Abstract
This article derives a method to estimate and correct the bias error of the shift vector’s absolute length in the presence of curved streamlines. The main idea is to identify the most likely streamline with constant curvature from the secondorder shift vector and its gradient. The work establishes a theoretical framework including the systematic errors of the firstorder and secondorder shift vector’s absolute value and angle. Synthetic images of a stationary vortex are used to validate the proposed method. The curvaturecorrection is also applied to a synthetic flow field with nonconstant curvature to demonstrate its potential for more realistic flow fields. The results reveal that secondorder accurate vector fields suffer from a biased shift vector length depending on the streamline’s curvature and on the shift vector length. The bias error is negligible for vector fields with a shift vector length below the streamline curvature radius. For large shift vectors or strong curvatures, the bias error can be significantly reduced with the developed method. The approach is very general and can be applied to any vector field obtained from windowcorrelation particle image velocimetry (PIV), singlepixel ensemblecorrelation PIV, particle tracking velocimetry or optical flow methods. It also works for all 3D extensions of the techniques, such as 3DPTV or tomographic PIV.
1 Introduction
Particle image velocimetry (PIV) is a nonintrusive measurement technique that estimates the firstorder velocity field in a plane, or even in a volume, by measuring the displacement of particles in a certain time interval \(\Updelta t.\) Due to the nature of the recording principle, each measured velocity vector represents a volumeaveraged mean motion of the discretized and quantized tracer particle’s diffraction images, rather than the actual velocity of the flow (Adrian and Westerweel 2010; Raffel et al. 2007). To maximize the bandwidth of PIV measurements, the dynamic spatial range (DSR) and the dynamic velocity range (DVR) must be maximized (Adrian 1997).
Singlepixel ensemblecorrelation allows for the reduction in the averaging volume in two dimensions, compared to standard windowcorrelation (Westerweel et al. 2004; Kähler et al. 2006; Scharnowski et al. 2012). Thus, singlepixel ensemblecorrelation yields a very large DSR, in particular for lowmagnified images (Kähler et al. 2012).
Eckstein and Vlachos (2009) as well as Sciacchitano et al. (2012) presented methods to improve the accuracy of PIV evaluation, which results in increased DVR values. Another possibility to achieve a large DVR, as well as to get accurate estimations of the velocity and quantities derived from it, especially for low magnifications, is to maximize the particle image shift by selecting a sufficiently large time delay between subsequent illuminations. On the other hand, a large time separation may cause a bias error in areas with curved streamlines, as illustrated in Fig. 1. Wereley and Meinhart (2001) analyzed a method for the estimation of the shift vectors with secondorder accuracy, and Scholz and Kähler (2006) applied this method to singlepixel vector fields. To achieve an estimation with higherorder accuracy, more than two time steps are generally required (Lele 1991). A multipulse system, as outlined in Kähler and Kompenhans (2000), can be combined with a multiframe evaluation method. Hain and Kähler (2007) as well as Persoons and O'Donovan (2011) describe such a method, developed for PIV, and Cierpka et al. (2012) recently reported an analogous idea, applicable to PTV. Using either combination would allow for higherorder accuracy velocity measurements.
Since multipulse techniques require more expensive equipment, the idea of this paper is to achieve higherorder accuracy from conventional doublepulse recordings by estimating the curvature of the particle path from the secondorder shift vector field. The reconstruction of the particle path from the local shift vectors and their gradients is mainly based on the assumption that neighboring streamlines do not cross. The article is divided into three main sections. Section 2 discusses the curvaturecorrection from an analytical point of view. In Sects. 3 and 4, the developed method is applied to synthetic PIV data of a flow fields with constant and nonconstant curvature, respectively, in order to verify the theoretical predictions. Finally, the approach is tested on a realistic flow field from the third PIV challenge (Sect. 5)
2 Mathematical description
2.1 Firstorder bias error
The evaluation of doublepulse doubleframe PIV images approximates the path of motion by a straight line, and the firstorder shift vector simply connects the location of particle images at time t and at time \(t+\Updelta t.\) The straight line is the shortest possible path; thus, for complex flows, the actual path is usually longer. Hence, the absolute value of the estimated shift vector is generally underestimated. Assuming an actual path with constant curvature, as shown in Fig. 2, this bias error can be computed from the difference of the arc length and the secant
where R is the radius of the curvature, \(\left\Updelta X^{(1)}\right\) is the absolute value of the firstorder shift vector and ξ the arc’s angle (see Fig. 2). Furthermore, the angle of the estimated firstorder shift vector is biased if the actual path is curved
In Eq. 2.2, π/2 is the angle between the tangent of the arc and the radius, whereas the arccos term is the angle between the first/secondorder shift vector and the radius. For \(\left\xi\right<\pi, \) Eq. 2.2 can be reduced to
Both the bias of the absolute value and the bias of the angle depend on the radius of the curved path and on the arc’s angle. However, these quantities are unknown and cannot be measured directly.
