011) 50: 825. doi:10.1007/s00348-010-0930-0
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The paper discusses bias errors introduced in Tomographic-PIV velocity measurements by the coherent motion of ghost particles under some circumstances. It occurs when a ghost particle is formed from the same set of actual particles in both reconstructed volumes used in the cross-correlation analysis. The displacement of the resulting ghost particle pair is approximately the average displacement of the set of associated actual particles. The effect is further quantified in a theoretical analysis and in numerical simulations and illustrated in an actual experiment. It is shown that the bias error does not significantly affect the measured flow topology as deduced in an evaluation of the local velocity gradients. Instead, it leads to a systematic underestimation of the measured particle displacement gradient magnitude. This phenomenon is alleviated when the difference between particles displacement along the volume depth is increased beyond a particle image diameter, or when the reconstruction quality is increased or when the accuracy of the tomographic reconstruction is improved. Furthermore, guidelines to detect and avoid such bias errors are proposed.
Tomographic-PIV is a recent technique for measuring the three-dimensional instantaneous velocity distribution in a fluid (Elsinga et al. 2006a). In this method, images of the tracer particles are recorded from several viewing directions simultaneously, after which a tomographic reconstruction algorithm is used to obtain the three-dimensional light intensity distribution associated to the tracer particle distribution within the measurement volume. The particle displacement field between subsequent recordings, hence velocity, then results from a cross-correlation analysis of two reconstructed light volumes analogue to planar PIV (Adrian 1991). Compared to other existing 3-D PIV methods, such as 3-D particle tracking (Maas et al. 1993) and digital holographic-PIV (Coëtmellec et al. 2001), Tomographic-PIV can operate at higher seeding densities, typically 0.05 particles per image pixel (Elsinga et al. 2006a), which is advantageous as this allows for a finer spatial resolution of the measurement. As a result, it is now being applied as a useful tool in fluid dynamics investigations of predominantly turbulent flows (e.g. Elsinga et al. 2007, 2010; Elsinga and Marusic 2010; Schröder et al. 2008a, b; Hain et al. 2008; Humble et al. 2009; Atkinson et al. 2009; Kühn et al. 2009; Ortiz-Duenas et al. 2010). Where the mean and RMS flow statistics have been evaluated in these experiments and compared against other data, they appear to agree to within approximately 0.3 pixel particle displacement (Elsinga 2008) showing that the method can be accurate.
Although the reported accuracy of the Tomographic-PIV technique is hopeful, very little is known about possible velocity bias errors that may appear when the experimental conditions are changed. Here, a distinction needs to be made between the errors introduced in the imaging and cross-correlation steps of the experimental procedure, and the errors coming from the tomographic reconstruction. The former are in common with planar PIV and have already been documented (for an overview see Raffel et al. 1998 or Adrian and Westerweel 2010), while the latter are yet to be investigated. What is known is that the tomographic reconstructions contain spurious light intensity peaks, which are referred to as ghost particles (Elsinga et al. 2006b). Their origin will be discussed in more detail in Sect. 2.
Surprisingly, first results from previous Tomographic-PIV measurements in a cylinder wake have indicated that the accuracy of the velocity vectors shows little to no dependence on the accuracy of the tomographic particle reconstruction, that is the number of ghost particles compared to the number of actual particles (Elsinga et al. 2006b; Elsinga 2008). This independence of the velocity and reconstruction accuracy applies at least within the experimental parameter range considered in that particular cylinder wake experiment, which included, however, a case where the ghost particles outnumbered the actual particles. Under that condition one would expect the falsely reconstructed particles to dominate the measurement and introduce errors in the particle displacement obtained by cross-correlation. This behaviour is clearly not observed in the actual measurements, which further illustrates the need for a better understanding of the ghost particle phenomenon.
In this paper, we will present a model describing the ghosts’ contribution to the velocity measurement (Sect. 3) and provide supporting evidence for it from numerical simulations and experiment (Sects. 4–5). It will be shown that ghosts can introduce an important velocity bias error under some circumstances. Further insight into the cross-correlation signal produced by the ghosts will be obtained, which enables us to formulate criteria and experimental procedures directed at preventing ghost particles from affecting the accuracy of the velocity field.
