On the uncertainty of digital PIV and PTV near walls
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Abstract
The reliable measurement of mean flow properties near walls and interfaces between different fluids or fluid and gas phases is a very important task, as well as a challenging problem, in many fields of science and technology. Due to the decreasing concentration of tracer particles and the strong flow gradients, these velocity measurements are usually biased. To investigate the reason and the effect of the bias errors systematically, a detailed theoretical analysis was performed using windowcorrelation, singepixel ensemblecorrelation and particle tracking evaluation methods. The different findings were validated experimentally for microscopic, longrange microscopic and large field imaging conditions. It is shown that for constant flow gradients and homogeneous particle image density, the bias errors are usually averaged out. This legitimates the use of these techniques far away from walls or interfaces. However, for inhomogeneous seeding and/or nonconstant flow gradients, only PTV image analysis techniques give reliable results. This implies that for wall distances below half an interrogation window dimension, the singepixel ensemblecorrelation or PTV evaluation should always be applied. For distances smaller than the particle image diameter, only PTV yields reliable results.
Keywords
Particle Image Bias Error Interrogation Window Particle Tracking Velocimetry Digital Particle Image Velocimetry1 Introduction
Digital particle image velocimetry (DPIV) has become one of the most widespread techniques for the investigation of flows because it allows for the instantaneous measurement of the flow field in a plane or volume without disturbing the flow or fluid properties (Adrian and Westerweel 2010; Raffel et al. 2007). Moreover, this technique presents the advantage that spatial flow features can be resolved and gradientbased quantities such as the vorticity can be calculated. In addition, correlation and spectral methods can be applied to analyze the velocity fields. In most cases, though, the technique is applied to efficiently measure average quantities such as mean velocity or Reynolds stress distributions because these are still the most relevant variables for the validation of numerical flow simulations and the verification or disproof of theories or models in fluid mechanics.
 1.
to sample the flow motion down to the wall with appropriate tracer particles that follow the fluid motion with sufficient accuracy, as discussed in Wernet and Wernet (1994), Melling (1997) and Kähler et al. (2002),
 2.
to use fluorescent particles as is typically done in microfluids (Santiago et al. 1998) or a tangential illumination along a properly polished wall (Kähler et al. 2006), such that the wall reflection can be suppressed,
 3.
to image the particles properly with a lens or a microscope objective such that the particle signal can be well sampled on a digital camera (Adrian 1997; Hain et al. 2007; Kähler et al. 2012) and
 4.
to estimate the particle image displacement with digital particle imaging analysis methods (Ohmi and Li 2000; Scarano 2001; Stanislas et al. 2003, 2005, 2008).
The first one is the windowcorrelation method that can be used to evaluate single image pairs in order to investigate instantaneous flow phenomena (Adrian and Westerweel 2010; Raffel et al. 2007) or to estimate mean flow properties as discussed in Meinhart et al. (2000). One drawback associated with this evaluation concept is the low dynamic spatial range (DSR), which is usually in the range between 20 and 250. Thus, the range of spatial scales which can be resolved with the technique is rather small. Other drawbacks result from the nonuniform particle image distribution in the vicinity of the wall and the spatial lowpass filtering that bias the location of the correlation peak in the case of nearwall flow investigations. Theunissen et al. (2008) reduced the problem of the nonuniform seeding by a vector reallocation on the basis of the gray levels within the interrogation window. Another approach was presented by Nguyen et al. (2010), who used a conformal transformation of the images at the wall and later correlated only 1D stripes of gray values on a line parallel to the surface. The velocity profile was later directly determined by a fit of the highest correlation peaks of all the lines. This method improves the resolution in the wallnormal direction at the expense of the resolution in the other direction.
The second evaluation method is the singlepixel ensemblecorrelation that estimates mean flow properties from an ensemble of image pairs. Here, the spatial resolution is improved in two dimensions, and the accuracy is significantly increased. This evaluation technique was first applied by Westerweel et al. (2004) for stationary laminar flows in microfluidics. In the last years, the approach was extended for the analysis of periodic laminar flows (Billy et al. 2004), of macroscopic laminar, transitional and turbulent flows (Kähler et al. 2006) and for compressible flows at large Mach numbers (Kähler and Scholz 2006; Bitter et al. 2011). Scholz and Kähler (2006) have extended the highresolution evaluation concept also for stereoscopic PIV recording configurations. Recently, based on the work of Kähler and Scholz (2006), the singlepixel evaluation was further expanded to estimate Reynolds stresses in turbulent flows with very high resolution (Scharnowski et al. 2011). In principle, the interrogation window size can be reduced down to a single pixel, but it was shown that the resolution is determined by the particle image size rather than the pixel size (Kähler et al. 2012). This leads to a dynamic spatial range of up to DSR = 2,000, which is an enormous improvement compared to windowcorrelation analysis. Although the temporal information is lost using this evaluation approach, important quantities such as the Reynolds normal and shear stresses can be extracted with improved resolution and precision, as outlined in Scharnowski et al. (2011).
