Error-based dynamic velocity range of PIV processing algorithms

Particle image velocimetry (PIV) is an established experimental fluid flow velocimetry technique which entails imaging the motion of tracer particles in a flow illuminated by a pulsed laser light sheet with known temporal separation \(\Delta t\) (Adrian 1991; Raffel et al. 2018). The fluid flow velocity field can be inferred from the motion of the imaged tracer particles using methods such as cross-correlation or optical flow velocimetry (OFV) (Raffel et al. 2018; Corpetti et al. 2006; Heitz et al. 2010; Liu et al. 2015; Cai et al. 2018; Schmidt and Sutton 2019; Stamhuis and Thielicke 2014; Thielicke and Sonntag 2021). One of the key distinguishing factors between PIV and OFV is the spatial resolution of the velocity field obtained. In cross-correlation-based PIV, cross-correlations among small interrogation windows in a pair of images are used to determine a single velocity vector for each window, whereas in OFV, a dense velocity field is estimated by calculating the displacement vector at each pixel using optical flow, a technique originating from the computer vision community (Horn and Schunck 1981; Beauchemin and Barron 1995). Accuracy of both the methods is highly dependent on the quality of the particle images, which is dictated by the design of the PIV experiment.

To obtain optimal velocimetry results using PIV, the general theory and design rules for PIV experiments are well established in the literature by, e.g., Keane and Adrian (1990), Keane et al. (1995) and Keane and Adrian (1990). As fluid flows generally contain a wide range of velocity scales, especially in wall-bounded flows where the velocity in a boundary layer can vary by over an order of magnitude (Jassal and Schmidt 2023), it is extremely important to optimize PIV systems—in terms of both the experimental parameters and the processing algorithm—to capture the full breadth of velocity magnitudes in the fluid motion. The criterion that is used to characterize the range of velocities that a PIV system is capable of resolving accurately is generally known as the dynamic velocity range (DVR), and typically, a high DVR is desired so that the full range of velocity magnitudes present in the flow can be accurately captured. In practice, knowledge of the DVR is very valuable because it acts as an experimental guideline that informs the choice of inter-frame time, which determines the particle displacements. It can alternatively be used to determine the size of interrogation window to be used, since the inter-frame time is typically set such that the maximum inter-frame displacement represents one quarter of an interrogation window (Westerweel 1997). The DVR allows the user to design their PIV experiment to properly capture the range of velocities desired, and to quantitatively characterize the uncertainty of velocity vectors with small magnitudes.

A definition of DVR based on the theoretical framework of PIV combined with the uncertainty principle was proposed by Adrian in 1997 (Adrian 1997). Since its introduction, it has been widely accepted that the DVR of a PIV system is given as the ratio of the maximum velocity present in the flow to the minimum resolvable velocity (Adrian 1997; Bharadwaj and Vybhav 2022; Persoons and O’Donovan 2010; de Silva et al. 2014). Mathematically, this reads as Eq. (1).

Here \(\sigma _{\max }\) is the largest velocity present in the flow and \(\sigma _{\textrm{V}}\) is the smallest resolvable velocity. \(\sigma _{\max }\) is typically assumed to be able to be measured accurately by appropriately choosing the size of the interrogation window and the time between laser pulses based on the inter-frame displacement of particles. More information on these design guidelines can be found in, e.g., Ref. Westerweel (1997), but often the interrogation window size is chosen such that at least 12 particles are included in each window on average, and \(\mathrm {\Delta } t\) is chosen such that the particles with the highest velocity are displaced by about a quarter of the largest interrogation window size. The minimum resolvable displacement \(\sigma _{\textrm{V}}\) is determined from the hypothesis that, due to noise inherent in the images and diffraction-limited imaging, there is a root mean square error \(\sigma _{\Delta X}\) in determining the position of the centroid of a particle. This uncertainty is present in both frames of an image pair, and also, it can be influenced by other factors such as non-uniformity between laser pulses. Following the derivation in Adrian et al., \(\sigma _{\textrm{V}}\) is mostly dependent on the imaging setup and is expressed as follows (Adrian 1997).

In Eq. (2), \(M_o\) is the optical magnification of the object plane, \(\Delta t\) is the pulse separation time, and \(\sigma _{\Delta X}\) is the root mean square error associated with the uncertainty of the algorithm used to determine the pixel displacement. It should be noted that \(\sigma _{\textrm{V}}\) is agnostic to the range of velocity magnitudes present in a PIV experiment and depends entirely on the uncertainty in the algorithm used to determine pixel displacements and the experimental setup that prescribes both \(M_o\) and \(\Delta t\). As stated above, this formulation for the minimum resolvable displacement is based on diffraction-limited imaging and in theory represents the smallest particle displacement that would produce a noticeable difference to a cross-correlation algorithm. However, this theoretical value can have little relevance to actual experiments. Even though a processing algorithm is theoretically capable of resolving a displacement with a certain minimal size, it does not mean that it can accurately calculate such a displacement in practice, when particles with larger displacements are also present in the same images and correlation windows. It is demonstrated in this work that the errors in vectors with small magnitudes are often many times larger than the theoretical minimum in actual flows. Therefore, a new definition of DVR is needed to provide a useful practical measure for characterizing the dynamic velocity range of a PIV system.