2.2 Secondorder bias error
For secondorder accurate PIV evaluation, the firstorder shift vector is moved to the middle position of the start point and the end point of the vector. This fully compensates for the angular error from Eq. 2.2 in the case of constant curvature, since the vector is aligned tangentially to the actual path. However, the vector’s length is still biased. Furthermore, the vector is pushed to a different streamline where the actual velocity is generally different. The bias error of the absolute velocity depends on the curvature, on the arc’s angle and on the change of the secondorder shift vector’s absolute value perpendicular to the streamline
where n is the unit vector perpendicular to the streamline. As before, the bias depends on the unknown parameters ξ and R.
2.3 Curvaturecorrection
In order to compensate for first and secondorder bias errors, the parameters ξ and R must be extracted from the shift vector field. Neighboring shift vectors that connect two points on the same streamline can be used to estimate the flow path’s curvature, as illustrated in Fig. 3. For the sake of clarity, the secondorder shift vector in Fig. 3 is moved such that its center falls on the original starting point. The same holds for the neighboring vector and for the curvaturecorrected one.
For the curvaturecorrection, the main idea is to find a neighboring vector in the vicinity (\(r \rightarrow 0,\) see Fig. 3) of the secondorder shift vector such that both vectors connect two points on the same circle. Under this condition, the following relation holds between the angles of the vectors \({\varvec{\delta}}_{\bf 1}\) and \({\varvec{\delta}}_{\bf 2}\) and the secondorder shift vector
Where \(\phi\) and \(\phi_n\) are the angle of the secondorder shift vector and the neighboring vector. The starting point of the neighboring vector must be close to the starting point of the shift vector in order to align the vector \({\varvec{\delta}_{\bf 1}}\) tangentially to the circle. Due to the finite vector spacing of PIV data points, a local (firstorder) series expansion for the neighboring vector
is required in order to interpolate between the discrete data points. The secondorder shift vector \(\Updelta{\bf X}^{\left(2\right)}\left(X_{0},Y_{0}\right)\) and a neighboring shift vector that fulfills the condition in Eq. 2.5 differ by the vector \({\varvec{\delta}_{\bf 2}}{\varvec{\delta}_{\bf 1}}\):
In order to find a suitable neighboring shift vector, vectors surrounding \(\left(X_{0},Y_{0}\right)\) are analyzed. For a fixed distance between the secondorder vector and the neighboring vector, the angle α is varied (see Fig. 3). The angles β _{1} and β _{2} depend on α as follows:
where \(\left(U_{0},V_{0}\right)\) are the components of the secondorder shift vector at \(\left(X_{0},Y_{0}\right),\) and U _{ X }, U _{ Y }, V _{ X }, and V _{ Y } are the partial derivatives with respect to X and Y, respectively. The angle \(\varphi\) is the orientation of the secondorder shift vector. It is important to note that β _{1} and β _{2} are not given explicitly in Eq. 2.8 and 2.9. However, it is possible to compute \(\beta_{1}\left(\alpha\right)\) and \(\beta_{2}\left(\alpha\right)\) for a set of α (ranging from \(\varphi\pi/2\) to \(\varphi+\pi/2\)) and find the solution for \(\beta_{1}\left(\alpha\right)=\beta_{2}\left(\alpha\right)\) numerically at fairly low computational cost. Once the angle β _{1} is found, the radius R of the circle can be computed
where the arc’s angle is given by
The sign of R defines the direction of the curvature: If R > 0, the streamline follows clockwise rotation, and for R < 0, the streamline follows counterclockwise rotation in a righthanded system. Finally, from the radius R and the arc’s angle ξ, the curvaturecorrection is applied to the secondorder shift vector field in three steps as follows:

the secondorder vector is shifted toward the streamline with constant curvature using the shift vector \({{\varvec{\delta}}}_{\rm cc}: \)

$$ {{\varvec{\delta}}}_{\rm cc}=R\cdot\left(1\cos\left[\frac{\xi}{2}\right]\right)\cdot \left(\begin{array}{c} \sin\varphi\\ \cos\varphi \end{array} \right) $$(2.12)

the secondorder shift vector is enlarged from the tangent’s length to the arc length in order to compensate for the bias error of the absolute value from Eq. 2.1

the corrected vector field is interpolated to the regular grid of the secondorder vector field
It should be emphasized that the estimation of the angle β _{1} is based on the gradients of the secondorder shift vector field (see Eqs. 2.8 and 2.9). Thus, for reliable curvaturecorrection, the gradients must be estimated accurately first.
