'Postage-stamp PIV': small velocity fields at 400 kHz for turbulence spectra measurements

The development of postage-stamp PIV has been performed using a supersonic jet exhausting into a transonic crossflow, a flow that has been studied thoroughly employing both conventional [28–30] and pulse-burst PIV [10, 15]. The present experimental configuration is identical. This work was conducted in Sandia's TWT, which is a blowdown-to-atmosphere facility using air as the test gas. In its solid-wall transonic configuration, the test section is a rectangular duct of dimensions 305  ×  305 mm² (12  ×  12 inch²). Mach 0.8 was tested exclusively for the data used herein at a fixed stagnation pressure of 154 kPa and stagnation temperature of 321 K  ±  2 K.

The jet-in-crossflow experiment was configured with a supersonic jet installed on one wall of the transonic test section upstream of its windows. The upstream location of the jet positioned the imaging region for measurement of the far-field of the jet interaction once it has developed, as shown in figure 1(a). The laser sheet was oriented in the streamwise plane and aligned to the center plane of the test section, which coincides with the center of the jet nozzle exit. A nitrogen jet exhausted from a conical nozzle with a design Mach number of 3.73, an expansion half-angle of 15°, and an exit diameter of 9.53 mm. In the present case, the jet was operated at a pressure of 3.9 MPa to produce a jet-to-freestream dynamic pressure ratio of J  =  8.1. The coordinate axes originate at the centerpoint of the nozzle exit plane; the u component is in the streamwise direction and the v component is positive away from the tunnel wall.

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The TWT is seeded by a thermal smoke generator (Corona Vi-Count 5000) whose output is delivered to the TWT's stagnation chamber upstream of the flow conditioning section through a series of pipes and tubes, in which agglomeration of the particles occurs. Previous measurement of the in situ particle response across a shock wave generated by a wedge reveal a frequency response of 500–700 kHz, corresponding to a particle size of 0.7–0.8 µm when sphericity is assumed. Stokes numbers have been estimated as at most 0.05 based on a posteriori analysis of PIV measurements, which by conventional analysis is sufficiently small to render particle lag errors negligible [31, 32].

However, TR-PIV deserves a more careful analysis as errors may concentrate towards high frequencies and impact specific spectral content even if the integrated response is adequate. A common if arbitrary descriptor of frequency response is to define the point at which the energy of the velocity fluctuations is reduced by a factor of two. Melling's analysis [32] predicts this will occur between 100–200 kHz for the present particle diameter and Mei [33] provides a more refined range of 160–200 kHz. As will be shown subsequently, under the best circumstances in the present experiments the noise floor emerges at about 120 kHz and therefore particle lag effects become significant only at frequencies beyond those measurable.

A burst-mode laser (QuasiModo-1000, Spectral Energies, LLC) with both diode- and flashlamp-pumped Nd:YAG amplifiers was used to produce a high-energy pulse train at 532 nm. The pulse-burst laser generates a burst of maximum duration 10.2 ms once every 8–12 s with a maximum repetition rate of 500 kHz. For postage-stamp PIV, it was operated to deliver single pulses at 400 kHz with an energy of about 8 mJ per pulse at 532 nm, creating a time between successive pulses of 2.5 µs. The laser sheet was narrowed to a slender width to concentrate the available energy in the small field of view and had a sheet thickness of 1 mm. A total of about 170 bursts of data were acquired.

Images were acquired using high-speed CMOS cameras (Photron SA-Z), which have an array of 1024  ×  1024 pixels at a full framing rate of 20 kHz. The SA-Z cameras were operated at 400 kHz, which necessitates that they are windowed down to a small array of only 128  ×  120 pixels; hence the use of the term 'postage-stamp PIV'.

