On the velocity of ghost particles and the bias errors in Tomographic-PIV

The imposed particle displacement is a linear function of the depth coordinate, representative of a shear layer. The motion field is given by:where α = (du/dz)Δt is the displacement gradient determining the actual particle displacement variation over the volume depth, z is the volume depth coordinate of the particle (in voxel length units) and x is the particle coordinate (in voxel lengths) in the homogeneous direction along the front of the volume slice. The particle displacement along the depth direction z is always zero.

To assess the effect of different gradients α and the other experimental parameters on the velocity measurement, Fig. 4 presents results obtained for a 4-camera system with a (1-D) particle image density of 0.15 ppp showing average velocity profiles along the depth coordinate z. The reconstruction quality factor Q is 0.75 for these conditions, which previously has been considered to be the lower limit for reconstruction accuracy (Elsinga et al. 2006a). These results clearly show a systematic underestimation of the velocity gradient consistent with the proposed model for the ghost particle velocities. As α is increased, hence the actual particle displacement variation along z, this modulation effect decreases, which happens at a faster rate for MART compared to the MLOS reconstruction. The MART and MLOS results also reveal a kink at either end of each profile, which are edge effects and not relevant to the discussion here. Additionally, a small discrepancy may be noticed between the input velocity and the particle displacement distribution obtained from the ideal reconstruction (especially for α = 0.01), which must be attributed to a bias introduced by the cross-correlation algorithm. Further, as predicted, the ghost particle displacement is the depth average of the actual displacement throughout the volume. Loss of ghost particle correlation is also evident from a decreasing cross-correlation signal-to-noise ratio with increasing α in the analysis of the ghost-only volumes (Fig. 5). This signal-to-noise ratio has been defined here as the ratio of the first and second correlation peak.

The effect of the ghosts on the cross-correlation is further illustrated by the average correlation maps (Fig. 6), where averaging has been performed in the homogeneous x-direction. Cross-plots through the correlation peak (Fig. 7) reveal that the correlation map for a MART reconstruction can be considered as the correlation map of an ideal reconstruction combined with the map from the ghost-only field, which shows that the ghost particle distribution is largely independent of the actual particle distributions (uncorrelated). This observation supports the assumption in Sects. 2 and 3 that the creation of ghosts can be described as a random process. It also allows a separate discussion of the ghost and actual particles contributions to the cross-correlation maps. The latter produce a single well-defined peak at the actual particle displacement, which is illustrated for two depth position, one near the edge of the volume at z = 25 voxels (Fig. 7-left) and one near the volume centre at z = 95 voxels (Fig. 7-right). In contrast, the ghosts produce a broad peak that is the same throughout the volume (compare Fig. 7-left with 7-right) and covers the full range of actual particle displacements over the volume depth. Given the linear displacement profile considered here, the maximum is attained at the depth-averaged actual displacement. For a relatively high displacement gradient (α = 0.05 voxels/voxel, Fig. 7-top), the ghosts peak ranges from 5 to 15 voxels displacement, which is the range of actual particle displacements inside this volume, with the correlation maximum in the middle at 10 voxels displacement, which is the depth-averaged actual displacement. Note that the ghosts average correlation map does not depend on the depth position z for a constant displacement gradient α. Only the peak from the actual particles varies with z. For smaller gradients (e.g. α = 0.01 voxels/voxel, Fig. 7-bottom), the range of displacements in the volume is smaller and consequently the ghost correlation peak narrows. Because the area under the ghosts’ peak is representative of the number of ghosts in the volume and the ghost intensity, which are both independent of the actual displacement, the ghosts correlation peak value increases with decreasing α. As a result, the relative importance of the ghosts in the correlation map increases resulting in a larger bias error as observed in the velocity profiles (Fig. 4).

Further simulations have shown less dramatic velocity gradient bias errors for lower particle concentrations, i.e. more accurate tomographic reconstructions. These results are summarized in Fig. 8 showing the displacement gradient modulation for varying ppp and displacement gradients for both MART and MLOS reconstructions. Here, the modulation is defined as the measured divided by the actual displacement gradient, where the former is obtained from a linear fit of the measured profiles such as those shown in Fig. 4. An attempt is made to collapse the modulation curves for seeding densities and displacement gradients using the reconstruction signal-to-noise ratio N _p/N _g ^* based on the estimated number of reoccurring ghosts, which is presented in Fig. 9. The variable l* needed to compute N _p/N _g ^* (Eq. 5) is taken as: l* = d _p/α. The collapse clearly is not perfect, which can be expected given the complexity of the MART iterations, the fact that the average lower peak intensity of the ghosts has been ignored and the other simplifying assumptions in deriving N _p/N _g ^* . Nevertheless, it can be concluded that above approximately N _p/N _g ^*  = 1 velocity bias errors are relatively small and appear to decrease even with increasing particle image density. This result may be a useful criterion to avoid bias errors from ghost particles.

The effect of the reoccurring ghost particles on counter rotating vortical structures representing a wake type flow is shown in Fig. 10 for a 1-D particle image density of 0.15 ppp. The particle displacement field in these simulations is given by the superposition of individual vortices with angular voxel displacement ξ _θ defined as:where β is the peak vorticity (in voxels/voxel) evaluated at the vortex axis, r is the radial distance from the axis, and R is the core radius. Vorticity in this section will be based on the particle displacement gradients, rather than the velocity gradients as in its true definition. The difference is simply the time separation Δt between the exposures. The gradients have been evaluated from the particle displacement using a second-order regression. All length scales are in voxel units.

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.

Profiles of the particle displacement in the x-direction through the vortex core (Fig. 11) reveal the same trends with increasing actual particle displacement variation over the volume depth as for the shear layer (Fig. 4). The ghost particles seem to take on the depth average velocity except for the highest peak vorticity case. There, however, the correlation signal-to-noise ratio is so small that the displacement may be affected by secondary effects, in particular the finite viewing angles.