For 3D PTV or tomographic PIV, the approach works in the same way: A neighboring shift vector that fulfills Eq. 2.5 allows for the curvaturecorrection. Additionally, the vectors δ_{1} and δ_{2} from Fig. 3 must lie in the same plane in order to estimate the most likely path of motion with constant curvature.
3 Synthetic example: LambOseen vortex
The Lamb–Oseen vortex is a frequently used vortex model in fluid dynamics. The circumferential velocity component \(V_{\varphi}\) of this vortex model is given by the following equation:
where \(\Upgamma\) is the total circulation and r _{ c } the vortex core radius. The radial velocity of a Lamb–Oseen vortex is zero; thus, the streamlines have a constant curvature. Therefore, this vortex is well suited to validate the capability of the developed method. The curvaturecorrection eliminates (in theory) the bias error completely due to constant curved streamlines. Thus, no bias error should be visible after correction. To demonstrate the performance of the developed method, synthetic images were generated, as described in Sect. 3.1, and evaluated using different methods: windowcorrelation for a single pair of PIV images (Sect. 3.2), ensembleaveraged windowcorrelation (Sect. 3.3) and singlepixel ensemblecorrelation (Sect. 3.4).
3.1 Image generation
The analysis based on synthetic images gives full control of all simulation parameters without any limitations regarding their range, and it is possible to detect sensitivities, which indicate what is relevant and what is not (Kähler et al. 2012; Stanislas et al. 2008); 10,000 synthetic image pairs, 512 × 512 px in size, with a stationary Lamb–Oseen vortex in the center, were generated for different total circulations \(\Upgamma.\)
A particle image density of dens = 25 % was applied, meaning that 25 % of the image area was covered by particle images. The digital particle image diameter was set to D = 3 px; hence, the number of particles per pixel was \(N_{\rm ppp}=4\cdot dens/\left(D^{2}\cdot\pi\right)=0.035\) on average. A Gaussian particle image shape was assumed
where \(\left(X_{0},Y_{0}\right)\) is the randomly chosen center position, and D is the diameter at 1/e ^{2} of the maximum intensity (4 times the standard deviation). The maximum intensity of the particle images was I _{0} = 2^{14}. From Eq. 3.2, the discrete pixel’s gray values were computed from the integral over the pixel’s area, corresponding to a sensor fill factor of 1. Nonuniform pixel response, as outlined in Kähler (2004), was not considered.
The intensity distribution of the first synthetic PIV frame A is given by the following equation:
where P and p are the number of particle images and the corresponding control variable, respectively. On top of the particle image intensities, a Gaussian noise with zero mean and a standard deviation of \(1\,\%\cdot I_{0}\) were added. Finally, the intensity distribution was converted into a 16bit unsigned integer matrix.
For the second frame, the center positions of the particle images were shifted such that the distances to the vortex center remained constant and the arc lengths were proportional to the circumferential velocity from Eq. 3.1. The image generation was performed using MatLab functions.
3.2 Windowcorrelation
To analyze the effect of curved streamlines on the accuracy of the estimated vector fields, single PIV image pairs of a synthetic vortex are processed. Figure 4a shows an instantaneous vector field computed with windowcorrelation using DaVis8.1 (by LaVision GmbH), including multipass evaluation with decreasing window size, iterative window shifting, image deformation and adaptive Gaussian window weighting. The interrogation window size was reduced from 64 × 64 px down to 16 × 16 px, and two passes were performed for each window size. Furthermore, an initial shift vector field, computed from the product of a constant and the field of a slowly rotating vortex, was used to overcome the difficulties due to the large shift vector length and the strong gradients. The background color in Fig. 4a represents the absolute value of the curvaturecorrected shift vector. Shift vectors with first and secondorder accuracy as well as curvaturecorrected ones are shown in different colors for X = − 4 px. The vortex core radius and the total circulation were r _{ c } = 20 px and \(\Upgamma=10^{4}\,\hbox{px}^{2},\) respectively. This results in a relatively large maximum particle image shift of 36 px. Figure 4b shows the circumferential shift vector component \(\Updelta X_{\Upphi}\) for three different circulations (\(\Upgamma=\left[1,000; 5,000; 10^{4}\right]\,\hbox{px}^{2}\)) estimated with first and secondorder accuracy and for curvaturecorrected vectors using a final interrogation window size of 8 × 8 px and 16 × 16 px.
