Spectral Improvement of Time Resolved PIV by FTEE in Homogeneous Isotropic Turbulence

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1 Spectral Improvement of Time Resolved PIV by FTEE in Homogeneous Isotropic Turbulence Thomas Earl 1*, Young Jin Jeon 1, Corine Lacour 2, Bertrand Lecordier 2, Laurent David 1 1: Institut P, Université de Poitiers - ENSMA, F86961 Futuroscope Chasseneuil Cedex,France 2: CORIA, UMR CNRS 6614, Université et INSA de Rouen, Saint Etienne de Rouvray Cedex, France * Correspondent author: thomas.earl@univ-poitiers.fr Keywords: PIV; Accuracy; Time-resolved; HIT ABSTRACT This work investigates the performance of the time-resolved PIV technique FTEE with an optimized 2-Frame PIV algorithm on both synthetic and experimental homogeneous isotropic turbulence (HIT). In the case of the synthetic tests, the algorithms are compared with a model that appropriately filters the full resolution HIT to replicate a perfect PIV treatment. Several turbulence quantities are computed such as turbulent kinetic energy, energy dissipation and integral length scale for both the experimental and synthetic experiments. Calculation of the energy spectra was carried out to analyse the algorithm s ability to resolve turbulent scales. It is shown that FTEE algorithm is able to resolve smaller scale structures by virtue of its ability to reduce the interrogation window (IW) size compared to the 2-frame algorithms and without adding high frequency noise to the velocity fields, unlike the 2-Frame implementation. This is mainly owing to the ensemble cross correlation of the cross-correlation maps that permits lower particle numbers in the IW and the modilisation of trajectories by a non-linear polynomial. 1. Introduction The aim of this work is to study the limitations and advantages of 2-frame and time resolved PIV measurement techniques on homogeneous isotropic turbulence (HIT). HIT is a challenging measurement case for PIV measurements owing to its wide range of length and velocity scales. Particular for the case where 2D measurements are being made, loss of particle pairs due to out of plane motion can set a ceiling on the time-step between image acquisition and thus reducing the velocity dynamic of the measurement. Recent developments of time-resolved (TR) techniques (such as FTEE, Jeon et al, 2014; Jeon et al, 2014a) in both two- and three-dimensional (2D and 3D) cases, have been demonstrated to increase the precision of TR-PIV measurements, as well as providing accurate material acceleration and thus pressure gradient data. This work will complement earlier studies by Lecordier et al (2001), Foucaut et al (2004) and Worth et al (2010). Considering the spectral response of the simulated HIT, we undertake an investigation of

2 the FTEE method for 2D cases in comparison to optimally set 2-frame PIV and push the techniques to the limit of their applicability. Spectral analysis provides an important measure of the scales resolved in the flow. The use of PIV interrogation windows acts as a sliding average filter and tends to limit the smallest resolvable spatial frequencies, in 2D PIV, by the relation (Foucaut et al, 2004) κ cut = 2.8 IW x where IWx is the physical dimension of the interrogation window. Lecordier et al (2001) showed that by introducing sub-pixel window-offsets, peak locking and high frequency noise could be reduced. This boundary was pushed further with the introduction of window deformation techniques such as WIDIM (Scarano, 2002) and weighting windows (Astarita, 2008). Accurately resolved smaller scales provide greater precision on pressure and force computations, which has is desirable from both practical and fundamental flow study perspectives. In cases such as turbulent wakes or HIT, cross-correlation methods tend to be more robust than particle tracking methods owing to shorter particle trajectories in highly three-dimensional regions. We simulate several cases with different laser sheet thicknesses to assess the optimal trade-off between particle loss due to out of plane motion for a thin laser sheet and the corruption of 2D flow statistics based on excess 3D particle motion in the laser sheet for a thicker laser sheet. This analysis is equivalent to the trade-offs made in a real 2D two-component PIV measurement in a 3D turbulent flow. Finally, a 2D PIV experiment was conducted in air of a turbulent flow that approaches the conditions of HIT. This real case will permit a thorough verification of the effectiveness of the FTEE algorithm and the reduction in cut-off frequency κcut. 2. Synthetic HIT investigation 2.1. Synthetic particle image generation A synthetic stationary velocity field that simulates HIT in terms of range of scales and spectral signature was used to displace randomly populated 3D particle fields as per the method in Lecordier et al (2001) (Figure 1). The total size of the simulated volume is a 70 mm 3, with integral length scale l0 = 8.3 mm and Kolmogorov scale h = 270 mm, turbulent kinetic energy Kt = 0.11 m 2 s -2, and with the simulated fluid being air. This field will be referred to as the full resolution simulation. The particles obtain a Langrangian motion as they are displaced in a stepwise manner by the velocity field, so that non-linear trajectories are obtained. The particles are then discretised on a voxel grid (X,Y,Z) = (52,52,13 mm 3 ) for each time-step, modelled as