In this study, two different synthetic PIV datasets are processed using state-of-the-art optical flow velocimetry and cross-correlation algorithms (PIVLab) to examine the accuracy of Eq. (1) for predicting the DVR of a PIV setup. While it is widely accepted that the DVR of a PIV system using cross-correlation is \(\mathcal {O} \left( {10^2} \right)\) (Persoons and O’Donovan 2010; Adrian 1997), it is demonstrated that the DVR observed is significantly lower than the widely accepted theoretical DVR in Eq. 1. To provide a more practical measure, an error-based DVR of the both the PIV processing algorithms is introduced in Sect. 4 and is evaluated for the synthetic test cases in Sect. 5. The DVR is then characterized experimentally using PIV data acquired in a cylinder wake flow.

The general goal of all PIV processing algorithms is to determine the particle displacements, \(\Delta \vec {x}\), which can be used to determine the flow velocity for a given inter-frame time, \(\Delta t\). This section describes two common approaches to determining \(\Delta \vec {x}\) from particle image data: cross-correlation and optical flow. One of the key distinguishing factors between these two methods is that cross-correlation computes each displacement vector independently, while optical flow typically computes a global displacement field for the entire particle image. It is worth noting that there are local optical flow methods such as that of Lucas and Kanade (1981); however, a global method is used in this study. This distinction between the two methods can affect the dynamic velocity range, as elaborated further in the following discussion.

2.1 Cross-correlation

Cross-correlation is a standard PIV processing algorithm that determines the particle displacement vectors by cross-correlating subregions of the images called interrogation windows (IWs) of two consecutive particle image frames. The reader is referred to one of several reference texts, e.g., Raffel et al. (2018), for further details. A simple explanation of a typical cross-correlation-based PIV processing algorithm is illustrated in Fig. 1. The particle images are first divided into IWs of a specific size, and for each of these regions, a cross-correlation matrix is computed using Eq. (3). This matrix essentially quantifies the similarity of a certain interrogation region in the first image to all possible regions in the second and hence can be used to determine the most statistically likely measure of how a group of particles moves from one image to the next. Once the cross-correlation map is obtained for all the interrogation windows, the displacement for each window is determined by identifying the peak in the correlation map of the corresponding interrogation window.

Numerous modifications such as sub-pixel Gaussian fitting, Fourier domain computations, deforming interrogation windows and multi-pass have been introduced into the method to increase its accuracy and robustness in modern algorithms (Thielicke and Sonntag 2021; Raffel et al. 2018). It has been shown that the displacement determined for each interrogation region is a spatial average of the particle displacements inside that particular window (Raffel et al. 2018). Therefore, by its mathematical construction, cross-correlation provides an average velocity vector over a region of the image and hence spatially down-sampled velocity field measurements, i.e., the spatial resolution of the velocity field obtained is always less than the spatial resolution of the original particle image data. This study employs a widely used state-of-the-art PIV cross-correlation package PIVLab (Thielicke and Sonntag 2021) for cross-correlation-based PIV processing.

A schematic explanation of a typical cross-correlation algorithm

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2.2 Optical flow

Optical flow-based particle image processing techniques, commonly referred to as optical flow velocimetry (OFV), are based on the optical flow problem originating from the computer vision community (Horn and Schunck 1981; Beauchemin and Barron 1995). As mentioned in Sect. 1, OFV produce a velocity vector at each pixel in the input images. However, it should be noted that the actual spatial resolution of optical flow is larger than one pixel, as the neighboring vectors at each pixel are correlated with each other by the constraints imposed on the solution, such as constraints on the regularity of the flow field. Under certain conditions, OFV has been demonstrated to yield a higher-resolution estimate of the velocity field with increased global accuracy compared to correlation-based PIV (Corpetti et al. 2006; Heitz et al. 2010; Liu et al. 2015; Cai et al. 2018; Schmidt and Sutton 2019; Jassal and Schmidt 2023, 2024), but an assessment of its dynamic range has not yet been made. A recent review on the application of OFV to fluid flows can be found in Jassal and Schmidt (2025).

