We present a new uncertainty estimation method for Particle Image Velocimetry (PIV), that uses the correlation plane as a model for the probability density function (PDF) of displacements and calculates the second order moment of the correlation (MC). The cross-correlation between particle image patterns is the summation of all particle matches convolved with the apparent particle image diameter. MC uses this property to estimate the PIV uncertainty from the shape of the cross-correlation plane. In this new approach, the Generalized Cross-Correlation (GCC) plane corresponding to a PIV measurement is obtained by removing the particle diameter contribution. The GCC primary peak represents a discretization of the displacement PDF, from which the standard uncertainty is obtained by convolving the GCC plane with a Gaussian function. Then a Gaussian least-squares-fit is applied to the peak region, accounting for the stretching and rotation of the peak, due to the local velocity gradients and the effect of the convolved Gaussian. The MC method was tested with simulated image sets and the predicted uncertainties show good sensitivity to the error sources and agreement with the expected RMS error. Subsequently, the method was demonstrated in three PIV challenge cases and two experimental datasets and was compared with the published image matching (IM) and correlation statistics (CS) techniques. Results show that the MC method has a better response to spatial variation in RMS error and the predicted uncertainty is in good agreement with the expected standard uncertainty. The uncertainty prediction was also explored as a function PIV interrogation window size, and the MC method outperforms the other uncertainty methods.
𝐼 ,, : Second order moment about x-axis 𝐼 --: Second order moment about y-axis 𝑅 ./01 : Gaussian convolved PDF plane 𝑁 344 : Effective number of pixels contributing to correlation 𝑒 67808 : Error in velocity measurements obtained using Prana processing 𝑒 98:;< : Error in velocity measurements obtained using DaVis processing 𝜎 => ? : Standard x uncertainty estimate using MC method 𝜎 => @ : Standard y uncertainty estimate using MC method 𝜎 A= : Standard uncertainty estimate using IM method 𝜎 >B : Standard uncertainty estimate using CS method
Particle Image Velocimetry (PIV) is a non-invasive quantitative fluid velocity measurement technique in which tracer particles are illuminated by a laser sheet, imaged by a high-speed camera, and the displacement of the particle patterns within an image sequence is estimated to resolve the velocity field. An overview of the development of PIV over the past 20 years is given by Adrian [1], and a comprehensive history can be traced in recent publications [2], [3]. Currently, the term PIV is used to encompass the extensive family of methods that are based on evaluating the particle patterns displacement using statistical cross-correlation of consecutive images with high number density flow tracers [2].
However, despite detailed investigation of potential error sources, the development of PIV methods did not involve simultaneous rigorous quantification of uncertainty for a given measurement. As a result there is currently no widely accepted framework for reliable quantification of PIV measurement uncertainty. The situation is exacerbated by the fact that PIV measurements involve instrument and algorithm chains with coupled uncertainty sources, rendering quantification of uncertainty far more complex than most measurement techniques.
Also, knowing the uncertainty bound on each PIV vector is crucial in comparing experimental results with numerical simulations. Therefore, developing a fundamental methodology for quantifying the uncertainty for PIV is an important and outstanding challenge.
Recent developments in this field have led to several uncertainty estimation methods which can be broadly classified into indirect and direct uncertainty estimation algorithms.
The indirect methods use pre-calculated calibration information to predict the measurement uncertainty. In the first such method published, Timmins et al. constructed an “Uncertainty Surface”(US) by mapping the effects of selected primary error sources such as shear, displacement, seeding density, and particle diameter to the distribution of the true errors for a given measurement [4]. This approach is analogous to a traditional instrument calibration procedure for standard experimental instruments. Ultimately, in order to comprehensively quantify the uncertainty, all possible combinations of displacements, shears, rotations, particle diameters, and other parameters must be exhaustively tested which can make this method computationally expensive. Moreover, many of the relevant parameters may not be easily obtained from a real experiment.
Charonko and Vlachos proposed an uncertainty quantification method based on the ratio of the primary peak height to the second largest peak (PPR) [5] in the correlation plane.
Using this method, the uncertainty of PIV measurement can be predicted without a priori knowledge of image quality and local flow conditions. Reliable uncertainty estimation results using a phase-filtered correlation (RPC) [6] were shown, however for standard cross-correlation (SCC) techniques the uncertainty estimates were not as good. Also, the approach depends, like the uncertainty surface method, on calibration of the peak ratio to the expected uncertainty.
Xue et al. [7] used an analogous approach to calibrate the measurement uncertainty with various
The uncertainty in a measurement can also be extracted directly from the image plane using the estimated displacement as a prior information. Sciacchitano et al. proposed a method to quantify the uncertainty of PIV measurement based on particle image matching (IM) or particle disparity [9]. The uncertainty of measured displacement is calculated from the ensemble of disparity vectors, which are due to incomplete matching between particle pairs within the interrogation window. This method accounts for random and systematic error; however peak-locking errors and truncation errors cannot be detected. In addition, the disparity can be calculated only for particles that are paired within the interrogation window, thus this method cannot account for the effects of in-plane and out-of-plane loss of particles. Finally, particle image pair detection can introduce additional sources of error and the method can be computationally expensive for high resolution images with higher seeding density.
Wieneke in his “Correlation Statistics”(CS) method computed the measurement uncertainty by relating the asymmetry in the correlation peak to the covariance matrix of intensi
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