3 Gaussian blobs of intensity. To simulate 2D PIV experiments, slices in the centre of the discretised volume of varying thicknesses ΔZS were extracted and projected orthogonally to make 2D images, without any image deformation effects. A Gaussian illumination intensity distribution was applied along the camera axis to replicate an experiment and to generate realistic particle images (Westerweel, 2000). The pixel size was approximately 0.1 mm. Fig. 1 Visualisation of HIT velocity field used to displace particles in both two-frame and pseudo time-resolved modes. (Left) full velocity field, skipping every 6th vector in each direction to aid visualisation; (Right) zoom of particles coloured by frame number n to show evolution of trajectories. The number of particles in each slice was kept constant. This is a reasonable measure as the particle seeding in an experiment will typically be modified to achieve a image particle density (or particles per pixel, ppp) irrespective of the laser sheet thickness. The particle image diameter approaches the ideal value of 2.5 px, and ppp The images were generated from slice thicknesses of ΔZS 1, 2, 4, and 6 mm. Figure 2 shows 64x64 px samples of each of these slice thickness. As the laser sheet thickness increases, there is a higher proportion of darker particle images. The images created also did not consider any background noise, although a systematic test of the influence of background noise for this analysis is underway. Naturally, the objective of a PIV measurement is to obtain the real flow field statistics, in this case those corresponding to the full resolution simulation. However it is not trivial to make a

4 comparison of velocity statistics with a 2D PIV field. To this end, to enable direct comparison with the PIV calculations, the full resolution velocity field was filtered with a Gaussian kernel in the Z-direction and equivalent weighting window of equivalent size to those of the PIV tests. Although applying a Gaussian filter on the particle intensity and on the velocity components is not equivalent (for example, particles at 50% intensity don t necessarily have a 50% contribution in the cross-correlation calculation and thus displacement calculation, this model appears to be a good approximation as is made evident by the lack of bias errors in the displacements and the behaviour of the spectral curves compared with the PIV calculations. Thus this model enables a comparison of the results obtained by PIV calculation with a known and equivalent velocity field. It also provides an indication of the best possible measure based on the filtering effect of the PIV process. This will be referred to as the IW filtered simulation hereon. (a) ΔZS = 1 mm (b) ΔZS = 2 mm (c) ΔZS = 4 mm (d) ΔZS = 6 mm Fig. 2 64x64 px samples of the projected images for different thickness of the volume PIV test regimes Sequences of temporally spaced particle volumes were selected, all for the same central slice in the volume. A later publication will increase the number of slices to improve the statistics. The FTEE algorithm was applied to a sequence of N = 3, 5 or 7 images (denoted here t4(n-1), i.e. t8, t16, and t24, where t is the time) to compute a trajectory. The polynomial fitting the trajectory Γ(x,n) was 2nd order, where x is the vector of coordinates of the IV centres, n the frame number. Taking the first derivative at the centre of the trajectory at n = 0 yields the displacement vector, s = d(γ(x;0))/dn. The two-frame implementation is computed over equivalent time steps, equivalent to a maximum in-plane displacement of 7 px. As mentioned previously, the 2-Frame PIV includes deforming wind off-set between iterations. Both 2-frame and FTEE algorithms presented herein use five iterations on the final IW size, with a Gaussian weighting window applied on the last pass, allowing for full convergence. There was no post-treatment of the PIV velocity fields, only intermittent Gaussian smoothing and median filtering. Both algorithms employed adaptive IWs size, starting from 32x32 px regardless of final IW size.