The present model builds on previous discussions of ghost particles in 3-D particle tracking (Maas et al. 1993) and aims at using the experimental parameters typically available at the design stage of an experiment. Alternative to our engineering approach, a rigorous mathematical treatment of ghost formation can be found in the literature describing the influence of the Null space in tomographic reconstructions (e.g. Louis 1981; Natterer 1986; Munshi 2002). These general works, however, are not specifically targeted at particles, nor do they contain information on ghost velocities.
2 Formation of ghost particles
It should, however, be kept in mind that this ratio alone does not account for the effect of the different peak intensity between ghosts and actual particles as previously discussed by the author (Elsinga et al. 2006a, 2008, histograms reveal a lower peak intensity for ghosts compared to the actual particles). In this particular case, the results will yield a conclusion from a conservative point of view whereby ghost and actual particles have the same peak intensity. Nevertheless, the mechanism under scrutiny is captured in its general features. From Eq. 3, it is readily seen that the signal-to-noise ratio in the reconstruction strongly depends on the particle image density expressed in ppp. With decreasing ppp the occurrence of random intersections of lines-of-sight corresponding to particle images rapidly decreases, hence the number of ghost particles drops. Furthermore, the reconstruction signal-to-noise ratio increases with the number of cameras N, because ppp < 1. The ratio Np/Ng furthermore depends on the thickness of the reconstruction volume, which is proportional to the interception length of the line-of-sight with the volume. Finally, the effective particle image area Ap is usually a constant in a given experimental setup, where the particles are in focus and their images are diffraction limited. The effects of the camera viewing angles are generally weak within the range commonly used in experiments (Elsinga et al. 2006a), and they are neglected in the present discussion. Camera calibration errors are also not taken into account, as they can be reduced to a fraction of a pixel by a volume self-calibration (Wieneke 2008).
The relation of Eq. 3 has been supported by results from 4-camera Tomographic-PIV experiments in which the particle image density and volume thickness varied (Elsinga et al. 2006b; Elsinga 2008; Michaelis and Wieneke 2008). In these studies, the effective particle image area Ap was obtained from a fit of Eq. 3 on the available reconstruction noise data indicating typical values around 2.5–3.0 pixel area in agreement with the individual particles images observed in the recordings. The reconstruction noise in these experiments was estimated by the ratio of the reconstructed light intensity inside and outside the light sheet (Michaelis and Wieneke 2008) or by the ratio of the number of reconstructed intensity peaks in these respective regions (Elsinga et al. 2006b).
3 Ghost particle pairs
The proposed mechanism has some important consequences. First, a velocity may be detected outside the illuminated domain related to the ghost particle pairs outside the light sheet. Whereas actual particles are reconstructed only within the laser sheet, ghosts formation is a random process that takes place throughout the entire reconstructed volume. Since the reconstructed volume is usually slightly larger than the laser sheet, the velocity can be affected by ghosts in some regions near the edges of the reconstructed volume (Elsinga et al. 2006b). There is, however, a simple remedy to this issue: detect the light sheet in reconstructed light intensity volumes (e.g. Elsinga et al. 2006a), so that the light position is known exactly and velocity vectors outside can be ignored. A second, more concerning consequence is the velocity modulation by the reoccurring ghost particles to which there is no straightforward solution. Therefore, this aspect will be studied further in the following section by means of numerical simulations.
4 Numerical simulations
Now the relevant parameters in ghost particle pair formation have been identified, their impact on the velocity measurement will be demonstrated by means of numerical simulations of a Tomographic-PIV experiment. Two flow cases will be considered here: a shear layer representing boundary layer and mixing layer flow, and an array of counter rotating vortices resembling the secondary flow structures in a separated wake.