It can be concluded from Fig. 4 that for the test case

for a shift vector length of <3.6 px, the maximum bias error of firstorder and secondorder evaluation is 0.01 px which is in the order of (or even below) the random error

for regions with large gradients (r < 30 px), small interrogation windows are required for reliable velocity estimations because the bias error due to the spatial averaging is large compared to the bias error caused by the curvature

the curvaturecorrection results in significantly decreased bias in the case of large shift vectors for 8 × 8 px interrogation windows.
3.3 Averaged windowcorrelation
While instantaneous velocity fields are very important for a deeper understanding of the flow physics, statistical values, such as the mean velocity and Reynolds stresses, are required for the validation of numerical flow simulations and for the proof of analytical theories in fluid mechanics. For the estimation of average quantities, the sumofcorrelation approach is more accurate (Wereley and Meinhart 2001). This is because of two reasons: First, the bias error arising from gradients can be reduced as smaller interrogation windows can be applied. Second, the error in estimating the maximum of the correlation is not summed up as in the case of classical cross correlation analysis.
To analyze the effect of curved streamlines on the accuracy of estimated shift vector fields by using sumofcorrelation evaluation, 100 PIV image pairs of a synthetic vortex were processed. Figure 5a shows an averaged vector field computed with windowcorrelation using DaVis8.1 (by LaVision GmbH) with the same processing parameters as before. Once again, the vortex core radius and the total circulation were r _{ c } = 20 px and \(\Upgamma=10^{4}\,\hbox{px}^{2},\) respectively. Figure 5b shows the circumferential shift vector component for three different circulations, estimated with first and secondorder accuracy using 8 × 8 px interrogation windows for the final pass as well as curvaturecorrected results.
Figure 5a, b show that the curvaturecorrection reduces the bias error significantly in the case of large shift vectors. However, for regions with large gradients (r < 30 px), a small systematic deviation remains even for the curvaturecorrection. This bias is caused by the final size of the interrogation windows. To reduce this bias error, singlepixel ensemblecorrelation analysis is required.
3.4 Singlepixel ensemblecorrelation
Singlepixel evaluation can be used for a large amount of PIV image pairs and results in significantly improved spatial resolution (Kähler et al. 2012). This is always of importance for the analysis of flow fields with strong gradients. Singlepixel ensemblecorrelation 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 et al. 2006), the singlepixel evaluation was further expanded to estimate Reynolds stresses in turbulent flows with very high resolution (Scharnowski et al. 2012).
For singlepixel ensemblecorrelation, the correlation functions C(ξ, ψ, X, Y) can be computed from a pair of PIV image sets A(X, Y) and B(X, Y) as follows:
where the standard deviation is given by:
(X, Y) are discrete coordinates of the pixel in question in both images and (ξ, ψ) are the coordinates on the correlation plane. The maximum position of the correlation function C represents the mean shift vector with firstorder accuracy. Secondorder accuracy can be achieved by moving the firstorder vectors to their middle position as demonstrated in Scholz and Kähler (2006). Furthermore, the curvaturecorrection discussed in Sect. 2.3 can be applied to the secondorder vector field (or directly from the firstorder field by applying some modifications to Eq. 2.8). In the case of a stationary laminar flow, the curvaturecorrection estimates the most likely streamline with constant curvature. On the other hand, for turbulent flows, the curvaturecorrection estimates the mean path with constant curvature which is not identical with the actual streamline of instantaneous vector fields.
