Assessment and application of wavelet-based optical flow velocimetry (wOFV) to wall-bounded turbulent flows

Optical flow describes the apparent displacement of brightness intensity patterns in an image sequence (Horn and Schunck 1981). The basic assumption in optical flow techniques is the conservation of a quantity in the image plane, typically brightness intensity along a point trajectory. This is expressed as the optical flow constraint equation (OFCE): where \(I({\varvec{x}},t)\) is the brightness intensity at pixel locations \({\varvec{x}}={\left({x}_{1},{x}_{2}\right)}^{T}\) in the image domain Ω and \({\varvec{U}}\left({\varvec{x}},t\right)= {\left({U}_{1}\left({\varvec{x}},t\right),{U}_{2}\left({\varvec{x}},t\right)\right)}^{T}\) is the two-dimensional displacement. Equation 5 is recognizable as a transport equation of a passive scalar in a divergence-free flow (Liu and Shen 2008). Assuming a constant velocity and a unit time interval between the image pair, Eq. 5 can be integrated to the displaced frame difference (DFD):

Equation 5 or 6 is known as the data term in optical flow literature. It establishes the relationship between a measurement in the image plane \(I({\varvec{x}},t)\) and the variable to be calculated \({\varvec{U}}({\varvec{x}})\). The data term is incorporated into a penalty function to be minimized, commonly a quadratic penalty as employed in the present study:

The data term however is ill-posed, as it relates a two-dimensional velocity to only one observed variable being the image intensity. This results in an ambiguous situation where only motion perpendicular to brightness gradient contours can be determined, known in literature as the aperture problem (Beauchemin and Barron 1995). Different methods of resolving the aperture problem exist (Barron et al. 1994). In the seminal work of (Horn and Schunck 1981), a variational approach was proposed to assimilate the data term together with an additional smoothness constraint known as the regularization term \({J}_{R}\), weighted by a scalar parameter \(\lambda\), into a minimization problem to solve for the image plane per-pixel displacement: where the caret (\(\widehat{}\)) denotes the final estimated quantity. The regularization term, which is solely a function of the velocity field, provides additional information about the velocity field to compensate for regions where motion cannot be determined from the data term alone, such as tangentially along image contours, as well as regions of constant, uniform image features where spatiotemporal image gradients vanish. \({J}_{R}\) affects the spatial coherence of neighboring velocity vectors and enforces a degree of regularity or visually perceived “smoothness” to the velocity field. Regularization also functions as a type of outlier rejection process during the minimization (Heitz et al. 2010) and reduces the susceptibility of the estimated vector field to noise and imaging imperfections. In the context of fluid velocimetry, regularization terms involving higher-order derivatives of the velocity field are preferred to better preserve velocity gradients and thus improve estimation of derived quantities such as vorticity and strain rate. The present work uses the Laplacian regularization in a quadratic penalty: with the continuous wavelet operator approximation described in (Kadri-Harouna et al. 2013). Laplacian regularization imparts a physically sound smoothing in a similar manner to viscosity in divergence-free two-dimensional flows (Schmidt and Sutton 2021). Using the Laplacian regularization provides nearly identical accuracy but with significantly less computing time than other high-order schemes such as the second-order divergence curl (Suter 1994) or viscosity-based regularization (Schmidt and Sutton 2021).

Variational optical flow techniques seek a per-pixel vector field transformation \(\widehat{{\varvec{U}}}\) that maps one image onto the subsequent that best: (1) conserves pixel brightness intensity and (2) enforces the regularity defined by \({J}_{R}\). The parameter \(\lambda\) in Eq. 8 establishes the relative importance of \({J}_{D}\) versus \({J}_{R}\) during the minimization process and determines the extent to which \({J}_{R}\) can deviate the estimated velocity field \(\widehat{{\varvec{U}}}\) from the constraint of brightness conservation in \({J}_{D}\). Lower \(\lambda\) values place a stronger emphasis on reducing \({J}_{D}\) during minimization, thus attempting to better match pixel intensities between \({I}_{0}\) and \({I}_{1}\) to reduce the DFD even if the intensity variations do not correspond to the true motion. This creates non-physical velocity fluctuations at fine scales visible as noise in \(\widehat{{\varvec{U}}}\). Increasing \(\lambda\) dampens the small-scale motion, however, a higher \(\lambda\) weighting can lead to excessive smoothing of the velocity field. Sensitivity analysis of this parameter is a key aspect in understanding the applicability of OFV as an alternative diagnostic technique to studying wall-bounded flows.