Briefly, optical flow velocimetry aims to invert the displaced frame difference equation given in Eq. (4) for the displacement field \(\Delta \vec {x} = \vec {u} \Delta t\). The direct inversion of the DFD equation is an ill-posed inverse problem due to having only one equation per pixel, the intensity conservation from Eq. (4), but two unknowns, i.e., the two components of the velocity field. A typical way to remedy the ill-posedness of an underdetermined problem is to apply regularization, which can be explicit, implicit or both (Schmidt and Sutton 2021). Such an approach is called a variational solution, and the interested reader is referred to the review article by Heitz et al. for a more complete discussion (Heitz et al. 2010).

In a variational approach, the velocity field is determined by minimizing a functional based on Eq. (4) via a global solution, i.e., the vector at each pixel location is not estimated independently, but rather are influenced by nearby pixels. The dependence on the neighboring pixels is governed by the details of the regularization strategy and the strength of the regularization relative to satisfying Eq. (4). Therefore, in the context of DVR, if there is a wide range of particle motion present in a certain region of the flow, the dynamic range depends not only on the uncertainty associated with the algorithm for a single particle, but also on the full range of motion present in that region. Typically for variational optical flow algorithms, inter-frame particle displacements larger than a few pixels are difficult to recover. To remedy this, an image pyramiding scheme proposed by Ruhnau et al. (2005) is implemented within the multi-resolution framework provided by wavelet-based optical flow in this work. Otherwise, the specific algorithm employed here is identical to the one developed by Schmidt and Sutton (2021) and is freely available as part of the PIVLab package.

To evaluate the DVR of the PIV processing algorithms, a set of synthetic particle images are generated that mimic ideal PIV experimental conditions with a known velocity field. As the ground truth is known for a synthetic dataset, it can be used to accurately quantify the errors in the PIV processing algorithms, which is a necessary step before proceeding to actual experimental data. Synthetic datasets are chosen such that they contain a wide range of particle motions and velocity gradients. To investigate the DVR experimentally, PIV images are acquired in the turbulent wake of a cylinder. A summary of all the PIV test cases is provided in Table 1.

3.1 Synthetic datasets

3.1.1 Lamb–Oseen vortex flow

A first synthetic PIV dataset is generated using Lamb–Oseen vortices, and further information regarding the dataset can be found in Ref. Schmidt and Sutton (2019). The dataset features a set of 32 non-temporally decaying Lamb–Oseen vortices of random size, strength, location and sign. The vortices create a temporally evolving flow by moving under one another’s influence, leading to spatially varying velocities in the flow. The dataset contains 14 sets of 50 image pairs of resolution \(256\times 256\) pixels. Each of the 14 sets has a different maximum inter-frame displacement ranging from 0.05 to 8 pixels to investigate the effect of largest velocity scale present in a flow field on the DVR of the PIV processing algorithms.

a An example synthetic particle image in the Lamb–Oseen vortex flow set, b LIC illustration of the DNS velocity field of a snapshot from the data and c vorticity field corresponding to the velocity field shown

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A sample particle image from the synthetic data along with a line integral convolution (LIC) visualization (Cabral and Leedom 1993; Howard et al. 2023) of the ground-truth flow field is shown in Fig. 2a and b. Finally, the corresponding vorticity field LIC snapshot is shown in Fig. 2c. The vorticity field illustrates the rapid spatial variations in the velocity due to the concentrated velocity gradients present in the form of viscous vortices. The flow thus contains a wide range of fluid velocities which makes it well suited for evaluating the dynamic range of the PIV processing algorithms.

3.1.2 Buoyancy-driven mixing flow

Another synthetic PIV dataset containing a wide range of known velocity scales is generated to avoid biasing the analysis toward a specific simulation. A set of 100 pairs of \(256 \times 256\)-pixel synthetic particle images were generated from a direct numerical simulation (DNS) of buoyancy-driven mixing flow with a turbulence Reynolds number of \(Re_t\approx 17.8 \times 10^3\), made available via the John Hopkins Turbulence Database (JHTDB) (Li et al. 2008; Perlman et al. 2007; Livescu et al. 2014). The images contain an average of \(15 \times 10^3\) particles per image, leading to a density of 0.2 particles per pixel. \(64 \times 10^3\) inertia-less particles were seeded randomly into the three-dimensional domain, and the particle trajectories were calculated using the Lagrangian tracking utility available within the JHTDB (Yu et al. 2012). The particles were then illuminated by a simulated laser sheet with a \(1/e^2\) width of 8 pixels, and the particle images were generated from the known particle positions using the method described by Schmidt and Sutton (2020).

a An example synthetic particle image in the buoyancy-driven mixing flow set, b LIC illustration of the DNS velocity field of a snapshot from the data and c vorticity field corresponding to the velocity field shown

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The dataset contains 13 sets of 100 image pairs of resolution \(256 \times 256\) pixels, with maximum displacements ranging from \(0.05-8\) pixels. A sample particle image from the synthetic data along with an LIC visualization of the true DNS flow field and the corresponding vorticity field is shown in Fig. 3. Like the Lamb–Oseen vortex flow detailed in Sect. 3.1.1, this flow contains a wide range of velocity magnitudes over short length scales and so is suitable for evaluating DVR.

