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Abstract Optical flow methods have been developed over the past two decades for application to PIV images with the goal of acquiring higher resolution measurements of the velocity field than conventional cross-correlation-based techniques. Numerous optical flow velocimetry (OFV) algorithms have been devised to solve the ill-posed optical flow problem, with various physics-inspired strategies to tailor them to fluid flows. While OFV can be applied to continuous scalar fields, it has demonstrated the most success on images of tracer particles, i.e.\ traditional planar PIV images. Compared to state-of-the-art cross-correlation algorithms, OFV methods have demonstrated an order of magnitude increase in spatial resolution and up to a factor of two improvement in overall accuracy when evaluated on synthetic data, at the cost of increased computational time. The requirements for particle seeding density, inter-frame displacement, and image quality are also more stringent for OFV methods compared to cross-correlation. OFV has been applied sparingly in experiments to date, but appears to offer the same advantages demonstrated on synthetic data. At this stage, OFV seems best suited to planar velocity measurements, although extensions to stereoscopic measurements have been demonstrated. 

Abstract The ability of PIV processing algorithms to accurately determine velocity vectors across the range of motion present in PIV images is characterized by the algorithm’s dynamic velocity range (DVR). Conventionally, the DVR of PIV is defined using the ratio between the maximum and minimum resolvable particle displacements, with the minimum based on the uncertainty in the location of a single particle in the optical system. In this work, it is demonstrated that this definition is inadequate in practice, as it ignores many factors which affect the accuracy of an algorithm when determining small displacements, and the error in vectors with small magnitudes in actual flows is often many times larger than the theoretical minimum. A more useful criterion for determining the DVR of a PIV setup is proposed that depends on conditional errors, using synthetic data to produce a known ground truth. The introduced error-based DVR accounts for the effect of multiple flow velocity scales present in a PIV experiment as well as multi-particle effects. It is found that the practical, error-based DVR of cross-correlation-based PIV is highly experiment-dependent and much lower than the widely accepted value of$$\mathcal {O} \left( {10^2} \right)$$ $O\left({10}^2\right)$, typically$$\mathcal {O} \left( {10^0} \right) - \left( {10^1} \right)$$ $O\left({10}^0\right)-\left({10}^1\right)$. The findings from the synthetic data results are corroborated using experimental PIV data to approximate the DVR via a deviation-based approach when the ground truth is unknown. 

Abstract The influence of several potential error sources and non-ideal experimental effects on the accuracy of a wavelet-based optical flow velocimetry (wOFV) method when applied to tracer particle images is evaluated using data from a series of synthetic flows. Out-of-plane particle displacements, severe image noise, laser sheet thickness reduction, and image intensity non-uniformity are shown to decrease the accuracy of wOFV in a similar manner to correlation-based particle image velocimetry (PIV). For the error sources tested, wOFV displays a similar or slightly increased sensitivity compared to PIV, but the wOFV results are still more accurate than PIV when the magnitude of the non-ideal effects remain within expected experimental bounds. For the majority of test cases, the results are significantly improved by using image pre-processing filters and the magnitude of improvement is consistent between wOFV and PIV. Flow divergence does not appear to have an appreciable effect on the accuracy of wOFV velocity estimation, even though the underlying fluid transport equation on which wOFV is based implicitly assumes that the motion is divergence-free. This is a significant finding for the broader applicability of planar velocimetry measurements using wOFV. Finally, it is noted that the accuracy of wOFV is not reduced notably in regions of the image between tracer particles, as long as the overall seeding density is not too sparse i.e. below 0.02 particles per pixel. This explicitly demonstrates that wOFV (when applied to particle images) yields an accurate whole field measurement, and not only at or adjacent to the discrete particle locations. 

Abstract High fidelity near-wall velocity measurements in wall bounded fluid flows continue to pose a challenge and the resulting limitations on available experimental data cloud our understanding of the near-wall velocity behavior in turbulent boundary layers. One of the challenges is the spatial averaging and limited spatial resolution inherent to cross-correlation-based particle image velocimetry (PIV) methods. To circumvent this difficulty, we implement an explicit no-slip boundary condition in a wavelet-based optical flow velocimetry (wOFV) method. It is found that the no-slip boundary condition on the velocity field can be implemented in wOFV by transforming the constraint to the wavelet domain through a series of algebraic linear transformations, which are formulated in terms of the known wavelet filter matrices, and then satisfying the resulting constraint on the wavelet coefficients using constrained optimization for the optical flow functional minimization. The developed method is then used to study the classical problem of a turbulent channel flow using synthetic data from a direct numerical simulation (DNS) and experimental particle image data from a zero pressure gradient, high Reynolds number turbulent boundary layer. The results obtained by successfully implementing the no-slip boundary condition are compared to velocity measurements from wOFV without the no-slip condition and to a commercial PIV code, using the velocity from the DNS as ground truth. It is found that wOFV with the no-slip condition successfully resolves the near-wall profile with enhanced accuracy compared to the other velocimetry methods, as well as other derived quantities such as wall shear and turbulent intensity, without sacrificing accuracy away from the wall, leading to state of the art measurements in the $y^{+}<1$region of the turbulent boundary layer when applied to experimental particle images. 

Optical flow velocimetry (OFV) is a method for determining dense and accurate velocity fields from a pair of particle images by solving the classical optical flow problem. However, this is an ill-posed inverse problem, which generally entails minimizing a weighted sum of two terms–fidelity and regularization–and the weights in the sum are parameters that require manual tuning based on the properties of both the flow and the particle images. This manual tuning has historically been a consistent challenge that has limited the general applicability of OFV for experimental data, as the calculated velocity field is sensitive to the value of the weights. This work proposes a hierarchical model for the weighting parameters in the framework of a maximum a posteriori-based Bayesian optimization approach. The method replaces the classical Lagrange multiplier weighting parameter with a new, less-sensitive parameter that can be automatically predetermined from experimental images. The resulting method is tested on three different synthetic particle image velocimetry (PIV) datasets and on experimental particle images. The method is found to be capable of self-adjusting the local weights of the optimization process in real-time while simultaneously determining the velocity field, leading to an optimally regularized estimate of the velocity field without requiring any dataset-specific manual tuning of the parameters. The presented approach is the first truly general, parameter-free optical flow method for particle image velocimetry (PIV) images. The developed method is freely available as a part of the PIVlab package.