The current wOFV implementation was proposed by Dérian et al. (2013), developing on the original wavelet-based optical flow methods of Wu et al. (2000) and Chen et al. (2002) for computer vision applications. Improvements in the form of symmetric boundary conditions (Schmidt and Sutton 2020) and efficient implementation of high-order and physically sound regularization terms (Kadri-Harouna et al. 2013; Schmidt and Sutton 2021) have furthered the robustness and accuracy of this technique. For brevity, only an overview of the wavelet-based optical flow method is presented. Details of the wOFV algorithm can be found in (Dérian et al. 2013; Kadri-Harouna et al. 2013; Schmidt and Sutton 2019, 2020).

The principle of wOFV, in contrast to other OFV techniques, is to perform the minimization in Eq. 8, not over the physical velocity field \({\varvec{U}}({\varvec{x}})\), but over the wavelet coefficients \({\varvec{\theta}}= {\left({\theta }_{1},{\theta }_{2}\right)}^{T}\) from its discrete wavelet transform (DWT) \({\varvec{\theta}}=\boldsymbol{ }{\boldsymbol{\Psi }}^{-1}\left({\varvec{x}}\right){\varvec{U}}({\varvec{x}})\), where \({\boldsymbol{\Psi }}^{-1}\left({\varvec{x}}\right)\) denotes the wavelet transform decomposition operator. The minimization problem is then expressed as:

Broadly speaking, a wavelet transform extracts the frequency content of a signal (or image in 2D) at different scales of resolution (Mallat 2009). The wavelet transformed velocity field coefficients \({\varvec{\theta}}\) are optimized sequentially in a multiresolution strategy. The wavelet coarsest-scale coefficients are estimated first, before estimating coefficients associated with progressively finer scales until the pixel scale is reached. Previous coarse-scale velocity estimates are included in every level of estimation, therefore, earlier spurious vectors from coarser-scale velocity estimates are corrected for as finer-scale motion is determined. Once the full minimization is complete, the velocity field in physical space is recovered by application of the DWT reconstruction operator \(\boldsymbol{\Psi }\left({\varvec{x}}\right)\) to the output wavelet coefficients \(\boldsymbol{\Psi }\left({\varvec{x}}\right)\widehat{{\varvec{\theta}}}=\boldsymbol{ }\widehat{{\varvec{U}}}\).

To cope with large displacements, traditional OFV methods commonly use multiresolution coarse-to-fine warping strategies (Heitz et al. 2010), which extend the achievable dynamic range. This approach, however, can lead to propagation of errors during the multi-scale estimation process with no possibility of posterior correction. Conversely, the multiresolution framework inherent in wavelet decompositions provides a natural scheme that is well-suited to represent the multi-scale nature of turbulence (Deriaz and Perrier 2009; Farge et al. 1996). Decomposition of the velocity field across the wavelet basis functions also allows for accurate implementation of high-order derivatives (Beylkin 1992) used in the calculation of \({J}_{R}\). Previous studies demonstrated wOFV to be among some of the more accurate existing modern OFV methods, see (Cai et al. 2018; Kadri-Harouna et al. 2013; Dérian et al. 2013; Schmidt and Sutton 2019).

In this work, the wOFV implementation uses the odd length biorthogonal nearly Coiflet wavelet basis (BNC 17/11) introduced by (Winger and Venetsanopoulos 2001). This basis has a nearly maximum number of vanishing moments possible for a given biorthogonal wavelet filter size. Maximizing the number of vanishing moments increases velocity estimation accuracy up to a degree (Dérian et al. 2013). This wavelet family is notable for the improved retention of fine details in its wavelet transform partial reconstructions which are implicit in the multi-scale estimation process in wOFV methods. Since the basis is biorthogonal, it is implemented using the non-expansive symmetric boundary condition described in (Schmidt and Sutton 2020) which eliminates boundary artifacts resulting from a lack of periodicity in the imaged motion.