3.2 Experimental dataset

3.2.1 Flow past a cylinder

In addition to investigating the DVR of PIV processing algorithms on synthetic data, it is also assessed on experimental particle images with appreciable locally varying velocity gradients in the turbulent wake of flow past a cylinder, acquired in the TRIREME water tunnel at Case Western Reserve University. The cylinder diameter is 5 cm, and the Reynolds number is \(1.55 \times 10^3\) with a free stream velocity of 0.4 m/s. Spherical hollow glass particles of diameter of \(\approx 8-10\) µm were seeded into the water, and image pairs were captured at a rate of 1 kHz using a Photron Nova S-12 camera equipped with an Olympus 85 mm lens at a resolution of \(1024 \times 1024\) pixels and an inter-frame time of 1 ms. Figure 4 shows the schematic of the experimental setup.

Setup for flow past a cylinder PIV experiments

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In total, 200 pairs of time-resolved particle images were captured in the region downstream of the cylinder in a field of view of \(5D \times 5D\) as indicated in Fig. 4. The flow field was illuminated using a dual-head Photonics Nd:YLF 527-nm laser, and the laser sheet is formed by a diverging lens with a focal length of \(f=-15\) mm. The data captured have maximum particle displacement \(\sigma _{\max } \approx 1.2-1.5\) pixels and a mean displacement of approximately 0.8 pixels. In order to investigate the DVR, frames are skipped when evaluating the images to produce larger inter-frame displacements. A sample image from the data acquired along with the LIC visualization of the corresponding velocity field obtained using wOFV is shown in Fig. 5.

a An example particle image from the experiment and b LIC illustration of the velocity field obtained using wOFV

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3.2.2 Jet flow

To further test the robustness of the dynamic range computation on experimental particle images, PIV imaging was also performed in a turbulent jet within the test section of the TRIREME water tunnel using the same tracer particles and laser as the above experiment. The jet diameter is 6.35 mm with a mean velocity of 1.2 m/s, giving a Reynolds number of \(5.8 \times 10^3\). Image pairs were captured at a rate of 5 kHz using a Photron Nova S-12 camera equipped with an a 75 mm Tokina AT-X Pro lens at a resolution of \(1024 \times 1024\) pixels and an inter-frame time of 0.2 ms. Figure 6 shows the schematic of the experimental setup.

Experimental setup for jet flow PIV experiments

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In total, 200 pairs of time-resolved particle images were captured in the region 20 D downstream of the jet in a field of view of \(25D \times 25D\) as illustrated in Fig. 6. The resulting data have a maximum particle displacement of \(\sigma _{\max } \approx 1-1.3\) pixels and a mean displacement of approximately 0.12 pixels. A sample PIV image (a) along with the LIC visualization (b) of the corresponding velocity field obtained using wOFV is shown in Fig. 7. The flow contains a wide range of particle displacements and strong gradients as well as three dimensionality.

a An example particle image from the jet flow experiment and b the LIC illustration of the velocity field obtained using wOFV

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As described in Sects. 1 and 2.1, the measure of DVR of a PIV system proposed by Adrian given in Eq. (1) does not provide a practically useful measure of the dynamic velocity range for designing a PIV experiment, as it fails to account for the wide range of motion present in fluid flows. This will be rigorously demonstrated in Sect. 5. An alternative measure of the dynamic range based on the error in the measurements is proposed in Eq. (5).

Here, similar to Eq. (1), the numerator \(\sigma _{\max }\) is the maximum displacement in the flow field. However, unlike in Eq. (1), the denominator \(\sigma _\tau\) quantifies the displacement at which the error in the measurement crosses a certain set error tolerance \(\varepsilon _\tau\), defined in terms of the velocity magnitude at the minimum relative error. Therefore, the minimum resolvable displacement given \(\sigma _\tau\) assesses the practical accuracy, rather than just the theoretical uncertainty in PIV processing given by \(\sigma _{\textrm{V}}\). A schematic for estimating \(\sigma _\tau\) is given in Fig. 8. First, the velocity field is determined for each image pair of the PIV dataset with a fixed maximum inter-frame displacement. As the ground truth is known, the exact relative error magnitude in the velocity field at each pixel is then computed for each of the estimated velocity vectors in the full dataset.