The third established method for the evaluation of DPIV images is particle tracking velocimetry (PTV). Since the resolution of PTV is not affected by the digital particle image diameter, as shown in Kähler et al. (2012), this technique is often superior to correlationbased methods. However, good image quality is required for reliable measurements and, due to the random location of the velocity vectors, the application of correlation and spectral methods for the vector field analysis becomes difficult and interpolation techniques are required for the estimation of quantities based on velocity gradients. Because of the lowered seeding concentration, the dynamic spatial range is comparable to windowcorrelation techniques for instantaneous fields. However, if averaged data are of interest, the dynamic spatial range can be increased even beyond the range of the singlepixel evaluation to the subpixel range (Kähler et al. 2012).
The combination of PIV and PTV was also proposed by several authors to combine the robustness of correlationbased methods with the spatial resolution of tracking algorithms (Keane et al. 1995; Stitou and Riethmuller 2001). However, all evaluation methods have their respective strengths and weaknesses. Therefore, guidelines are important to know the conditions under which each method performs best. In Sects. 2 and 3, synthetic DPIV images are evaluated using the different methods to assess systematic errors in the vicinity of the wall. Section 4 shows the impact of the resolution limit on the estimation of the nearwall velocity for a macroscopic turbulent boundary layer flow experiment as well as for a microscopic laminar channel flow. Finally, all findings will be summarized and guidelines will be given in Sect. 5.
2 Synthetic test cases
In this section, a fundamental analysis of a synthetic image set is performed for three reasons: first, it gives full control of all parameters considered for the simulation (as opposed to experiments where many uncertainties exist such as local density, temperature, viscosity, flow velocity, particle properties, illumination power and pulsetopulse stability, local energy density in the light sheet, imaging optics, recording medium, and bias effects due to data transfer that are unknown or cannot be precisely controlled as can be done with simulations). Second, the variation of single parameters is possible (which is often difficult to do in experiments because of the mutual dependence of the parameters like light intensity and signaltonoise ratio, optical magnification and lens aberrations). Third, the range of the parameters can be increased beyond the experimentally accessible range (higher shear rates and turbulence levels, higher particle concentrations…).
The major drawback of the synthetic image approach is that not all physical effects can be simulated properly because of a lack of physical knowledge and the fact that each experimental setup is unique. Thus, the idealized assumptions and approximations that are used in simulations lead to deviations from experimental results. To keep the deviations small, the important physical effects must be considered, while the higherorder effects, which are below the resolution limit of the techniques, can be neglected. As this requires an a priori knowledge, experiments are always necessary to prove the main predictions and sensitivities of the simulations and to estimate the uncertainty of the simulation relative to the experiment. This will be done in Sect. 4.
A displacement profile with a constant gradient of \(\partial \Updelta X^{*}/\partial Y^{*}=0.1 \,\hbox{px/px}\) was simulated to illustrate the effects and main sensitivities. Since spatial gradients are assumed to be negligible as long as the gradient multiplied with the interrogation window dimension is less or comparable to the particle image diameter, this implies that for a 8 × 8 px and 16 × 16 px window, the chosen gradient should be irrelevant for particle image diameters of 2–3 px, while for a 32 × 32 px window, it becomes slightly larger than recommended (Keane and Adrian 1992). The surface was located several pixels away from the border of the DPIV images and was slightly tilted (1:20) with respect to the image boarder to simulate the wall location at random subpixel positions. (X, Y) corresponds to the image coordinates, while \((X^{*}, Y^{*})\) denotes the wallparallel and wallnormal coordinates, respectively. Only particle images with a random center position above the simulated surface were generated. Even though all particles are located above the wall, their images can extend into the region below the wall. This effect is illustrated in Fig. 1, where the synthetic images in the nearwall region of the boundary layer are shown for different digital particle image diameters D. This implies a virtual velocity at negative \(Y^{*}\)locations. In real experiments, this leads to the problem that the wall location cannot be estimated reliable from the velocity profile. However, for some experiments, the particle images are mirrored at the surface; in this case, the wall location can be determined precisely from the velocity profile (Kähler et al. 2006).
Frequently used variables and their meaning
Quantity  Symbol  Unit 