Three SA-Z cameras were used to acquire two independent measurements. One camera was aligned for a conventional two-component measurement. The other two cameras combined for a stereoscopic measurement with each camera oriented at a half-angle of 20°. The cameras each were equipped with 200 mm focal length lenses and anti-peak-locking diffuser plates [34]. Scheimpflug mounts were used to ensure focus of the two oblique images. The two-component configuration provides a safe measurement without possibility of bias due to an inadequate calibration using such a tiny imaging array, but a successful stereo measurement offers three-component data for evaluation of the turbulent kinetic energy spectra.

Data were processed using LaVision's DaVis v8.3.1. In each case, image pairs were background corrected, intensity normalized, and then interrogated with an initial pass using 64  ×  64 pixel interrogation windows, followed by two iterations of Gaussian-weighted 24  ×  24 pixel interrogation windows. A 50% overlap in the interrogation windows was used as well. The resulting vector fields were validated based upon signal-to-noise ratio, nearest-neighbor comparisons, allowable velocity range, and a median filter in both space and time. Close attention to the seeding stability yielded a valid vector rate in excess of 99%, which is particularly important for measurement of high-frequency spectral content. Low valid-vector rates degrade high-frequency accuracy and are not well corrected by vector interpolation.

Much of the data analysis concerns power spectral densities (PSDs) of the velocity fluctuation energy. A standard Welch periodogram fast Fourier transform with a Hamming window was used, except that spectra were computed on each burst of data without further windowing and then all bursts were averaged together. The frequency resolution was 100 Hz limited by the burst length of 10.2 ms.

A customized single-plane calibration plate was used to calibrate all three cameras for the two PIV measurements. The target was an aluminum plate 3.05 mm thick machined such that each surface was very flat and parallel, drilled with normal fiduciary holes every 1.59 mm. The target was carefully aligned to known coordinates then translated through the volume of the laser sheet. Seven calibration stations were imaged by each camera and calibration was accomplished using a third-degree polynomial fit by the DaVis software. The two-component measurement was calibrated simply by determining a scale factor.

A unique challenge arises when performing a stereo calibration on an image only 128  ×  120 pixels in extent. In the field of view for postage-stamp PIV, only a 5  ×  4 matrix of fiduciary dots is visible, which limits the precision of the calibration. In contrast, the field of view of the full 1024  ×  1024 array of the camera captures the entire calibration target and can fit through many more points better differentiated in space. A stereo calibration using the full array and all fiduciary dots could be applied to the postage-stamp images formed by only the central 128  ×  120 pixels. However, this seemingly superior stereo calibration did not produce significantly different spectral data than the limited calibration based upon the 5  ×  4 array of dots.

Self-calibration is similarly a challenge on such a small field of view. Correlations for self-calibration typically must be applied on much larger interrogation windows than velocity vector interrogation, which exceed the size of the postage-stamp array. A sufficient array of disparity vectors could not be produced to accurately correct rotation but even a single disparity vector can correct translation. Stereo calibrations adjusted by incorporating self-calibration estimates did not yield significantly different spectra. Therefore, the resulting velocity data appear robust to the quality of the stereo calibration, at least for spectral content.

The power spectra of the velocity fluctuations that are measured by the two-component postage-stamp PIV are shown in figure 3. Two PSD plots are shown averaged from all 170 bursts, one for the streamwise component of velocity fluctuations and the other for the vertical component. Also given in figure 3 are the velocity spectra obtained by supersampling the 25 kHz PIV data from Beresh et al [15]. A supersampling factor of 16 was used to reflect the number of vectors between successive vector fields, reducing the time between snapshots to 2.5 µs, which still exceeds the frequency response limitation of the 2.0 µs time between paired laser pulses. Slopes for  −1 and  −5/3 power law dependencies are provided as well.

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In the present case, the properties of the jet interaction change very little within the far-downstream field of view and it suffices to examine the spectra at a single point near the center of the field of view where no edge effects interfere with computing correlations as particles may exit the field of view. Though a single vector location is used, an adequate spatial extent must be provided to both capture the convection of particles between exposures and to provide a sufficient spatial description of the velocity gradients to serve the image deformation step within the interrogation process. Despite the slowly evolving flow, spectra are not averaged over adjacent vectors because PIV uncertainty correlates spatially due to common flow features and therefore neighboring vectors are not statistically independent (e.g. [35]).

