Accurate turbulence level estimations using PIV/PTV
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Abstract
The estimation of the turbulence level of flow facilities is very important for the comprehensive description of experimental results. While for low flow velocities various measurement techniques can be used (for example hotwire, LDV, PIV) the task becomes difficult in the case of compressible flows as temperature and density fluctuations bias the measurement of the velocity fluctuations. In this work, we analyze the freestream flow of the trisonic wind tunnel Munich (TWM) by means of particle image velocimetry (PIV) and particle tracking velocimetry (PTV). The goal is to determine the flow quality, i.e. the turbulence level, over the operating range of the facility without bias due to temperature and density variations. The capability of PIV/PTV for the estimation of small velocity fluctuations is investigated in detail. It is shown that a small particle shift on the measurement plane in combination with a large particle image displacement on the image plane allows for precise velocity measurements. Furthermore, a variation of the time separation between the PIV double images, \(\varDelta t\), enables the measurement uncertainty to be determined, which was estimated to be as low as \(0.04\%\) of the mean displacement for a mean displacement of \(\varDelta x=100\,\) pixel and an interrogation window size of \(32\times 32\,\) pixel. Regarding the wind tunnel turbulence, it was found that the turbulence level generally decreases with increasing Mach number for the TWM facility, starting with \(1.9\%\) at \(Ma=0.3\) and reaching \(0.45\%\) at \(Ma=3.0\). With this analysis, a methodology exists to perform accurate turbulence measurements in incompressible and compressible flows.
Graphical abstract
1 Introduction
Accurate turbulence measurements require a measurement system that has an uncertainty that is well below the velocity fluctuations of interest. Additionally, the measurement volume that each velocity vector is estimated from must be small compared to the size of the smallest turbulent structures. Furthermore, the number of samples N must be sufficiently large to ensure that the average velocity is fully converged. Hotwire probes are well suited for precise velocity measurements with high temporal resolution (Alfredsson et al. 1988; Hutchins et al. 2009; Hultmark et al. 2013). However, hotwires have several drawbacks. To mention a few, they are intrusive measurement methods, results can be biased if other velocity components also have large fluctuations, temperature fluctuations influence the results, and extracting the velocity from the measured signal requires knowledge about the flow density. Furthermore, at supersonic Mach numbers, the shock waves generated by the probe will influence the measurement. Therefore, the application of hotwire probes is usually limited to incompressible flow. A nonintrusive alternative measurement technique is the LaserDoppler velocimetry (LDV) (George and Lumley 1973; Shirai et al. 2006). Due to its working principle LDV does not require knowledge about the flow density and is, therefore, suited for measurements in compressible flows even in the case of temperature fluctuations. Hotwire probe and LDV are measurement techniques with the capability of high temporal resolution, however, they only provide pointwise information about the flow. To analyze the spatial organization of turbulent structures planar or volumetric measurement techniques are needed. Particle image velocimetry (PIV) provides such spatial information which corresponds up to MHz sampling locally taking the convection velocity into account. However, PIV is known to have higher measurement uncertainties than hotwire probes and LDV, in general (Wilson and Smith 2013; Neal et al. 2015; Timmins et al. 2012). Although systematic errors would not affect the estimation of Tu, according to Eq. (1), it is important to quantify random errors, since they increase the estimated velocity fluctuations. This is discussed in detail in Sect. 3.5.
It was already shown in the past that timeresolved sequences of PIV and particle tracking velocimetry (PTV) recordings allow for improved measurement uncertainty (Hain and Kähler 2007; Schröder et al. 2015; Cierpka et al. 2013; Sciacchitano et al. 2012; Willert et al. 2018; Scarano and Moore 2012). However, timeresolved measurements are limited to flow velocities around 10 m/s, for a recording rate in the low kHz range. In order to achieve several 100’s of kHz, as needed for a detailed analysis of supersonic flows, the number of images and/or the image size is usually rather limited (Wernet and Opalski 2004; Beresh et al. 2017). Furthermore, reliably estimating flow statistics from timeresolved measurements requires very long sequences leading to an amount of data that is challenging to handle and to evaluate.
To perform reliable turbulence level estimations by means of PIV/PTV, the measurement setup and the evaluation procedure must be selected carefully to minimize the measurement uncertainty. In the following the test facility, the measurement setup and the data evaluation are described in detail. Further on, the effect of the particle image displacement and the interrogation window size on the estimated turbulence level is evaluated. Finally, conclusions are drawn from the presented results.
2 Measurement setup