Once the relative error field has been determined, the data are sorted into bins based on the ground-truth displacement magnitudes \(|\vec {u}|\), such that a histogram of the relative errors is generated for each displacement bin. If one assumes that the errors in the velocity components are normally distributed and uncorrelated from each other, the error in the velocity magnitude is Rayleigh distributed, since a Rayleigh distribution results from the Euclidean norm of two uncorrelated normal distributions. For each histogram containing data from pixels with similar velocity magnitudes, a Rayleigh distribution is fitted to the relative error within that displacement bin, and the most probable error (\(\hat{\varepsilon _\sigma }\)) for each bin is computed. The results are then plotted as illustrated in Fig. 8. That is, the mode of each histogram obtained at different displacements is plotted against the pixel displacements. Then, for a certain tolerated relative error \(\varepsilon _\tau\), the corresponding pixel displacement \(\sigma _\tau\) can be easily determined from this plot.

A schematic explanation of estimating error-based DVR

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The value of \(\sigma _\tau\) so determined is essentially a measure of the smallest displacement the algorithm is able to measure up to a certain tolerated error \(\varepsilon _\tau\) in the presence of \(\sigma _{\max }\) in the flow field. This provides a much more practical measure of the smallest displacement resolved when compared to \(\sigma _{\textrm{V}}\), as \(\sigma _{\textrm{V}}\) only accounts for the theoretical limit, assuming that the full breadth of the motion is resolved with equal accuracy. In practice, for a given \(\sigma _{\max }\), an algorithm with a lower \(\sigma _{\tau }\) is desired as then the dynamic range, and hence, the accuracy of the range of velocity field resolved is maximized. Therefore, error-based dynamic range, \(DR^{\varepsilon }_V\), provides a practical and quantifiable measure of the range of velocities that are accurately resolved by a PIV processing algorithm below a set tolerated error \(\varepsilon _\tau\). The choice of \(\varepsilon _\tau\) is highly user- and experiment-dependent, but for this work we consider relative tolerated errors of 10-20% to be realistic.

A schematic explanation of estimating deviations in the experimental data for approximating dynamic velocity range

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As the ground truth is unknown in an experimental setting, evaluating the exact dynamic velocity range from experimental data is not possible. However, an alternative strategy based on deviation rather than the error can be used to approximate the DVR. The approach is similar to the one described in Fig. 8, except that error fields are replaced with deviation fields relative to a reference. The procedure for estimating the deviation fields is outlined in Fig. 9. First, time-resolved PIV data are acquired such that the maximum pixel displacement is small (\(\approx 1-2\) pixels). The acquired data are then temporally subsampled to obtain particle image pairs with larger maximum inter-frame displacements. The resulting particle image dataset now essentially represents the same PIV experiment with a different maximum particle displacement. The velocity fields are estimated using both the original and the subsampled PIV data. From the estimated velocity fields, the deviation fields for the temporally subsampled data with respect to the sum of the corresponding series of velocity fields from the original data can be computed. The resulting deviation fields are essentially a measure of the change in the velocity measurements as the inter-frame displacements are increased in the same flow, and serve as a surrogate model for the error. The intuition is that if a PIV processing algorithm is highly uncertain about a certain range of particle motion estimated, the deviations observed would be large for those displacements and there would likely be higher errors associated with those vectors. Therefore, by construction, the deviation-based method defines high DVR as the case when the uncertainty in the measurements is minimized. It should be noted that the sensitivity of the approach is experiment-dependent, as the range of motion present in the experiment affects the accuracy of the processing algorithms and the deviation-based metric is expected to track with the error-based DVR. An analysis similar to the one described for error-based DVR can be conducted on the deviation fields, by specifying a maximum deviation that one is willing to tolerate in the experiment to obtain an approximate measure of the dynamic velocity range of the processing algorithm. It is acknowledged that this characterization of the DVR is not perfect, as temporal nonlinearity in the velocity field may be responsible for some disagreement between the subsampled data and the fully time-resolved data. However, because the displacements are still quite small even for the subsampled data it is not expected that nonlinearity would be a significant source of error in these demonstration experiments.

5.1 Synthetic data

The results of computing the error-based DVR proposed in Sect. 4 for the two datasets described in Sect. 3 are presented in this section. For cross-correlation processing, all the results presented were computed at the settings (interrogation window size, number of passes, etc.) that maximized the accuracy of the velocity fields obtained, that is, minimized \(\sigma _\tau\). Similarly, as the ground truth is known, the regularization parameter is chosen to minimize the overall RMSE of the flow field in wOFV. Results are omitted when the error in all the computed displacements is greater than or equal to the tolerated error. This typically occurs when \(\sigma _{\textrm{max}} \ll 1\) or \(\sigma _{\textrm{max}} \gtrsim 10\). In the results that follow, CC refers to cross-correlation.

