Test case 3 Circular jet

For this test case 26500 spherical Gaussian particles with 3 voxels diameter are distributed in a volume of 19.2 x 19.2 x 57.6 mm3_{, again with a resolution of}

(i.e. 384 x 384 x 1152 voxels, with approximately 1.5% of voxels with non-zero intensity, and about 17 particles per IV, being the final IV a cube of 48 voxels for each linear dimension). The displacement field of a circular jet directed along the z direction with a peak displacement of 16 pixels is simulated. The

used equation of the jet-like profile is:

_(5.9)

where is a parameter to modulate how steep is the descent from the peak velocity to zero and is the station where the jet reaches a velocity that is equal to

(set to 96 voxels).

The structure of the processing algorithm is reported in Tab. 5.3. Since the flow field is characterized by strong gradients (whose intensity is controlled by ), the modulation effects influence the predictor estimation, determining the corrector peak to be above the search radius of ±0.5 pixels for a wide number of IVs. For this reason an hybrid method has been tested, i.e. the cross-correlation is evaluated using block FFT (BFFT from this point on) in the refinement steps, and one of the proposed solutions for the fast final corrector computation (for this test-case the 1D DC method is adopted; the lower efficiency of the method with respect to that of the block direct cross-correlations is of relative importance, since most of the processing time is related to the peaks falling above the search radius, i.e. the full correlation maps to be computed again with FFT; on the other hand, 1D DC enables to use rigorously weighting windows).

Chapter 5 – Efficient 3D PIV interrogation algorithms

Fig. 5.9 Speed-up for each step as a function of the processing step for (a) and (b). TH and

BL indicate Top Hat and Blackman weighting window for the cross-correlation map, respectively.

Fig. 5.10 Percentage of re-computed full cross-correlation maps with FFT for (a) and (b).

TH and BL indicate Top Hat and Blackman weighting window for the cross-correlation map, respectively. Step [ ] Binning factor IV size [vox] Effective IV size

[vox] Grid Overlap

CC evaluation method 1 4x 243 ₉₆3 _{0% FFT-Based} 2 4x 243 ₉₆3 _{50% BFFT/DC} 3 2x 243 ₄₈3 _{50% BFFT/DC} 4 1x 483 ₄₈3 _{75% BFFT/DC} 5 1x 483 ₄₈3 _{75% DC} 6 1x 483 ₄₈3 _{75% DC}

Table 5.3 Processing algorithm: in the refinement section the CC evaluation method can be different

The velocity profiles of the jet-like velocity field are reported in Fig. 5.8. In the case of the results show that with a standard approach without using weighting windows (say, a top hat moving average, indicated with TH) the obtained profile is only slightly modulated with respect to the original one, the peak displacement being 2% smaller; the application of a Blackman (BL) weighting window in both cross-correlation and dense predictor averaging (over a window of the same size of the IV) reduces this modulation to less than 0.5%. When the modulation by the TH approach is much more evident; the profile obtained by using BL in the process is closer to the exact one.

The speed-up for each step (i.e. the reduction of the processing time relative to the time needed to execute the iteration with the full FFT approach) of the process is reported in Fig. 5.9. The first step is performed about 75 times faster thanks to 4x binning; when a Blackman weighting window is applied, the speed-up is of about 27 times with respect to the standard FFT approach. For the case of , and no windowing applied, the approach based on block FFT during the grid refinement is consistently faster, since a broader search area increases the number of "fast- detected" peaks; this occurs due to modulation effects, placing the corrector displacement peak beyond the limit of the search radius of 0.5 pixels, as shown in Fig. 5.10, where the percentage of re-computed full cross-correlation map due to direct correlation failure is reported as a function of the iteration number (note that in the first iteration all the correlations are executed using the FFT, i.e. 100% of the correlation is non-direct; on the other hand, since no repetition of the calculation is performed, all the curves start fictitiously from 0 at the first iteration). Altogether, when no weighting windows are applied, the processing time is reduced by 18 and 11 times for the method employing BFFT in the first steps, and the other based only on direct cross-correlation with pre-calculation of sums along segments, respectively. When weighting windows are applied, the speed-up reduces to a factor of 12 and 9 for the two cases. However, one should consider that if the standard process with application of weighting windows in the computation of FFT is considered as a reference, the process with weighting window is accelerated by 27 and 21 times, respectively.

The case of is quite different, since the region affected by velocity gradients is smaller, but the slope of the velocity profile is much higher. In this case for all the 4 presented methods a not negligible percentage of non-direct cross correlations is still present even in the last iterations because of the modulation of the velocity profile; this reduces the maximum allowed speed-up of the cross- correlation algorithm to 6.4 and 5.9 times for the methods without the aid of weighting windows, with or without BFFT, respectively (the corresponding speed- up in case of filtering of the cross-correlation map with a Blackman weighting window is 3.9 for both the methods). One should note that the percentage of non- direct cross correlations for the method without using weighting windows increases with the number of iterations; this occurs because of aliasing. This aspect

Chapter 5 – Efficient 3D PIV interrogation algorithms

is made clear in Fig. 5.10b. The approach with block FFT implies a higher percentage of correlation maps to be recomputed in the 5th _{step (when direct cross-}

correlation is employed) due to the effects of the imposed periodicity, reducing the accuracy of the estimated displacement field.

These issues are less critical if a Blackman weighting window is employed in computing the cross-correlation coefficients for all the proposed approaches. In this case, the frequency response is more similar to that of an ideal low pass filter (Astarita 2007), and no aliasing is present, so that the percentage of non direct cross-correlations decreases in the final iterations.