For the PIV measurements, the flow was seeded with DiEthylHexylSebacat (DEHS) tracer particles with a mean diameter of \(1\,{\upmu \mathrm{{m}}}\), as described by Kähler et al. (2002). The size of the tracers is a compromise between visibility and the ability to follow the flow. The response time of these droplets is about \(2\,{\upmu \mathrm{{s}}}\) (Ragni et al. 2011), which is considered to be sufficient for investigations at the selected Mach and Reynolds numbers since no strong turbulent fluctuations are to be expected in the freestream. The tracers were illuminated by a \(500\,{\upmu \mathrm{{m}}}\) thick light sheet in a horizontal plane at the center of the test section. The scattered light of the tracer particles was imaged onto the sensor of a sCMOS camera with double image capability and global shutter by means of a 100 mm lens in combined with a \(2\times\) teleconverter. The fnumber of the lens aperture was set to 8 in order to reduce aberrations and to ensure sufficiently large particle images. Double images were recorded at a frequency of \(15\;\) Hz. The resulting size of the field of view is \(26\;\) mm \(\;\times \,22\;\) mm and the resulting scaling factor is \(10.4\,{\upmu }\)m/pixel. With this PIV setup, a mean particle image diameter of about 3.4 pixel was achieved. From the autocorrelation function of the images a background noise level with a standard deviation of about 50 counts was estimated with the method presented in Scharnowski and Kähler (2016b). This noise level leads to a lossofcorrelation due to image noise of \(F_{\sigma }\approx 0.91\) and to a signaltonoise ratio of SNR \(\approx 3.2\), which is considered to be well suited for accurate PIV evaluation, according to Scharnowski and Kähler (2016b). Figure 3 shows a small section of a typical PIV image pair and the corresponding correlation function. It can be seen from the image that most particle images can be paired although the mean displacement was set to \(\langle \varDelta x\rangle =100\,\) pixel. Consequently, a correlation function with a well detectable peak was computed.
3 Results and discussions
3.1 Wind tunnel stability
3.2 Effect of number of recordings
3.3 Effect of \(\varDelta t\) and window size
To analyze the effect of the time between the double images \(\varDelta t\) and the squared interrogation window size \(D_\mathrm {I}\) on the estimated turbulence level, these parameters were varied for a Mach number of \(Ma=0.3\) and a total pressure of \(p_0 = 1.5\;\) bar. Based on the findings of Sect. 3.2, 1000 double images were recorded for nine different \(\varDelta t\) between \(0.2\,{\upmu \mathrm{{s}}}\) and \(20\,{\upmu \mathrm{{s}}}\), corresponding to a mean particle image shift between \(\varDelta x\approx 2\,\) pixel and \(200\,\) pixel. The PIV recordings were evaluated using stateoftheart PIV software from LaVision GmbH (DaVis 8.3) including multipass image deformation and Gaussian window weighting. The final window size was varied between \(12\times 12\) pixel and \(64\times 64\) pixel. For a window size of 12 or 16 pixel no window overlap was applied, while for larger \(D_\mathrm {I}\) a window overlap of \(50\%\)was used.
The results for the variation of the particle image shift \(\varDelta x\) and different interrogation window sizes \(D_\text {I}\) are illustrated in Fig. 6 by means of example velocity fields. For each column in the figure the same PIV recordings were evaluated. It can be seen from the top left image in the figure, that with decreasing displacements and decreasing interrogation window size the noise is amplified. It is obvious that under these conditions the measurement uncertainty is too high to reliably estimate the velocity fluctuations. For increasing \(\varDelta x\) as well as for increasing \(D_\mathrm {I}\) the vector fields appear much smoother and are, therefore, better suited for turbulence level estimations. The fields are less noisy because the uncertainty of the estimated velocity is reduced by the decreased absolute uncertainty of the particle image displacement \(\sigma _{\Delta {\mathrm x}}\) (in the case of larger \(D_\mathrm {I}\)) or by decreased relative shift vector uncertainty due to enlarged mean displacement \(\langle \varDelta x\rangle\). However, care must be taken when increasing \(\varDelta x\) or \(D_\mathrm {I}\) for two reasons. First, the larger the particle image displacement becomes the higher is the risk of introducing bias errors due to curved streamlines, which are not captured in doublepulse PIV experiments (Scharnowski and Kähler 2013). These bias errors would also affect the Tu estimation, since they vary within the flow field. Second, larger interrogation windows cause increased spatial lowpass filtering of the velocity field, leading to smoothed results which might filter out smallscale turbulent structures (Scharnowski et al. 2012). To minimize these two effects, a small field of view was selected such that the interrogation window sizes as well as the particle image displacement are small when projected on the measurement plane.
3.4 Peak locking
One important error source that must be considered for PIV and PTV measurements is the socalled peaklocking effect (Raffel et al. 2018). Peak locking is a bias error that shifts the estimated displacement values towards the closest integer pixel positions and is caused by very small particle images. In the case of peak locking, the probability density function of the displacement shows peaks at integer pixel values (Kähler 1997; Christensen 2004; Overmars et al. 2010; Raffel et al. 2018). As a result, the velocity is overestimated or underestimated depending on the subpixel length of the shift vector on the image plane.
To evaluate the significance of the peaklocking effect for a specific measurement, it was demonstrated in Nerger et al. (2003) that velocity histograms for measurements with different \(\varDelta t\) differ strongly in the case of peak locking. Only if peak locking is negligible the velocity histograms appear similar as only the random error varies. For reliable velocity measurements it is important to avoid peak locking during the data acquisition using appropriate optical setups (Michaelis et al. 2016; Kislaya and Sciacchitano 2018), short observation distances or cameras with small pixel size.
3.5 Turbulence level estimation