Particle diameter  d _{ p }  μm 
Particle image diameter  d _{τ}  μm 
Digital particle image diameter  D  px 
Dynamic spatial range  DSR  m/m 
Optical magnification  M  m/m 
Spatial resolution  res  m 
Wallparallel shift vector component  \(\Updelta X^{*}\)  px 
Wallnormal shift vector component  \(\Updelta Y^{*}\)  px 
Wallnormal image coordinate  \(Y^{*}\)  px 
Wallshear stress  τ_{ w }  N/m^{2} 
Friction velocity  u _{τ}  m/s 
Normalized wallnormal coordinate  y ^{+} = \({y\cdot u_{T}}/v\)  – 
Boundary layer thickness  δ_{99}  mm 
interrogation window height  W _{ Y }  px 
3 Comparison of evaluation techniques
3.1 Windowcorrelation
The windowcorrelationbased evaluation was performed for a digital particle image diameter of D = 3 px, which is close to the optimal value to achieve low RMSuncertainties (Raffel et al. 2007; Willert 1996). Four different interrogation window sizes ranging from 8 × 8 px to 64 × 64 px were applied. The evaluation was performed using a commercial software (DaVis by LaVision GmbH) with a sumofcorrelation approach. For each interrogation window size, 100 image pairs were analyzed.
It is important to note that shift vectors below the surface are computed although no particle positions were generated in this region. This is because the interrogation windows centered below the surface are still partly filled with the images of the particles located above the surface. When the wall location is known, the vectors can be easily rejected. However, in the case of a tangential illumination or fluorescent particles, the wall is not visible at all and the wall detection may become problematic. It is obvious that a proper reallocation of the vectors must be made and a suitable alignment and optimized size of the interrogation windows must be used to minimize these effects. However, this is difficult to achieve close to solid surfaces and interfaces as the ideal conditions (constant flow gradient, homogeneous particle image distribution, straight walls…) do not generally hold in real experiments.
3.2 Singlepixel ensemblecorrelation
It is interesting to note that the wallnormal component \(\Updelta Y^{*} \) is also biased in the nearwall region (Fig. 3, bottom) because the particle images further away from the wall broaden the correlation peak only on the side facing away from the surface. In the case of flows with constant gradients and ideal conditions (homogeneous particle image distribution…), this effect is averaged out as long as the particle is at least \(Y^{*}\) > D/2 away from the wall. In case of instantaneous flow measurements with windowcorrelation PIV, this effect cannot be avoided and results in an increased random error.
For negative \(Y^{*}\) values, the effect causing the systematic motion in the positive \(Y^{*}\) direction can reverse, see Fig. 3 at D = 10 and \(Y^{*}\) < −2.5 px. This bias error is due to the normalization of the correlation by its variance. For regions lower than \(Y^{*}\) < D/2, the gray value distribution over the ensemble can only have values lower than the maximum intensity and only for particles that are within a range of \(Y^{*}\) < D. If these ensembles are correlated with ensembles further away (+) from the wall, more signal peaks (due to the uniform seeding) with large magnitudes up to the maximum intensity are present. Some of them might give a good correlation. However, the peak is averaged by its variance that is quite high for these signals, and thus, the correlation value itself is low.
Nevertheless, τ_{ w } can be estimated directly from the first values above the wall that are not biased according to Eq. (9). Now the experimenter has to determine whether the positions of these first reliable vectors are close enough for the estimation of the wallshear stress, that is they belong to the viscous sublayer of a turbulent boundary layer flow, for instance, or not and whether the mean particle image displacement is large enough for reliable measurements with low uncertainty.
3.3 Particle tracking velocimetry
3.4 Nearwall gradient