For both velocity components in figure 3, the postage-stamp results well match the supersampled data to nearly the end of their range, at which differing noise floors account for their deviation. The noise floor begins to show at about 150 kHz for the supersampled data, but emerges at about 120 kHz for the postage-stamp PIV and has a greater value. The most consequential of the spectra is the vertical component in figure 3(b). The spectra peak at 4 kHz, which corresponds to one full period between pairs of turbulent eddies convecting through the flow [15]. Following the peak, the spectra then exhibit a  −1 slope dependence to about 60–70 kHz before gradually transitioning to an apparent  −5/3 slope dependence. The presence of the  −5/3 region is well-known and expected as a consequence of the inertial subrange of turbulence decay (e.g. Pope [1]). The  −1 dependence, on the other hand, is an unanticipated discovery and extends over more than one full decade, from approximately 4–60 kHz.

The streamwise component in figure 3(a) also appears to be supportive of a  −1 slope dependence but its lack of a peak means this region does not initiate until close to 10 kHz. The  −5/3 slope dependence at high frequencies is somewhat less convincing for the streamwise component than for the vertical component. This is because the overall intensity of the streamwise velocity fluctuations is reduced compared to the vertical velocity fluctuations, and therefore the impact of the noise floor is felt sooner.

The close agreement of the postage-stamp PIV data with the supersampled data from the original 25 kHz pulse-burst PIV data validates the efficacy of Scarano and Moore's supersampling algorithm [23], at least for the jet-in-crossflow experiment. This flowfield is well-suited for use of supersampling because in the far-field where measurements are acquired, the flow is dominated by convective motion nearly aligned with the freestream direction. Therefore, the out-of-plane motion that cannot be accounted by the supersampling algorithm in a planar measurement is minimal. Out-of-plane velocity fluctuations are small as identified by a spanwise turbulence intensity $\sqrt{\overline{{{w}^{\prime 2}}}}/{{U}_{\infty}}$ less than 0.1 [30]. Unlike supersampling, spectra obtained using postage-stamp PIV are not subject to assumptions regarding the convection of frozen turbulence or the absence of out-of-plane data in a planar measurement, and therefore postage-stamp PIV is suitable for a much greater range of applications. Still, the agreement between the two techniques supports the veracity of the postage-stamp PIV approach.

The frequency response of the postage-stamp PIV is coupled to the size of the interrogation window. The vector spatial resolution is about 1.2 mm based simply on the size of the final interrogation window (neglecting Gaussian weighting), which corresponds to a frequency of 220 kHz at the convection velocity. This exceeds the maximum possible frequency response of the 400 kHz data rate, given Nyquist considerations. In comparison, the integral length scale of the flow at the present location is about 40 mm [29] and inertial subrange scales initiate at about 7 mm [1], indicating that the present measurements capture the range of dominant turbulence eddies in the flow. Still, this covers only a portion of the turbulence spectrum as the Kolmogorov scale can be estimated on the order of 0.01 mm, and therefore only onset of dissipation scales can be expected to be measured.

Although a smaller final interrogation window nominally will improve the spatial resolution and the frequency response associated with it, the noise floor will increase due to the fewer particle pairs in the correlation, and this counteracts the spatial benefit. Conversely, a larger interrogation window reduces the bandwidth of the measurement and restricts the frequency response below that needed to observe the initiation of the inertial subrange. The 24  ×  24 pixel windows were found to be an optimal value for these competing considerations.