Relative error at maximum displacement vs maximum displacement evaluated on a Lamb–Oseen vortex flow and b buoyancy-driven mixing flow

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Figure 10 shows the percent error obtained at the maximum inter-frame displacement for (a) Lamb–Oseen vortex flow and (b) buoyancy-driven mixing flow plotted against the maximum displacement present in the flow field. The results show the trends that are well documented in the literature (Heitz et al. 2008; Liu et al. 2015), i.e., the error increases in both cross-correlation and OFV with the increase in maximum pixel displacement. For the Lamb–Oseen dataset, the error grows rapidly after about \(\sigma _{\max } \approx 5\) pixels is reached, which agrees with the results presented by Liu et al. (2015) and Schmidt and Sutton (2019). This is in line with commonly accepted guidelines for PIV processing with \(16 \times 16\)-pixel interrogation windows (Raffel et al. 2018). The results also show that wOFV estimates velocity fields with a higher accuracy when compared to cross-correlation (CC), which is typical for clean, synthetic 2D data (Schmidt et al. 2024).

Pixel displacement at different cutoff errors (\(\varepsilon _\tau\)) vs maximum pixel displacement evaluated on a Lamb–Oseen vortex flow and b buoyancy-driven mixing flow

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Figure 11 shows the displacement \(\sigma _{\tau }\) at three different tolerated errors \(\varepsilon _\tau\) for cross-correlation and OFV evaluated for (a) Lamb–Oseen vortex flow and (b) buoyancy-driven mixing flow plotted against maximum flow displacement. If the traditional definition of DVR provided in Eq. (1) gave a practical measure of the range of velocities accurately resolved, then it is expected that \(\sigma _\tau \approx \sigma _{\textrm{V}}\), and should remain constant with respect to \(\sigma _{\max }\), i.e., the minimum resolvable displacement should merely depend on the uncertainty in the processing algorithm’s ability to resolve motion of a particle given the imaging parameters and not the maximum displacement in the flow. However, Fig. 11 demonstrates that the minimum resolvable displacement \(\sigma _\tau\) increases with the maximum displacement \(\sigma _{\max }\) in the flow. Hence, as the maximum displacement in the flow field becomes larger, the processing algorithms struggle to accurately measure the small displacements at the same time. Therefore, if in a PIV experiment, the flow has a wide range of small and large displacements, additional care must be taken when choosing \(\Delta t\) to accurately measure the full breadth of the velocity range present in the flow. As expected, as the tolerated cutoff error \(\varepsilon _{\tau }\) is increased, the minimum resolvable displacement also increases at a fixed \(\sigma _{\max }\).

Error-based dynamic range vs maximum pixel displacement evaluated on a Lamb–Oseen vortex flow and b buoyancy-driven mixing flow

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The findings above are further buttressed by the results presented in Fig. 12, which shows the error-based dynamic velocity range proposed in Eq. (5) computed for the (a) Lamb–Oseen vortex flow and (b) buoyancy-driven mixing flow at different tolerated errors \(\varepsilon _\tau\), plotted against maximum pixel displacement in the flow. Theoretically, according to the conventional DVR in Eq. (1), the DVR should increase linearly with \(\sigma _{\max }\), as the minimum resolvable displacement \(\sigma _{\textrm{V}}\) is solely determined by the ability to detect the motion of a particle, and remains constant for a fixed imaging setup. However, as demonstrated by the results presented in Fig. 11, the minimum accurately resolvable displacement \(\sigma _\tau\) increases with \(\sigma _{\max }\). That is, the ability of the PIV processing algorithms to accurately resolve small displacements decreases with an increase in the maximum displacement in the flow field. Therefore, as Fig. 12 demonstrates, the dynamic velocity range actually stays nearly constant with respect to \(\sigma _{\max }\).

Furthermore, as the tolerated error \(\varepsilon _\tau\) increases, as shown in Fig. 11, the accurately resolvable minimum displacement \(\sigma _\tau\) decreases. Hence, as shown in Fig. 12, the error-based dynamic velocity range increases with an increasing \(\varepsilon _\tau\) at a fixed \(\sigma _{\max }\). In an experimental setting, this suggests that if a flow field has a wide range of velocity scales, as that is generally the case for turbulent flows, one should choose \(\Delta t\) such that both the large and the small velocity scales are accurately resolved, i.e., both the small-scale and the large-scale displacements are within the dynamic velocity range of each other. Figure 12 shows that wOFV typically exhibits a higher dynamic velocity range compared to cross-correlation, perhaps due to the global nature of the velocity field estimated.

The results show a nearly constant dynamic range for both PIV processing algorithms, which is unreported in the literature. The constant depends on the particular details of the image data and velocity field being estimated, as \(DR^{\varepsilon }_V\) depends on the error in the velocity estimates. Contrary to the results available in the literature, the practical dynamic velocity range of cross-correlation-based PIV processing is not \(\mathcal {O}\left( {10^2} \right)\), but is rather \(\mathcal {O}\left( {10^1} \right)\). Therefore, when performing a PIV experiment, care should be taken when drawing conclusions from vectors with small magnitudes because the smallest resolvable displacement does not remain constant as the largest displacements are increased by increasing \(\Delta t\).