half the interrogation window size in the case of windowcorrelation (without vector reallocation),

half the particle image diameter in the case of singlepixel ensemblecorrelation and

the uncertainty of the estimated particle image position in the case of PTV.
4 Experimental verification
To prove the findings of the previous section, three experiments were performed at magnifications of M = 0.1, 2.2 and 12.6 to cover the imaging range from the macroscopic to the microscopic domain. Experiments at lowmagnification realize a large field of view with small particle images and a high particle image density, in general. Thus, lowmagnification experiments are well suited for singlepixel ensemblecorrelation or windowcorrelation depending on the number of image pairs. Increasing the magnification results in larger particle images and does not gain much spatial resolution in case of correlationbased DPIV evaluation, as discussed in detail in Kähler et al. (2012). However, PTV evaluation results in increased spatial resolution in the case of a large number of image pairs, as the resolution is only limited by the error in the determination of the particle image location and the particle image displacement.
4.1 Large field DPIV investigation at low magnification
The first experiment was performed in the largescaled Eiffel type wind tunnel located at the Universität der Bundeswehr München. The facility has a 22mlong test section with a rectangular crosssection of 2 × 2 m^{2}. The flat plate model is composed of coated wooden plates with a superelliptical nose with a 0.48mlong semiaxis in the streamwise direction. The flow was tripped 300 mm behind the leading edge of the plate by a sandpaper strip. Three DEHS particle seeders producing fog with a mean particle diameter of d _{ P } ≈ 1 μm (Kähler et al. 2002) were used to sample the flow. The light sheet for illuminating the particles was generated by a Spectra Physics QuantaRay PIV 400 Nd:YAG doublepulse laser. The light sheet thickness was estimated to be 500 μm. For the flow measurements, a PCO.4000 camera (at a working distance of 1 m) in combination with Zeiss makro planar objective lenses with a focal length of 100 mm was used. The results presented here are taken at 6 m/s free stream velocity, which corresponds to a Reynolds number based on momentum thickness at the measurement location of \(Re_{\delta_2} \) = 4,600. A detailed description of the experimental setup is outlined in Dumitra et al. (2011). In order to resolve the complete boundary layer velocity profile, a large field of view was selected that extends almost 250 mm in the wallnormal direction. The particle image concentration is close to 100 % (illuminated area), and the digital particle image diameter is in the range of \(D \approx 2\ldots3\,\hbox{px}\). These conditions are well suited for the singlepixel ensemblecorrelation, according to Fig. 7 and the analysis in Sect. 3.2. On the other hand, the small particle image size and the high concentration, which causes a large amount of overlapping images, would lead to large errors for PTV, according to Fig. 6. Thus, this lowmagnification data set is evaluated using singlepixel ensemblecorrelation.
4.2 Longrange microscopic DPIV
In order to resolve the wallnormal gradient within the viscous sublayer, the magnification was increased using a longdistance microscope system (K2 by Infinity). The light sheet thickness and the working distance were again 500 μm and 1 m, respectively. The magnification was set to M ≈ 2.2 resulting in particle images with a size of \(D \approx 8\ldots10\,\hbox{px}\). Due to the high magnification, the seeding concentration is rather sparse. The density of particle images, in terms of illuminated area, was less than 3 %. At this seeding density, a huge number of image pairs would be required for singlepixel ensemblecorrelation. Additionally, the large particle images limit the resolution to >6 pixel, according to Kähler et al. (2012), which corresponds to 25 μm. On the other hand, PTV evaluation seems to be well suited for this kind of data, since the resolution is not limited by the particle images size and the low density allows for reliable detection and tracking of the particle images.
Figure 14 shows the velocity profile in normalized coordinates, where the usual normalization was used, that is \(y^{+} = y \cdot u_{\tau} / \nu\) and \(u^{+}=\bar{u}/u_{\tau}\) with the kinematic viscosity ν. The logarithmic region, 30 ≤ y ^{+} ≤ 200, was approximated by an exponential equation shown in Fig. 14. The von Kármán parameters, κ and B, served as dependent variables. The estimated values are in agreement with the range presented in the literature (Zanoun et al. 2003).