However, translating the interrogation window size into a frequency response limit is not straightforward due to its sensitivity to numerous details of the interrogation algorithm that couple with one another [36, 37]. Moreover, the modulation is gradual, typically a sinc function, rather than a definitive threshold (e.g. [37, 38]); therefore some loss of amplitude will occur over a range of frequencies near the cutoff. Nogueira et al [37] suggest that with 50% overlap or more between interrogation windows in multi-iteration analysis and a particle displacement not exceeding half the window size, the minimum detectable fluctuation wavelength is on the scale of the final interrogation window. This indicates that the frequency cutoff due to the spatial resolution of a vector is indeed around 220 kHz, which exceeds the point in the spectra at which the noise floor becomes dominant. Therefore, the spatial resolution of the PIV is not the dominant limitation on the frequency response for the present parameters.

An additional concern is that aliasing might erroneously shift energy from unmeasurably high frequencies to contaminate lower ranges. However, Gamba and Clemens [39] determined that the spatial averaging of the PIV interrogation window serves as an effective anti-aliasing filter, and indeed the impact on the spectrum is dominated by the spatial limitations with little energy redistribution due to aliasing. Given that the current PIV spatial resolution is not the limiting factor for the measurement frequency response, aliasing effects are expected to be negligible. This conclusion is supported by the close agreement with the supersampled data before the noise floor arises, since Scarano and Moore [23] report that aliasing is well mitigated by the supersampling algorithm.

It is important to note that peak locking can have a noticeable impact on the noise found in the velocity spectra. Figure 4 replicates the data from figure 3 that were acquired using an anti-peak-locking diffusion filter behind the camera lenses, denoted here as 'anti peak locking'. It also includes data acquired without the filter, named 'higher peak locking'. Figure 4 indicates that the noise floor in the spectra increases considerably when peak locking is more prevalent. In this case, the slope in the inertial subrange also can be seen to decrease as the noise floor is approached, which is due to additive noise from peak locking affecting turbulent motions at these high frequencies. The increased thickness of the spectral line for the peak-locked data is simply a consequence of having acquired many more bursts of data for the anti-peak-locked measurements, and hence improved statistical convergence in the latter case. The improvement in peak locking is verified using probability density functions of the vertical velocity component in figure 5. High levels of peak locking in the no-filter cases are apparent from the split peaks; conversely, hardly any peak locking is evident in data generated using the diffusion filters. Figure 5 confirms that the diffusion filters successfully remove peak locking, and that this correlates with the reduction in the noise floor in the spectra.

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A further reduction in the noise floor requires an improvement in the data processing accuracy of TR-PIV. One means to accomplish this is by utilizing multiple-frame image interrogation rather than the image-pair interrogation of conventional PIV [40–46]. The pyramid correlation of Sciacchitano et al [41] probably is the most appropriate, in which an ensemble-averaged correlation is generated based on more than the usual two image pairs to improve correlation precision. Correlations are created between adjacent time steps as well as longer intervals within a maximum range such that an optimization may be achieved between dynamic range and signal-to-noise ratio. Here, the pyramid correlation was tested with n  =  ±2 images and n  =  ±3 images and the results are included in figure 4. The comparison is made against the 'higher peak locking' data rather than the more recent data employing the diffusion filters.

For both velocity components, use of the pyramid correlation distinctly reduces the noise floor of the spectra by about one-half an order of magnitude. However, the spectra begin to roll off at much lower frequencies. This is attributable to a loss in signal bandwidth due to the use of multiple images per correlation, which corresponds to a longer period of time and therefore lower frequency response. The minimal difference between n  =  2 and n  =  3 likely is a result of the algorithm attempting to optimize the data processing with smaller image displacement regardless of the length of the available image sequence. Figure 4 demonstrates that the pyramid correlation is not well suited to data sets such as the present one because it is not temporally oversampled. The use of additional time steps cannot be achieved without compromising the frequency response. When frequency content is desired very close to the camera framing rate, the pyramid correlation is unhelpful.

Improvements in TR-PIV data accuracy also may be found by incorporating a noise-reduction algorithm into post-processing [39, 47, 48], but current methods are unsuitable to the present case. This matter is discussed in section 4 subsequently.