Deviation-based and error-based dynamic range evaluated for Lamb–Oseen dataset computed using a CC and b wOFV at different cutoff error

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To investigate the feasibility of using the deviation-based DVR model proposed for experimental data, both error-based and deviation-based approaches are compared in Fig. 13 for the Lamb–Oseen vortex field dataset. Figure 13 presents the DVR computed using (a) CC and (b) wOFV results, at cutoff errors of 10% and 20%. It is observed that while the deviation-based approach consistently overestimates the DVR, it is still found to be close to the error-based DVR results and follows the same trend. Hence, it can serve as an approximate surrogate model, if perhaps an optimistic one, in experimental contexts where the error cannot be directly determined due to the lack of a ground truth.

In an experimental setting, traditionally, \(\Delta t\) is set such that the maximum particle displacements are a quarter of the interrogation window size (Raffel et al. 2018), which is approximately the largest displacement possible before correlation errors begin to occur. This would maximize the DVR if it were true that the minimum resolvable displacement \(\sigma _{\Delta X}\) governed the dynamic range, i.e., the DVR would scale with the maximum displacements. One quarter of an interrogation window is the practical limit beyond which cross-correlation algorithms become unstable and can fail to find accurate vectors. Therefore, operating close to this limit can lead to accuracy problems in practice and requires care. However, since the results presented in this manuscript indicate that the error-based DVR is nearly unchanged with respect to \(\sigma _{\max }\) for a given experiment, the user can select a smaller value for \(\Delta t\) than what would produce displacements of one quarter of an interrogation window, hence improving algorithmic stability, without fear of sacrificing dynamic range. It is also worth noting that practical tolerance level is experiment-dependent. For instance, if an experimentalist is merely concerned with estimating the bulk motion, it is possible that a large error might be tolerable. However, if the goal is to obtain high accuracy not only in the velocity field but also in the derivative quantities, or higher-order statistics, it is advised to lower the measurement error as much as possible. Practically speaking, from the results presented, a deviation error of 10–20% is realistic. Typically, lowering the error further requires additional care when conducting experiments.

5.2 Experimental data

To experimentally investigate the findings presented in Sect. 5.1, the DVR of cross-correlation and wOFV was approximated using the deviation-based method discussed in Sect. 4 for the experiments described in Sect. 3.2.1. The data acquired at 1 kHz were subsampled 4 times, i.e., to effective rates of 500, 250, 125 and 75 Hz to obtain maximum inter-frame displacements of about 4, 6, 8 and 10 pixels, respectively. The datasets obtained were processed using both cross-correlation and wOFV. A final interrogation window size of \(32\times 32\) pixels with \(50\%\) overlap was used for cross-correlation processing. The standard cross-correlation options such as window deformation, sub-pixel fitting and outlier removal were used to increase the accuracy of the results. For wOFV, the regularization parameter is adjusted based on user expertise to avoid over- or under-regularization.

a Percent deviation at maximum displacement and b \(\sigma _\tau\) at different cutoff deviation levels vs maximum displacement evaluated on the cylinder wake experimental data

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Figure 14a shows the effect of an increase in the maximum pixel displacement \(\sigma _{\max }\) on the percent deviation observed at \(\sigma _{\max }\). As \(\sigma _{\max }\) increases, an increase in the percent deviation at the maximum displacement is observed for both wOFV and cross-correlation. This is analogous to an increase in relative error with an increasing pixel displacement as presented in Fig. 10 in Sect. 5.1 for the synthetic data. This suggests that the uncertainty in the velocity estimates increases for both the processing methods as the pixel displacement is increased. It is also observed that wOFV shows less deviation in the results compared to cross-correlation suggesting a lower uncertainty in wOFV velocity estimates.

Trends in the smallest resolvable displacements presented in Fig. 11 for the synthetic data are also observed for the experimental data. Figure 14b demonstrates the effect of increasing \(\sigma _{\max }\) on the estimate of the smallest resolvable displacement \(\sigma _\tau\) obtained at different values of maximum tolerated deviation \(\varepsilon _\tau\). As the maximum displacement in the flow increases, the uncertainty in resolving smaller displacements increases. Therefore, \(\sigma _\tau\) grows with growing displacements. As the tolerance on the deviations is increased, \(\sigma _\tau\) decreases as a higher uncertainty is tolerated in the displacements estimated, as expected. It is again noted that wOFV consistently results in a smaller \(\sigma _\tau\) at given \(\varepsilon _{\tau }\) and \(\sigma _{\max }\) when compared to cross-correlation; hence, the uncertainty in the velocity fields obtained using wOFV is likely lower relative to cross-correlation results.