4.3 Microscopic DPIV
In order to validate the different evaluation methods for microscopic flow applications, an experiment in a straight micro channel was performed. The channel was made from elastomeric polydimethylsiloxane (PDMS) on a 0.6mmthick glass plate with a crosssection of 514 × 205.5 μm^{2}. A constant flow rate was generated by pushing distilled water through the channel using a highprecision neMESYS syringe pump (Cetoni GmbH). The flow in the channel was homogeneously seeded with polystyrene latex particles with a diameter of 2 μm (Microparticles GmbH). The particle material was premixed with a fluorescent dye and the surface was later PEG modified to make them hydrophilic. Agglomeration of particles at the channel walls can be avoided by this procedure, allowing for longduration measurements without cleaning the channels or clogging.
For the illumination, a twocavity frequencydoubled Nd:YAG laser system was used. The laser was coupled with an inverted microscope (Zeiss Axio Observer) by an optical fiber. The image recording was performed with a 20x magnification ojective (Zeiss LD PlanNeofluar, NA = 0.4) using a 12bit, 1,376 × 1,040 px, interline transfer CCD camera (PCO Sensicam QE) in doubleexposure mode. With the relay lens in front of the camera, the total magnification of the system was M = 12.6. The time delay between the two successive frames was set to \(\Updelta t = 100\) μs. 8,000 image pairs were recorded at a depth of z = 93 μm with an intermediate seeding concentration of \(5 \times 10^{5}\) infocus particles per pixel, which corresponds to 0.4 % of the sensor area covered by particle images or approximately 70 particles per frame. The mean infocus digital particle image diameter was around 10 px.
In the lower part of Fig. 17, the bias error for the wallnormal component, caused by the nonuniform seeding concentration at the wall, can be seen. An artificial velocity component toward the channel center is observed in case of singlepixel ensemblecorrelation. It should be emphasized that the apparent motion of the seeding particles toward the channel’s center is a pure systematic error of the evaluation approach, according to Fig. 5, and not a result of the Saffman effect (Saffmann 1965), which describes a lift force of spherical particles in a shear flow. This is evident from the PTV analysis that does not show any motion away from the wall.
The theoretical flow profile in a rectangular channel is given by a series expansion of trigonometric functions (Bruus 2008). To estimate the wall gradient, the profiles were fitted by a sixthorder polynomial resulting in a wallshear stress of τ_{ w } = 0.0131 ± 0.0002 N/m^{2} for singlepixel ensemblecorrelation and τ_{ w } = 0.0147 ± 0.0002 N/m^{2} for particle tracking, while the theoretical value is τ_{ w } = 0.0158 ± 0.0001 N/m^{2}, where the uncertainty is estimated from the 95 % confidence level of the fit parameters. However, the paper in hand enables to judge, to which distance the values close to the wall are biased. Using only the data points in the range of 8 μm < y < 506 μm results in τ_{ w } = 0.0143 ± 0.0001 N/m^{2} for the singlepixel ensemblecorrelation. From the difference between the experimental and theoretical results, it cannot be concluded that the measurements are biased. More likely, the theoretical fitting is erroneous as the real channel geometry may differ from the assumed one. The same holds for the flow rate. For such a simple geometry, where the biased region can be properly determined, the estimation of the wallshear stress using both methods works quite well. For more complex geometries or when a distinct wallnormal velocity is present, this is not the case.
Since the whole channel is illuminated in micro fluidics, outoffocus particles also contribute to the correlation and bias the velocity estimate (Olsen and Adrian 2000; Rossi et al. 2011). If a normalized correlation is used, this bias is larger due to the sparse seeding, for details we refer to Cierpka and Kähler (2012). Since infocus particles show higher intensity values, they contribute much more to the correlation peak in singlepixel evaluation and the bias due to the depth of correlation is decreased. However, the velocity is still underestimated as can be seen in Fig. 16. For the PTV evaluation, the particle image size was evaluated as well and the velocity estimation was performed later using only infocus particles. As can be seen in Fig. 16, the velocity profile is closer to the theoretical one.
5 Summary