In order to analyze the effect of curved streamlines on the accuracy of estimated shift vector fields using singlepixel ensemblecorrelation, 10,000 synthetic PIV images were generated and evaluated. Figure 6a shows an averaged vector field computed with singlepixel ensemblecorrelation using Eq. 3.4. As before, the vortex core radius and the total circulation were r _{ c } = 20 px and \(\Upgamma=10^{4}\,\hbox{px}^{2},\) respectively. Figure 6b shows the circumferential shift vector component for three different circulations estimated with first and secondorder accuracy as well as with curvaturecorrection. Figure 6b shows that the curvaturecorrection causes the expected bias error reduction for the singlepixel evaluation. Furthermore, even for regions with strong gradients (r < 30 px), the thirdorder accurate data points fit the theoretical curve closely. Thus, the remaining bias of the windowcorrelation approach could be eliminated completely, as expected from the theory.
Figure 7 shows the difference of the shift vector’s absolute value (left) and angle (right) for firstorder (top) and secondorder (bottom) accuracy with respect to the curvaturecorrected shift vector. The simple firstorder approximation underestimates the circumferential velocity (Fig. 7a) and causes a strong inwardfacing radial component, indicated by the positive sign of the angular bias in Fig. 7b. The secondorder approximation gives the right values for the shift vector’s angle (Fig. 7d) but the circumferential component is still biased. Figure 7c shows that the velocity in the core (r < 20 px) is overestimated, while the velocity outside the core is underestimated. The curvaturecorrected estimation does not suffer from any bias error for the tested Lamb–Oseen vortex. These findings are in agreement with the theoretical predictions in Sect. 2.
Figure 8 shows the relative bias error of the secondorder shift vector’s absolute value with respect to the arc’s angle ξ. From the figure, it can be concluded that for the tested cases, the bias error is negligible for angles smaller than ξ < 10°. Only for large angles ξ, which require strongly curved streamlines, combined with large shift vectors, are the curvaturecorrection necessary.
4 Nonconstant curvature
So far, the developed method was applied to synthetic shift vector fields of stationary vortices, where the curvature along the streamlines is constant. This section analyzes vector fields with nonconstant curvature. Again, synthetic images are used to have full control of all involved parameters. The test case is a flow along sinusoidal streamlines, as shown in Fig. 9. The shape of a streamline centered at Y _{0} is given as follows:
where a and b are the amplitude and the period, respectively. The streamline’s curvature 1/R changes periodically with respect to X
At the inflection points, the curvature is zero, where it reaches values of up to \(4\cdot a\cdot\pi^{2}/b^{2}\) at the extrema of the sin function in Eq. (4.1). To analyze the effect of this nonconstant curved streamlines on the accuracy of the estimated shift vector field, synthetic image pairs were generated, as discussed in Sect. 3.1. A constant shift vector length \(\Updelta S\) along the streamlines was simulated. Therefore, for each point \(\left(X_{0},Y_{0}\right),\) the displacement in the X direction \(\Updelta X\) is computed from the curve’s length
and the displacement in the Y direction from
using the definition in Eq. (4.1). The resulting shift vector field is shown in Fig. 9 for a = 20 px and b = 128 px.
Synthetic PIV image pairs, 256 × 256 px in size, with sinusoidal streamlines were generated and evaluated with averaged windowcorrelation. Figure 10 shows cross section plots of the estimated shift vector length for different simulated \(\Updelta S\) ranging from 5 to 50 px. Each point in Fig. 10 represents the mean displacement (absolute value) averaged over Y computed from 100 image pairs with sumofwindowcorrelation using a final interrogation window size of 8 × 8 px.
The upper part of Fig. 10 clearly shows a significant bias error: The shift vector length is underestimated in regions with strong curvature. The bias error is largest for large simulated shift vectors and reaches a maximum of 12 % for \(\Updelta S=50\,{\text {px}}\) at the extrema of the sinusoidal function, which illustrate the importance of this correction for precise measurements. The curvaturecorrection, shown in the lower part of Fig. 10, results in significantly reduced bias error. The maximum bias error is only 5 %. Similar results were achieved by applying windowcorrelation to single image pairs (resulting in a higher random error) as well as by applying singlepixel ensemblecorrelation to 1,000 image pairs.
The mean bias error, averaged over the image area, as a function of the shift vector length is illustrated in Fig. 11; it can be seen that the curvaturecorrection results in reduced bias error for all tested shift vector lengths. Additionally, the mean random error is shown in Fig. 11. For the averaged windowcorrelation of the test case, the mean random error is always lower than the mean bias error. Thus, the shift vector gradients can be reliably detected, and curvaturecorrection results in improved accuracy.