5 2.3. Synthetic PIV Results Figure 3 shows extracted velocity fields for ΔZS = 6 mm. Figure 3a shows the central slice in Z, extracted from the full resolution simulation. This enables a visualisation of the flow scales and structures that are present in the flow field. The remaining images show the IW filtered synthetic field, 2-frame and FTEE PIV results, calculated from a 32x32 px IW with 75% overlap. Visually, the difference between the velocity fields is almost negligible. Careful analysis of relative errors reveals that for the parameters used in this test case, the 2-Frame PIV was able to resolve the flow correctly limited only by the IW size, as there is insignificant difference between the PIV approaches. (a) Synthetic full resolution (b) Synthic field filtered by IW (c) 2-Frame PIV (d) FTEE PIV Fig. 3 2D velocity field representations with contours of U. (a) Single slice of synthetic volume at full resolution; (b) synthetic velocity field integrated over depth ΔZS and filtered with equivalent interrogation window size; (c) 2-Frame PIV from 2D images from projected particles in volume slice; (d) FTEE PIV from 2D images from projected particles in volume slice.

6 2.4. Turbulence statistic comparison We are interested in the effect of the IW size, time separation between particle images and the effect of the laser sheet thickness on the ability of the PIV algorithms to obtain flow characteristics. Figure 4 summarises these contributions on the 2 component (2C) turbulent kinetic energy values Kt for ΔZS = 2 and 6 mm. In both cases, increasing the IW size globally reduces Kt, owing to smoothing by a larger averaging kernel, shown in both the filtered DNS field and the PIV algorithms. Thus smaller interrogation windows are highly desirable. We note that the 2C Kt of the full resolution data is approximately 0.05 m 2 s -2, which is still off the scale at this measurement resolution. It can be seen the FTEE align very well with the IW filtered DNS for the 3 time separations shown (indicated by the colours of the symbols). The 2-Frame levels of Kt approach the filtered DNS reducing the IW size, however at window sizes of 8x8 and 12x12 px the cross-correlation was very noisy owing to too few particles in the window. This points to a key advantage of FTEE, which by ensemble averaging the cross-correlation signals allows for fewer particles in the IW to obtain valid fields. For the thicker laser sheet profile, there is an additional averaging and thus a reduction in the calculated Kt levels compared to the thinner laser sheet. A larger variation on the PIV results compared to the IW filtered simulation can also be observed, likely owing to reduced cross-correlation signal from the lower intensity particles. (a) ΔZS = 2 mm (b) ΔZS = 6 mm Fig. 4 Turbulent kinetic energy budget Kt component of velocity over time, comparing DNS filtered, 2-Frame and FTEE results. Black, red, blue, green markers represent t8, t16, t24, t32, respectively. In Figure 5 continues the comparison, this time on the value of the integral length scale l0 and the 2C energy dissipation ε. These important turbulent statistics rely on velocity field autocorrelation analysis and spatial gradient correlations respectively. The calculation of the l0 is not

7 trivial, as shown by (Soria et al 2005), particularly for small spatial domains. The values presented here were computed by integrating over the maximum length of the auto-correlation signal. These results show that for both the 2F and FTEE calculations, the integral length scale is relatively stable, with the 2F at worse having a 2% relative difference to the IW filtered result. The 2C ε was computed from the 2D velocity components by ε = ν {2 ( ( u 2 x ) + ( v 2 y ) ) + ( u 2 y ) + ( v 2 x ) + 2 (( u y 2 v x ) )} The IW size has a very strong influence on the calculation of ε, as from 32x32 px to 8x8 px IWs increases the computed value by a factor of 2.5, compared to l0 which over the same range changes by no more than The FTEE results correspond well to the IW filtered simulation across the range of IW sizes except for the smallest size. By this IW size however the 2-Frame calculated values are off the scale, at 12x12 px the difference exceeds 30%, reducing to 10% at 16x16 px. On the other hand, the FTEE results are overlaid on the IW filtered simulation; (a) ΔZS = 2 mm (b) ΔZS = 2 mm Fig. 5 (a) Integral length scale l0 and (b) 2C energy dissipation ε, comparing DNS filtered, 2- Frame and FTEE results. Black, red, blue, green markers represent t8, t16, t24, t32, respectively Energy Spectrum comparison A strong indicator of the resolved scales of the measured fields is given by a spectral analysis of the kinetic energy in the flow. Figure 6 shows the influence of IW size on the PIV energy spectra for the laser sheet thickness ΔZS = 4 mm. The dashed lines show the 2C energy spectrum of the full velocity field, i.e the target spectrum. The vertical lines are the κcut for the corresponding PIV energy spectrum curves for the IW size noted in the legend. The solid black lines overlaying the