The stereoscopic postage-stamp PIV measurements offer turbulent kinetic energy spectra rather than the 1D spectra examined in figure 3. This is important because turbulent theory typically is based upon 3D fluctuations rather than 1D. Prior to calculating these three-component spectra, figure 6 shows the spectra of the streamwise and vertical velocity components extracted from the stereo measurements and compares them to the two-component measurements from figure 3.

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The comparison between the two-component spectra and the stereo spectra is disappointing. Although the noise floor is similar in both cases, the power-law slope is different in the inertial subrange for the stereo data as compared to the two-component measurement, and its onset is earlier. This difference in slope in the inertial subrange is attributable to poorer spatial resolution in the stereo data because of the oblique camera images and the registration of two cameras, which increases the practical size of each interrogation window and thereby lowers the associated frequency response. For the present 24  ×  24 pixel interrogation windows, the spatial extent of a single vector increases from 1.2 mm in the two-component data to 1.4 mm in the stereoscopic data, with the caveat that the true spatial resolution of a PIV vector is somewhat ambiguous owing to the repeated warping and filtering of the images [36] and by the oblique view of a finite-thickness laser sheet [49]. The larger spatial extent attenuates the signal at high frequencies in comparison to the two-component measurement. The effect appears stronger for the vertical component because the magnitude of these velocity fluctuations is greater. At low to mid-range frequencies, the agreement between the two-component and stereo data is excellent.

In figure 7, 1D spectra for each of the three components of the stereo measurement are shown, as is a single 3D spectrum derived from the total velocity fluctuation, ${{k}^{\prime}}=\sqrt{{{u}^{^{\prime}2}}+{{v}^{^{\prime}2}}+{{w}^{^{\prime}2}}}$. Again, the stereo results are disappointing. The spectrum of the out-of-plane component, w, exhibits a considerably higher noise floor than the two in-plane components, u and v. Moreover, values are elevated at mid-range frequencies as well, which is likely inaccurate as w fluctuations should not be more intense than u or v fluctuations in this frequency band alone. The greater noise in w will, of course, propagate into the spectrum of k, and this is evident simply comparing the relative noise floors.

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The elevated noise in the w component of the stereo measurements can be attributed to the propagation of precision uncertainties from the two constituent correlations into in-plane and out-of-plane components. Classical analyses predict that for the present camera angles, the precision uncertainty in the out-of-plane component w is approximately 2–3 times greater than the in-plane components u and v [50, 51]. A recent, more sophisticated analysis is agreeable [52]. These estimates would be consistent with the noise floor differences in figure 7, for which the w floor has magnitude of 0.01 and the u and v floor is 0.004, or a ratio of 2.5. Uncertainty estimation drawn from the actual postage-stamp PIV images using the correlation statistics method [35] is given in figure 8. The uncertainty magnitudes are very similar for the two in-plane components despite mildly differing distributions, but the uncertainty in w is clearly much larger. Mean uncertainty values are 2.6 and 2.4 m s⁻¹ respectively for u and v but 6.1 m s⁻¹ for w, confirming that the uncertainty in the w component is consistently about 2.4 times greater than that in the u or v components. This corroborates that the elevated noise floor of the w component is simply a reflection of the expected division of precision uncertainty into the three velocity components.

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The intensity of the w spectrum remains elevated above the v spectrum in the range of 30–100 kHz by about 0.010 to 0.018, which suggests that this effect constitutes additive noise of a magnitude consistent with the noise floor. The v and w components should be most similar because both are induced by the counter-rotating vortex pair. Taken together, these observations imply that the aberrant nature of the w component is a result of increased correlation noise in the out-of-plane component of stereo PIV.

The clear presence of increased noise in the w component and its apparent distortion of the spectrum suggests that the same effect may be plausibly found even for the lower noise levels in u and v, though to a lesser extent. These effects are likely to become significant at frequencies greater than the approximately 30 kHz at which the w component begins to express deviance, but it is unclear how much greater. Therefore, despite the additional challenges in high-speed flows of discriminating noise from signal, a denoising algorithm is needed.