Figure 12 shows the normalized maximum bias error as a function of the arc’s angle ξ. The curvature at the extrema of the sinusoidal streamline is relatively large (\(1/R_{\rm min}=4\cdot a\cdot\pi^{2}/b^{2}=4\cdot20\cdot\pi^{2}/128^{2}\approx 0.05\,\hbox{px }^{1},\,R_{\rm min} \approx 20\,\hbox{ px}\)); thus, the secondorder accurate shift vectors are strongly biased. The curvaturecorrection reduces the maximum bias error significantly. For arc angles ξ ≤ 20°, the maximum bias error of the secondorder shift vector is below 1 %, which is likely to be smaller than the random error for real PIV images. Thus, curvaturecorrection becomes important for ξ > 20°.
5 Flow field example
In the following, the curvaturecorrection is applied to a realistic flow field example: test case B of the third PIV challenge (Stanislas et al. 2008). Case B is a synthetic sequence of 120 images with equidistant time intervals of a laminar separation bubble. The image size is 1,440 × 688 px, the particle image diameter is around 2 px and the number of particles per pixel is N _{ppp} = 0.024.
Images 10 and 11 were evaluated with windowcorrelation using DaVis8.1 (by LaVision GmbH), including multipass evaluation with decreasing window size, iterative window shifting, image deformation and adaptive Gaussian window weighting. A final interrogation window size of 16 × 16 px was chosen. The curvaturecorrection, as discussed in Sect. 2.3, was applied to the secondorder accurate vector field.
Figure 13a shows the estimated curvaturecorrected shift vector field. The background color in the figure represents the absolute value of the shift vector, which is characterized by a very large dynamic range. Additionally, characteristic streamlines are drawn in Fig. 13a. Several regions with very strong curvature can be identified from the streamlines. Thus, the curvaturecorrection approach should help to reduce bias errors.
However, the bias does not only depend on the streamline’s curvature but also on the local shift vector length. The arc’s angle ξ (see Fig. 2) is an indicator of the necessity of curvaturecorrection. The spatial distribution of the angle ξ is illustrated in Fig. 13b. It reaches a maximum of ξ _{max} = 1.75° at X = 1,140 px and Y = 452 px. However, according to the findings of Sects. 3 and 4, values of ξ < 10° result in negligible bias errors. This is also the case here, as can be seen from the difference of the shift vector length for the secondorder accurate vectors and for the curvaturecorrected ones, illustrated in Fig. 13c. The estimated maximum bias error is 5 orders (!) of magnitude smaller then the maximum shift vector length. It can be concluded that the developed method of curvaturecorrection does not improve the accuracy for this particular test case. In order to have a useful correction, the shift vectors must be much larger (factor of 5 or more). This would, on the other hand, increase the amount of loss of paired particle images due to out of plane motion for this 3D flow field. However, for all volumetric PIV and PTV techniques, where out of plane loss of pairs is not possible, this approach can enhance the quality of the results.
6 Conclusion
The presented method allows for the estimation and correction of the bias error due to curved streamlines from PIV data. This result is of great importance for the accurate estimation of velocity fields from PIV data acquired at low optical magnification, as required for high dynamic spatial range. The developed method was validated by analyzing synthetic PIV images of a stationary Lamb–Oseen vortex and of a flow with sinusoidal streamlines. The curvaturecorrected vector fields demonstrated the benefit for standard windowcorrelation (instantaneous and ensembleaveraged) as well as for singlepixel ensemblecorrelation. Due to the curvaturecorrection, the bias error of the shift vector’s absolute value and angle is significantly reduced.
The approach is limited to vector fields with low noise level, since the spatial derivatives of the secondorder shift vector must be estimated reliably in order to perform the curvaturecorrection. Furthermore, the detectable curvature is limited by the averaging area: The interrogation window size must be significantly smaller than the streamline radius for reliable secondorder accuracy and curvaturecorrection. The accuracy of a secondorder accurate shift vector field that suffers from bias errors due to limited spatial resolution cannot be improved by applying the curvaturecorrection.
On the other hand, it was shown that the bias error due to curved streamlines is usually negligible for flow fields with moderate shift vector lengths and curvatures (\({\Updelta X}<20\,\hbox{ px}, \xi<10^{\circ}\)). Thus, it can be concluded that secondorder accurate PIV evaluation reveals reliable results for most practical applications. However, for highly accurate measurements of the mean velocity field with up to singlepixel resolution, the approach outlined in the paper is recommended to be applied.