8 PIV results are the IW filtered simulation spectra. The deviation of all of the computed spectra at low frequencies shows that the solutions are not yet converged, owing to the singular velocity field considered. Future work will contain averaged results over independent slices from the full resolution case. The first key observation to make here is that the 2-frame PIV case is unable to resolve the flow down to IWs of 8x8 px unlike FTEE which resolves the energy spectrum well to at least its corresponding cut-off frequency κcut. Also, the 2F case tends to have additional noise particularly at the high frequencies. (a) ΔZS = 4 mm (b) ΔZS = 4 mm Fig. 6 Energy spectra of (a) 2-Frame PIV and (b) FTEE PIV at 75% overlap for the laser sheet thickness ΔZS =4 mm. The solid black lines overlaying the PIV results are the IW filtered simulation spectra. The dashed lines show the energy spectrum of the full velocity field. 3. Application to Experimental Measurements In this section, an analysis of experimental data that creates HIT is performed. The test bench is composed of a fully transparent constant volume cell (100x60x60 mm 3 ) connected to a compression volume separated by a perforated block (Lacour et al, 2013). After being introduced into the both volumes at atmospheric pressure, the seeded air is compressed by means of the rapid motion (85 ms total travel time) of a pneumatic piston. The turbulence is generated in a constant volume cell by means of the rapid compression pushing the air through a block perforated with 25 cylindrical channels of 2.5mm in diameter. At the end of the compression step (t0+85 ms), the turbulent flow generated in the constant volume cell during the compression step is in a decay regime with a large range of turbulent scales and a null mean velocity. The TR-PIV setup is composed of a high speed phantom V10 camera and a Quantronix Darwin Duo laser of energy 2x30 mj. The laser light sheet of thickness 1 mm is formed at the median

9 plane of the volume with spherical (f = 1 m) and cylindrical lenses (f = -50 mm). The investigation area is limited to 8.9x8.9 mm 2 with a reduced region of interest (ROI 336x336 px) to ensure a 10 khz acquisition frame rate with the camera. The two laser cavities operate at 10 khz and are both synchronized from the camera internal from rate. The image recording sequence starts at the setting in motion of the piston, t0, and lasts for a duration of 1 s, corresponding to the full decay of turbulence in the chamber. (a) 2-Frame, IW 16x16 px (b) 2-Frame, IW 24x24 px (c) FTEE, IW 16x16 px (d) FTEE, IW 24x24 px Fig. 7 Influence of IW size on the velocity field and Z component of vorticity, comparing 2-Frame and FTEE computation Figure 7 shows four calculated velocity fields at 0.2 s after the experiment initialisation using both 2-Frame and FTEE algorithms. For both images, a vector skip of 4 is applied to aid visualisation. The Z-component of positive and negative vorticity is shown in the contours. As shown in Figure 3, the full resolution synthetic case seems to be noisier owing to the smaller

10 fluctuations that were smoothed by the spatial filtering of the IW windows. Similarly, the 16x16 px IW results here appear to be noisy compared to their larger IW counterparts. We undertake a systematic approach to analysing the flows to ascertain the quality and reliability of these results Turbulence statistic comparison In a similar analysis to that performed in the synthetic case, the IW size is reduced as much as possible, as this enables the resolution of smaller scales in the flow. Figure 8 shows Kt and ε calculated for instantaneous velocity fields in the experimental case. Each coloured symbol represents one velocity field in the sequence of velocity fields separated by dt, in order to provided statistical weight to the results by sampling the large and medium sized persistent structures, while the two columns are separated by 0.1 s, to show the results surrounding two distinct and statistically independent flow events with high and low Kt in the decay. Fig. 8 Influence of IW size on turbulent kinetic energy Kt and 2C energy dissipation ε. Each symbol represents the value for a single velocity field in a sequence, differentiated by colour. (Left Column) 2 s after acquisition time, (Right Column) 3 s after acquisition time.