In between the vortex cores, the actual particle displacement in the z-direction (depth) varies very little causing the ghost particle to displace coherently in the same direction as can be seen in Fig. 10 even for relatively strong vortices. Again the measured displacement profiles reveal an important modulation (Fig. 12), which demonstrates that also the velocity component in the depth direction can be affected by biases due to the ghosts.

From the profiles of Fig. 11 (and 12), a modulation coefficient is computed as the maximum measured vorticity (or the maximum z-displacement difference) for MART relative to ideal reconstruction, which is shown in Fig. 13. Because the actual displacement variation through the vortex core is larger than in between cores, the modulation coefficient is much lower in the latter indicating a larger bias error.

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 A _p = 3.2 pixels (the area is obtained from the image auto-correlation), the ratio of actual particles versus ghosts is estimated at N _p/N _g = 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 N _p/N _g estimate is very sensitive to uncertainty in the particle image density ppp and effective particle image area A _p for large N (Eq. 3).

In the experiment, the measurement volume depth direction, z, corresponds to the wall-normal direction in the flow. Therefore, the average velocity profile in z-direction is expected to return the mean shear in the boundary layer, similar to the above simulations of a simple, non-turbulent shear layer. However, as before, the measured profiles reveal a small velocity gradient modulation for the smallest time separation, i.e. smallest particle displacement variation (Δt = 0.33 ms, Fig. 14-left). This modulation decreases as the displacement variation increases until the profiles converge to what is considered to be the actual velocity distribution (Δt = 3.0 and 4.0 ms). These effects are most notable near the edges of the measurement volume due to the decreasing light intensity there, which increases the relative importance of the ghost particles in terms of their intensity. Based on the velocity profile for Δt = 4.0 ms, the maximum difference in actual particle displacement over the volume depth is 0.25 and 3.0 voxels for the time separations Δt = 0.33 ms and 4.0 ms, respectively. Clearly, a maximum difference of only 0.25 voxels (Δt = 0.33 ms) causes the ghost particles, however, few, to reoccur in both exposures and affect the velocity measurement. On the other hand, a maximum difference of 3.0 voxels (Δt = 4.0 ms) is large enough for the ghosts to de-correlate and not contribute to a velocity error.

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 bias error affects not only the mean but the RMS velocity fluctuations as well. However, it is not as easy to detect, because the RMS contains both signal and noise. Measurement noise is approximately constant in terms of pixels particle displacement, and therefore its relative contribution depends on the magnitude of the actual particle displacement, hence on the time separation Δt. With decreasing Δt, a larger contribution from the measurement noise is expected, which opposes the RMS reduction due to the ghosts’ bias error. The auto-correlation function can be used to separate the two effects as illustrated in Fig. 15 for the u-component of velocity. The DC peak in the auto-correlation contains both noise and signal, but when the shift Δx is equal or larger than an interrogation volume (the first point away from the peak in Fig. 15) the measurement noise de-correlates and only the signal remains. Hence, the signal contribution to the DC peak may be roughly estimated by extrapolating data away from the peak to Δx = 0 (dashed lines). The corrected peak value is then related to the noise-free RMS fluctuations according to u _rms = R _uu (Δx = 0)^1/2. It is seen from Fig. 15 that the corrected RMS value and the auto-correlation around the peak increase with decreasing time separation Δt, which is consistent with the expected behaviour for the ghosts’ bias error.

Further, from a visual inspection of the spatial distribution of vorticity and other local topology detection criteria (such as the Q-criterion, Hunt et al. 1988), it is found that main topological features of the turbulent flow are preserved at small Δt (see for example Fig. 16) even though their amplitude clearly is decreased due to the bias error. At the same time, the measurement noise is increased with decreasing Δt due to the decrease in the dynamic range of the particle displacement. For completeness, it is noted that the presented velocity gradients have been determined using a second-order regression.

The observed similarity in topology can be quantified by the correlation of the spanwise vorticity distribution ω _y for Δt = 0.33 and 0.67 ms with the actual vorticity distribution, which is approximated by ω _y 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 ω _y 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.

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.

First, the variation of N _p/N _g ^* with the particle image density ppp needs to be established for the specific experimental conditions, i.e. the number of cameras N = 4, the effective particle image area A _p = 2.4 pixels, volume thickness l _z  = 150 voxels, and the length scale l* for which the particle displacement normal to the viewing direction varies to within an effective particle image diameter d _p = (4A _p/π)^0.5 = 1.7 pixels. The length l* can be assessed based on the measured instantaneous velocity distributions such as shown in Fig. 17-left. Suppose the accuracy of the velocity around the secondary vortices is of interest here. Then, the length l* is obtained directly from a profile of the particle displacement taken along the z-axis and through the vortex core. In particular, the displacement component normal to the viewing direction, i.e. z-direction, needs to be considered (Fig. 17-right). [For simplicity the viewing direction is taken as the depth direction, z.] In this plot, the region where the displacement varies within d _p = 1.7 pixels is indicated by the horizontal blue lines. Presently, l* is approximately 0.5D, where D ≈ 150 voxels is the cylinder diameter, which leads to l*/l _z  = 0.5.

Substitution of all parameters in Eq. 5 yields the result in Fig. 18, which shows N _p/N _g ^* for varying ppp and l*/l _z . For l*/l _z  = 0.5, it is found that N _p/N _g * remains sufficiently large even for the largest particle image density encountered in these experiments, ppp = 0.08. This explains why no significant differences have been observed in the results for the different particle image densities over the range 0.02 < ppp < 0.08.

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).