A 2-D ensemble cross-correlation analysis (Meinhart et al. 2000) combined with iterative multi-grid and window deformation (Scarano and Riethmuller 2000) is performed of the MART, MLOS, ideal reconstructions and the ghosts to obtain the particle displacement in each of these volume intensity fields. The final correlation window size is 31 × 31 voxels and each ensemble contains 31 volume slices, which resembles a 313 correlation volumes in case of a real 3-D measurement. Finally, the velocity fields are averaged over four independent realizations of such correlation averaging results to suppress correlation noise.
Overview of the particle image densities used in the simulations
Source density Ns = ppp·Ap
ppp in 1-D images (Ap = 3 × 1 pixels)
Corresponding ppp in 2-D images (Ap = 3 × 3 pixels)
MART reconstruction quality factor Q
As mentioned, the present 1-D images can be regarded as a pixel line taken from the full 2-D recordings. Similarly, the reconstruction here represents a slice out of the complete 3-D volume. Such conditions may occur in actual experiments where the cameras are all positioned along a single line and placed at a large distance. Then, the magnification over the volume depth may be approximated as constant and the lines-of-sight from each camera become almost parallel. This appears a reasonable assumption, as these parameters are changing very gradually over the measurement volume, commonly. Nevertheless, most Tomographic-PIV setups feature a rectangular camera arrangement, not a linear one. This is likely to have a minor effect on the reconstruction noise levels comparable to changing the cameras viewing directions. The effect remains to be quantified, but a previous fully 3-D simulation using a rectangular arrangement did yield a reconstruction quality Q comparable to the linear camera arrangement (Elsinga et al. 2006a).
4.1 Shear layer
4.2 Counter rotating vortices
As for the shear layer, a modulation of the particle displacement gradient is visible resulting in significantly reduced vorticity level for the case of a lower peak vorticity (0.03 voxel/voxel, Fig. 10-left). Please note here that the measured local flow topology, as represented by the general shape of the vorticity contours, is still in good qualitative agreement with the actual flow pattern. This is important as Tomographic-PIV is mainly used at present to investigate flow topology, i.e. coherent structures. For a large peak vorticity (0.20 voxel/voxel, Fig. 10-right), the actual particle displacement variation over the volume is larger so that the ghosts again decorrelate and do not affect the correlation signal significantly. Consequently, the velocity field is measured with a higher accuracy.
This part of the paper presents evidence of ghost bias errors in real measurements of a turbulent boundary layer and a validation the overall trends observed in the numerical simulations. Also, a demonstration is given on how these errors can affect the turbulence statistics. Further, a procedure is proposed to assess Np/Ng*, that is the relative occurrence of ghost pairs, in general measurements, which is illustrated for a cylinder wake flow (Sect. 5.2).
5.1 Validation of the numerical simulations
For the experimental validation, time-resolved boundary layer data is used (see Schröder et al. 2008b for details on the experimental arrangement). The high image recording rate in that experiment (3 kHz) allows for a control of the displacement variation over the volume depth, while maintaining the same turbulent flow inside the measurement volume, by changing the time separation between the exposures used in the cross-correlation analysis (at increments of 0.33 ms). This increment is indeed much smaller than the smallest timescales in the flow, which are reported to be approximately 11 ms (Schröder et al. 2008b). To recall the other main experimental details: the Reynolds number based on momentum thickness was 2,460 and the boundary layer thickness was 37 mm. A system of six high-speed cameras (N = 6) was used to measure the velocity distribution in a 60 × 60 × 12 mm3 volume. The spatial discretization in the images and the reconstructed objects was 12 voxels/mm, while the particle image density was approximately 0.06 ppp. For these experimental conditions and Ap = 3.2 pixels (the area is obtained from the image auto-correlation), the ratio of actual particles versus ghosts is estimated at Np/Ng = 7. Even if this ratio is high, a modulation effect is still expected for very small particle displacement variation over the volume depth. Note also that the Np/Ng estimate is very sensitive to uncertainty in the particle image density ppp and effective particle image area Ap for large N (Eq. 3).
Similarly, the accuracy of the velocity depends strongly on the number of cameras used in the tomographic reconstruction. Reducing this number from 6 to 3 cameras leads to increasing bias errors (Fig. 14-right), which is the result of a decrease in the reconstruction quality and consequently an increase in the number of ghost particle pairs (at constant time separation Δt = 4.0 ms).