References
Adrian RJ (1997) Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Meas Sci Tech 8:1393 doi:10.1088/09570233/8/12/003
Adrian RJ, Westerweel J (2010) Particle image velocimetry. Cambridge University Press, Cambridge
Billy F, David L, Pineau G (2004) Single pixel resolution correlation applied to unsteady flow measurements. Meas Sci Tech 15:1039–1045 doi:10.1088/09570233/15/6/002
Bitter M, Scharnowski S, Hain R, Kähler CJ (2011) Highrepetitionrate PIV investigations on a generic rocket model in sub and supersonic flows. Exp Fluids 50:1019–1030 doi:10.1007/s0034801009888
Cierpka C, Lütke B, Kähler CJ (2012) Higher order 4pulse particle tracking velocimetry. In: 16th international symposium on applications of laser techniques to fluid mechanics, Lisbon, Portugal, July 9–12
Eckstein A, Vlachos PP (2009) Digital particle image velocimetry (DPIV) robust phase correlation. Meas Sci Tech 20:055401. doi:10.1088/09570233/20/5/055401
Hain R, Kähler CJ (2007) Fundamentals of multiframe particle image velocimetry (PIV). Exp Fluids 42:575–587 doi:10.1007/s0034800702666
Kähler CJ (2004) The significance of coherent flow structures for the turbulent mixing in wallbounded flows. PhD thesis, GeorgAugustUniversität, Göttingen. http://webdoc.sub.gwdg.de/diss/2004/kaehler/kaehler.pdf
Kähler CJ, Kompenhans J (2000) Fundamentals of multiple plane stereo PIV. Exp Fluids 29:70–77 doi:10.1007/s003480070009
Kähler CJ, Scholz U (2006) Transonic jet analysis using longdistance micro PIV. In: 12th international symposium on flow visualization, Göttingen, Germany
Kähler CJ, Scholz U, Ortmanns J (2006) Wallshearstress and nearwall turbulence measurements up to single pixel resolution by means of longdistance microPIV. Exp Fluids 41:327–341 doi:10.1007/s0034800601670
Kähler CJ, Scharnowski S, Cierpka C (2012) On the resolution limit of digital particle image velocimetry. Exp Fluids 52:1629–1639. doi:10.1007/s003480121280x
Lele SK (1991) Compact finite difference schemes with spectrallike resolution. J Comput Phys 103:16–42 doi:10.1016/00219991(92)90324R
Persoons T, O'Donovan TS (2011) High dynamic velocity range particle image velocimetry using multiple pulse separation imaging. Sensors 11:1–18. doi:10.3390/s110100001
Raffel M, Willert CE, Wereley ST, Kompenhans J (2007) Particle image velocimetry: a practical guide. Springer, New York
Scharnowski S, Hain R, Kähler CJ (2012) Reynolds stress estimation up to singlepixel resolution using PIVmeasurements. Exp Fluids 52:985–1002 doi:10.1007/s0034801111841
Scholz U, Kähler CJ (2006) Dynamics of flow structures on heaving and pitching airfoils. In: 13th international symposium on applications of laser techniques to fluid mechanics, Lisbon, Portugal
Sciacchitano A, Scarano F, Wieneke B (2012) Multiframe pyramid correlation for timeresolved PIV. Exp Fluids 53:1087–1105. doi:10.1007/s003480121345x
Stanislas M, Okamoto K, Kähler CJ, Westerweel J, Scarano F (2008) Main results of the third international PIV challenge. Exp Fluids 45:27–71 doi:10.1007/s003480080462z
Wereley ST, Meinhart CD (2001) Secondorder accurate particle image velocimetry. Exp Fluids 31(3):258–268 doi:10.1007/s003480100281
Westerweel J, Geelhoed PF, Lindken R (2004) Singlepixel resolution ensemble correlation for microPIV applications. Exp Fluids 37:375–384 doi:10.1007/s003480040826y
Acknowledgments
Financial support from the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Sonderforschungsbereich Transregio 40 is gratefully acknowledged by the authors. The authors also would like to thank Rodrigo Segura for technical language revisions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Scharnowski, S., Kähler, C.J. On the effect of curved streamlines on the accuracy of PIV vector fields. Exp Fluids 54, 1435 (2013). https://doi.org/10.1007/s0034801214359
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s0034801214359
Keywords
 Particle Image Velocimetry
 Bias Error
 Shift Vector
 Particle Image Velocimetry Data
 Particle Image Velocimetry Image