11 As per the synthetic data, for all cases, the 2-Frame PIV is well resolved and thus the turbulence data obtained are essentially indistinguishable. Reducing the IW size, the FTEE algorithm performs similarly to the results obtained in the synthetic data: The Kt gradually increases owing to a smaller averaging kernel and likewise for the ε, owing to better resolution. This behaviour indicates that the FTEE algorithm is able to resolve the flow correctly at the smallest IW size of 16x16 px. On the other hand, the 2-frame PIV algorithm introduces significant measurement noise at the smallest IW, owing to reduced correlation quality Energy Spectrum comparison Figure 9 shows instantaneous energy spectra at times corresponding to the data presented in Figure 8, i.e. at 0.2 s and 0.3 s after experiment initialisation. Only the two smallest IW sizes are shown, and the theoretical frequency cut-offs are represented by the solid and dotted vertical lines for the 24x24 px and 16x16 px, respectively. For the 24x24 px IW, the 2-F follows the FTEE spectrum reasonable well with a slight increase in high frequency noise. For the 16x16 px IW, the FTEE curve follows a straight trajectory at least until the theoretical cut-off, whereas the 2-Frame PIV at 0.2 s peels off the spectrum very early and at 0.3 s exhibits significant noise. The decay of the low frequency can be seen comparing the two instants in time, i.e. the level of Kt that was also shown in Figure 8. All methods resolve the largest scale structures identically. (a) (b) Fig. 9 Energy spectra of (a) 2-Frame PIV and (b) FTEE PIV at 75% overlap for the laser sheet thickness ΔZS = 4 mm. The solid black lines overlaying the PIV results are the IW filtered simulation spectra. The dashed lines show the energy spectrum of the full velocity field.

12 4. Conclusion and Outlook Homogeneous isotropic turbulence provides a challenging data set to test the limits of algorithm performance owing to the large range of velocity and length scales. In this article we used synthetic HIT to displace 3D particle fields to test 2D PIV algorithms. We tested an optimised 2- Frame algorithm with deforming window offset and Gaussian weighting window with a similarly configured time resolved PIV technique, namely FTEE. A main objective of this article was to explore the improvement to the energy spectrum and retrieval of turbulence quantities by these measurements. The two PIV algorithms were compared with the full resolution 3D velocity field, while to obtain a perfect data set to compare the signal cut off, the full resolution field was filtered with equivalent sized IWs and laser profile. FTEE was firstly shown to obtain more accurate turbulence quantities than the 2-Frame. Importantly, we were able to reduce the IW size more with FTEE, and thus resolve smaller turbulence scales, as the ensemble averaging of the crosscorrelation peak provides more robust measurements despite less than optimal number of particles in the IW. The laser sheet thickness was also varied to investigate the compromise between minimising loss of pairs due to particles leaving a thin laser sheet or signal pollution owing to integrating 3D motions along a thick laser sheet. In both cases, FTEE showed superior performance. The thinner light sheets can be explained by the fact that if particle pairs are lost, the correlation maps at the ends of the trajectory do not contribute to the trajectory calculation, which relies on centred and sampled, sequential cross-correlation maps. Finally, an experimental measurement was described and systematic analysis following that of the synthetic case was undertaken. Combining an analysis of the turbulence quantities and the energy spectrum, and comparing the tendencies with the synthetic cases, lead to the conclusion that the FTEE was once again able to resolve smaller scales in the flow. Besides the advantage of being able to simply calculate the trajectory acceleration, time resolved PIV has been demonstrated to increase measurement dynamic and resolve smaller flow structures by employing smaller windows, made possible through the ensemble averaging of the technique. Future work will look at extending this to a super-resolution approach, where ensemble correlations on single particles could enable a descent even further along the energy spectrum curve into high frequency spatial measurements.

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