The observed similarity in topology can be quantified by the correlation of the spanwise vorticity distribution ωy for Δt = 0.33 and 0.67 ms with the actual vorticity distribution, which is approximated by ωy for the case Δt = 4.0 ms. These correlation coefficients yield 54 and 72%, respectively, which demonstrates a reasonable correspondence given the high noise level, especially at small Δt. This noise level is estimated at 40% (Δt = 0.33 ms) and 20% (Δt = 0.67 ms) based on the increased drop off in the ωy auto-correlation peak w.r.t. the reference case (Δt = 4.0 ms). All the above trends in the experimental results are fully consistent with the proposed model and results from numerical simulations.
5.2 Cylinder wake revisited
The above results can now help to understand the invariance of the velocity and vorticity statistics in the cylinder wake with changing particle image density (Elsinga et al. 2006b; Elsinga 2008), which is part of the initial motivation for this work as discussed in the introduction. Moreover, the procedure that will be outlined here may be useful in a general measurement to check whether ghost bias errors are expected to be important.
It should be noted that the above l* is specific to the core region of the secondary vortex structure, i.e. the particle displacement profile used in Fig. 17-right, and that other values may be obtained in other regions of the flow. In the outer flow region, the flow is more uniform and smaller particle displacement variations over the volume depth are expected resulting in larger values for l*. Hence, the very small velocity gradients in these more uniform outer flow regions may be underestimated due to the ghosts, but in the absolute sense this is a small error, which is obscured by the measurement noise and uncertainty (0.005 voxels/voxel for the displacement gradient in this experiment, Elsinga 2008).
The mechanism responsible for the formation of ghost particle pairs, i.e. reoccurring spurious intensity peaks, in Tomographic-PIV has been described. These reoccurring ghosts are present in both reconstructed light intensity volumes used in the cross-correlation analysis and are displaced in between by approximately the average displacement of the actual particles from which they have been formed. Therefore, they have the effect of smoothing and reducing the measured particle displacement variations (and gradients) over the volume thickness. The number of ghost pairs can be estimated from the experimental parameters as outlined in Sect. 3.
Results from numerical simulations and experiments further illustrated this phenomenon showing a decrease in the measured average shear in a shear layer and peak vorticity in vortices due to the ghost pairs. The local flow topology, however, remained largely unaffected. Furthermore, the simulations showed that the bias errors decrease with increasing reconstruction accuracy as well as with increasing actual particle displacement variation (or displacement gradient) along the line of sight, in agreement with theory. Hence, it is important to monitor the reconstruction accuracy and the displacement variation over the volume depth in real experiments. The former can be done by evaluation of the average reconstructed intensity inside and outside the laser sheet (Elsinga et al. 2006a; Michaelis and Wieneke 2008). A procedure to check for sufficient displacement variation has been presented in Sect. 5.2, but as a rule of thumb it can be said that the range of displacements normal to the viewing direction should exceed a particle image diameter. Alternatively, a comparison of the flow statistics with results obtained by other experimental techniques may be used to identify important bias errors. Examples of the relevant quantities in that respect are the mean velocity variation over the volume depth and the RMS velocities.
The required minimum particle displacement variation over the volume thickness can be translated into a minimum value for the time separation between (laser) light pulses without exceeding limits imposed by the cross-correlation algorithm.
The knowledge gained on the reoccurring ghosts can further be used to improve the accuracy of the individual tomographic reconstruction (e.g. Novara et al. 2010). There are also anticipated applications related to separating ghost from actual particle trajectories in time-resolved Tomographic-PIV or 3-D PTV measurements (e.g. Schröder et al. 2008b). The ghost trajectories will be relatively short compared to the actual particle trajectories, which is due to the individual ghost particle disappearing with increasing time, i.e. increasing displacement variation of the corresponding actual particles. The actual particles, on the other hand, are never lost and can, in principle, be tracked through the entire measurement domain.
The authors thank F.F.J. Schrijer for his contribution in the ensemble cross-correlation analysis. This work is supported by the Dutch Stichting FOM.
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