interrogation window dimension W _{ Y } (windowcorrelation analysis)

particle image diameter D (singlepixel ensemblecorrelation)

uncertainty in the estimation of the particle image position σ_{PTV} (PTV).
In Figs. 6 and 7 of this paper, it is shown that the measurement uncertainty of PTV can be below 0.01pixel for low seeding densities. This is a result of the high signaltonoise ratios, which can be easily achieved experimentally, and the relative large digital particle image diameters (3 px < D < 15 px) that allow for a precise detection of the intensity maximum. In addition, the noise of the digital camera is uncorrelated from pixel to pixel. On the other hand, a SNR > 5 is difficult to achieve for windowcorrelation analysis (if 6–8 particle images are considered for the correlation). Furthermore, the noise induced by the correlation of nonpaired particle images is correlated over a distance ∼D. This enhances the random errors as the interference of the signal peak with a noise peak is likely to happen.

≈250 for windowcorrelation with 16 × 16 pixel interrogation windows,

≈2,000 for singlepixel ensemblecorrelation with 2–3 pixel particle image diameters and

>25,000 for PTV in case of low seeding densities and high signaltonoise ratios.
To evaluate the uncertainty of mean quantities, a detailed analysis was performed. In the case of homogeneous seeding distribution and constant flow gradients, it is shown that the bias errors caused by the gradient are averaged out and windowcorrelation, singlepixel ensemblecorrelation and PTV show identical results far away from walls (Fig. 9).

\(\delta \Updelta X^{*}_{\rm wall}=W_Y/4 \cdot \partial\bar{\Updelta X^{*}}/\partial Y^{*}\) at the wall (\( Y^{*} \) = 0) and decreases to zero at \( Y^{*} \) = W _{ Y }/2 (windowcorrelation),

\(\delta \Updelta X^{*}_{\rm wall}=D/(4 \cdot \sqrt{ \pi }) \cdot \partial\bar{\Updelta X^{*}}/\partial Y^{*} \) at the wall (\( Y^{*} \) = 0) and decreases to zero at \( Y^{*} \) = D/2 (singlepixel ensemblecorrelation) and

\( \delta \Updelta X^{*}_{\rm wall} \approx \sigma_{\rm PTV}\) for \( Y^{*} \) < σ_{PTV} (PTV, see first point in Fig. 10).
Furthermore, it can be concluded that PTV image analysis techniques should always be used in the case of inhomogeneous seeding and/or nonconstant flow gradients since correlationbased methods are always biased under these conditions.
This implies that the instantaneous estimation of the particle image displacement from a single image pair, calculated using windowcorrelation techniques, is always biased in the case of flow gradients (even for constant ones) since the particle image distribution cannot be homogeneous for a small number of randomly distributed particle images. In effect, the averaging of windowcorrelation results obtained from a set of image pairs leads to a larger uncertainty compared to the singlepixel ensemblecorrelation analysis. This happens because the bias error due to the inhomogeneous particle image distribution appears as an increased random error on average.
For this reason, it can be concluded that the singlepixel ensemblecorrelation should be used instead of windowcorrelation approaches for the estimation of averaged flow quantities at high seeding densities. However, since the particle images further away from the wall broaden the correlation peak only on the side facing away from the wall (see Fig. 3), \( \Delta Y^{*} \) is also biased for \( Y^{*} \) < D/2 in the case of singlepixel ensemblecorrelation. Therefore, PTV should always be used near walls. This is usually possible as the particle image density decreases toward walls down to an acceptable level for an accurate PTV analysis.

Although the velocity estimation with LDV is usually more precise than particle imaging techniques, the localization of the measurement volume in physical space is seldom better than a fraction of a millimeter [or around 60 μm for sophisticated approaches under ideal conditions (Czarske 2000)], while in PTV, it is only a few micrometers or even less for highmagnification imaging.

In the case of flows with gradients, this position error causes significant bias errors for LDV or other measurement probes, similar to that present in the windowcorrelation results shown in Fig. 9.

Uncertainties raising from mechanical translation stations or thermal elongation do not need to be considered for PTV, while they must for LDV or other probes that are traversed to measure a profile.

Errors due to model or equipment vibrations can be completely accounted for in case of particle imaging techniques, while for LDV and other probes, this becomes difficult. In effect, the spatial resolution is further reduced using these singlepoint techniques.
In summary, the analysis shows that all the evaluation techniques considered in this article have their specific strengths. Therefore, a hierarchical evaluation concept that puts together the benefits of all techniques is desirable to achieve the best possible results. Unfortunately, PTV requires low particle image densities, while windowcorrelation and singlepixel ensemblecorrelation perform best for high seeding concentrations, as this leads to small interrogation windows or moderate number of image pairs for the singlepixel ensemblecorrelation. For this reason, the development of PTV image analysis techniques with low uncertainty at high particle image densities is necessary.
Notes
Acknowledgments
Financial support from German Research Foundation (DFG) in the framework of the Collaborative Research Centre—Transregio 40 and the Individual Grants Programme KA 1808/8 is gratefully acknowledged by the authors. The authors also would like to thank Rodrigo Segura for technical language revisions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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