Since several value pairs are available for the mean shift vector and the measured fluctuations, the unknown Tu and \(\sigma _{\varDelta \mathrm {x}}\) can be estimated from a fit function, according to Eq. (4). The results for the different \(\varDelta x\) and \(D_\text {I}\) are shown in Fig. 9. For comparison PTV results are also shown.
The symbols in the figure represent the spatial mean of the temporal standard deviation and the error bars correspond to the spatial variations within the field of view. The dashed lines indicate the distribution of a fitfunction using Eq. (4). Values for the fitparameters Tu and \(\sigma _{\varDelta \text {x}}\) are given in the figure. As expected, the estimated turbulence level increases with decreasing interrogation window size indicating that some of the turbulent structures are smaller than the interrogation windows. However, since the changes in the Tu values are very small it can be concluded that most structures are larger than 64 pixel (\(=670\,{\upmu \mathrm{{m}}}\)). Furthermore, the PTV results, which do not suffer from spatial lowpass filtering, confirm the PIV results of the smallest tested interrogation windows.
For the particle tracking velocimetry (PTV) evaluation, the 2D locations of the particle images was identified by applying 2D Gaussian fitting functions to the particle images, rejecting particle images that exceed a defined intensity threshold or min/max pixel size. The tracking of corresponding particle image pairs was performed by a noniterative double frame particle tracking approach (Fuchs et al. 2017). Due to the high seeding concentration which was optimized for PIV, overlapping particle images appear frequently. This has a strong effect on the measurement uncertainty (Kähler et al. 2012). Furthermore, it has to be taken into account that PIV requires just one peak center estimation while PTV requires two, which raises the uncertainty in this PTV evaluation.
In the last years several approaches for determine the uncertainty of PIV measurements were developed (Xue et al. 2015; Wieneke 2015; Timmins et al. 2012; Sciacchitano et al. 2013, 2015; Sciacchitano and Wieneke 2016; Neal et al. 2015; Wilson and Smith 2013; Charonko and Vlachos 2013; Christensen and Scarano 2015). If the uncertainty of the PIV velocity fields is known a variation of \(\varDelta t\) is not necessary for the estimation of Tu, in principle. However, the different approaches seem to have varying sensitivities and result in different uncertainty bounds. A variation of \(\varDelta t\) is more complex than a single measurement but the present analysis indicates that this is a reliable way.
3.6 Wind tunnel characterization
Based on the presented sensitivity analysis, the parameters for a reliable estimation of the wind tunnel turbulence level over the full range of possible Mach numbers and Reynolds numbers were selected. To minimize the wind tunnel run time, only one run with 500 densely seeded image pairs and a particle image shift of \(\langle \varDelta x\rangle \approx 100\,\) pixel was performed for each wind tunnel condition. According to the findings in Fig. 9, a direct estimation of the turbulence level from only one \(\varDelta x\) is possible for such a large mean particle image shift. The PIV images were evaluated with a final interrogation window size of \(32\times 32\,\) pixel. Due to the slightly higher measurement uncertainty of PTV at densely seeded flows it was decided to use PIV for further evaluation. Alternatively, lower seeded images could be evaluated with PTV but to reach the same statistical convergence of the higher seeded flow more wind tunnel blow downs (and refills) would be needed.
3.7 Spectral analysis
The velocity fields measured with PIV suffer from poor temporal resolution, since the acquisition rate was only 15 Hz, however, the small field of view and the relatively large camera sensor results in a high spatial resolution. The spatial information of the PIV results can be used to perform spectral analysis in time by applying Taylor’s hypothesis of frozen turbulence (Taylor 1938). This is done by multiplying the spatial frequencies with the mean velocity.
4 Summary and conclusions
The turbulence level of the trisonic wind tunnel Munich was determined to be \(\mathrm{{Tu}} = 1.9\%\) for a Mach number of \(Ma=0.3\) and it decreases with increasing Mach number to \(\mathrm{{Tu}}= 0.45\%\) for \(Ma = 3.0\).

The PIV interrogation window size \(D_\text {I}\) projected on the measurement plane must be small compared to the dominant turbulent structures. Otherwise the application of PTV is recommended.

The particle image displacement \(\varDelta x\) must be large on the image plane to reduce the relative uncertainty.

In contrast, the particle displacement on the measurement plane must be small to avoid bias errors due to velocity gradients. Which means the magnification must be sufficiently large.

A variation of the particle image displacement \(\varDelta x\) allows to determine the measurement uncertainty.

A variation of the interrogation window size \(D_\text {I}\) allows bias errors due to spatial filtering to be detected.
To avoid timeconsuming and expensive variations of \(\varDelta t\) the turbulent structures must be well resolved, such that a large displacement on the image plane as well as a large interrogation window can be used. Thus, for accurate PIV/PTV measurements it is recommended to use camera sensors with large number of pixels or multiple camera approaches which allow for high resolution and large dynamic spatial range at the same time. Finally, a variation of the interrogation window size is always recommended to achieve confidence on the estimated results.
Notes
Acknowledgements
This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures funded by the German Research Foundation (Deutsche Forschungsgemeinschaft—DFG) under the project number KA1808/211.
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