Deviation-based dynamic range vs maximum displacement evaluated for the experimental flow past a cylinder PIV data

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Similar to the findings presented for the synthetic data in Sect. 5.1, the results presented in Fig. 14b also suggest that the DVR of the processing algorithms does not increase with respect to \(\sigma _{\max }\), as \(\sigma _{\tau }\) also increases with an increasing maximum inter-frame displacement. The DVR approximated using the deviation-based approach for experimental data is presented in Fig. 15, which shows the deviation-based DVR plotted against the maximum displacement at different tolerated deviations for the experimental cylinder wake flow computed using cross-correlation and wOFV. The results are mostly consistent with the synthetic data results presented in Fig. 12 in Sect. 5.1, i.e., the DVR stays nearly constant, although it slightly decreases with increasing \(\sigma _{\max }\) in the experiment. wOFV again shows a slightly higher DVR compared to correlation.

To further investigate the robustness of the deviation-based DVR approach proposed, the deviation-based DVR was computed on another PIV experimental dataset imaged downstream of a turbulent jet flow, as described in Sect. 3.2.2. Figure 16 shows the deviation-based DVR (a) and \(\sigma _\tau\) (b) computed at different cutoff errors. The trends in Fig. 16 are consistent with the results presented in Fig. 15 in that it is again observed that the DVR of wOFV is larger than cross-correlation, and stays nearly constant with respect to \(\sigma _{\max }\), and the minimum accurately resolvable displacement \(\sigma _\tau\) increases with \(\sigma _{\max }\). It should be noted that a larger deviation-based DVR is observed in the case of jet flow compared to flow past a cylinder because the DVR proposed in the current work is flow- and experiment-dependent. In this case, the larger DVR is due to the fact that the flow contains two distinct regions: the jet core and the ambient surrounding fluid. The DVR in both regions individually is \(\mathcal {O} \left( {10^1} \right)\), but the overall DVR is larger because it is the ratio of the high velocities in the jet core to the smallest resolved vectors in the ambient region.

a Deviation-based DVR b \(\sigma _\tau\) at different cutoff deviation levels versus maximum displacement evaluated on the jet flow experimental data

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The analysis on the experimental data presented in this section corroborates the findings presented for the synthetic data in Sect. 5.1. The practical dynamic velocity range of cross-correlation-based PIV processing is again found to be \(\approx \mathcal {O}\left( {10^0} \right) -\mathcal {O}\left( {10^1} \right)\), rather than the commonly accepted value of \(\mathcal {O}\left( {10^2} \right)\). Therefore, the practical considerations provided at the end of Sect. 5.1 hold in an experimental context as well.

It is discussed that the theoretical definition of dynamic velocity range proposed by Adrian (1997) originating from uncertainty analysis of cross-correlation is not practically useful when designing a PIV experiment. This is because it fails to account for multiple experimental fluid flow effects such as the range of motion present in an experiment, multiple particles, etc. which affect the quality of the velocity estimates obtained for small-magnitude velocities. It is demonstrated that the smallest resolvable displacement, if it is to be measured with any degree of accuracy, is highly dependent on the maximum inter-frame displacement and not independent of it as predicted by the conventional model. Therefore, a new error-based criteria is proposed to determine the dynamic range of a PIV processing algorithm.

The error-based dynamic range \(DR^{\varepsilon }_V\) is evaluated for two synthetic test cases where the ground truth is known, and hence, the error in the velocity estimates can be accurately determined. It is found that the minimum accurately resolvable displacement is not the uncertainty limit imposed by the imaging system, but is rather dependent on the maximum displacement present in the flow through the processing algorithm. The findings from the synthetic data are further investigated and reinforced using a deviation-based approximate DVR model on the data acquired by PIV experiments conducted in the wake of a flow past a cylinder. The experimental investigation presented in Sect. 5.2 supports the findings from the synthetic data described in Sect. 5.1. The results also show that wOFV has a higher dynamic range compared to cross-correlation, although both are on the same order. Ultimately, it is found that the widely accepted and reported dynamic range of cross-correlation-based PIV is unreliable in facilitating the design of PIV experiments, as the practical dynamic velocity range of cross-correlation-based PIV processing is not \(\approx \mathcal {O}\left( {10^2} \right)\), but is rather \(\mathcal {O}\left( {10^0} \right) -\mathcal {O}\left( {10^1} \right)\). It is also found that the DVR does not change appreciably with respect to the maximum displacement, so it is not necessary to maximize the inter-frame displacement with respect to the interrogation window size, and thereby potentially incur inaccuracies do to algorithmic stability, in order to optimize the dynamic range. It is likely that factors such as the particle density and the signal-to-noise ratio of the images have an effect on the DVR as well, the characterization of which could be a subject of future research.