Table 1 describes the simulation parameters of the JHTDB, which are stored in a nondimensional form based on the half-channel height \(h\). One hundred temporally correlated velocity fields are extracted with a time separation of 2.5 \(\delta t\) (stored) DNS database timesteps from a subset of the DNS domain that includes the no-slip velocity grid point of the lower wall. The fields are of nondimensional size \(0.17h\times 0.17h\times \left(5\times {10}^{-4}\right)h\) and sampled from the database at a grid resolution of 1024 \(\times\) 1024 \(\times\) 3 using fourth-order Lagrange polynomial interpolation (Berrut and Trefethen 2004).

Once the velocity fields from the DNS are extracted, it is necessary to determine tracer particle displacements between frames of each image pair as they are advected by the DNS velocity. For the initial frame of each image pair, synthetic particle tracer locations are initialized from a random distribution for each of the extracted DNS velocity fields. The velocity field is assumed to be constant between consecutive images and the tracers assumed to be spherical and massless. The displacement of each particle in each second frame is computed numerically using an explicit Runge–Kutta scheme (Dormand and Prince 1980) and a modified Akima spline interpolation for the velocities at particle locations (Akima 1974). The velocity fields and particle displacements are then scaled from the nondimensional DNS units to pixel displacements per unity interframe time interval (\(dt=1\)) such that the maximum image plane velocity magnitude corresponds to ~ 3.5 \(px/dt\), and the maximum out-of-plane displacement is ~ 0.8 \(px/dt\).

The particle image pixel intensities are determined using classical methods of synthetic particle image generation (Raffel et al. 2018). The maximum particle intensity is governed by its diameter \({d}_{p}\) and out-of-plane position \({x}_{3,p}\) within a Gaussian profile synthetic laser sheet:

The laser sheet position \({x}_{3,LS}\) is centered in the middle of the extracted DNS domain. The standard deviation of the laser sheet profile is set to \({\sigma }_{LS}=2\) such that the \(1/{e}^{2}\) profile thickness is equal to the out-of-plane \({x}_{3}\) thickness of the DNS volume subsection. In this way, the out-of-plane particle displacement is less than 1/4 of the laser sheet thickness as recommended by (Adrian and Westerweel 2011). Each particle is randomly assigned a diameter that is drawn from a log-normal distribution of values: with parameters \(\mu =0.90\) \(px\) and \(\sigma =0.76\) \(px\) for the mean and standard deviation, respectively. The particle seeding density is 0.03 particles per pixel² (PPP), representative of that estimated from the experimental data presented in Sect. 5. The in-plane pixel intensity is computed from the integral form of the Gaussian function solved analytically using error functions. This is a more representative method of how a camera integrates the light intensity over individual pixels compared to simply using the analytical Gaussian expression. Finally, after the pixel intensities have been determined, the values are scaled to the dynamic range of an 8-bit camera sensor and rounded to integers to mimic discretization.

Once the particle images are rendered, the images are vertically shifted upwards by 160 pixels to create a masked wall region of zero intensity. This vertical shift avoids having the flow region near the bottom of the image where boundary conditions in the wavelet transforms of wOFV can affect the near-wall velocity estimates. Moreover, this shift of the flow region from the image boundary is consistent with experimental images presented in Sect. 5. As this masked region in the images has 0 intensity, it does not contribute to the DFD and is effectively ignored in the minimization (Schmidt and Woike 2021). An example of a rendered particle field image is shown in Fig. 1.

This section evaluates wOFV’s ability to estimate the mean velocity behavior within the boundary layer by analyzing the normalized velocity profiles depicted by \({u}^{+},{y}^{+}\). The effect of lambda on wOFV to accurately calculate \(\langle {U}_{1}\rangle\) and the near-wall velocity gradient \(\gamma\) is first assessed, since these variables are necessary to calculate \({u}^{+}\) and \({y}^{+}\). Subsequently, the fidelity of wOFV to resolve the various turbulent boundary layer regions is evaluated. The ensemble mean velocity field presented in this section is composed from 100 velocity images and is evaluated separately for each \(\lambda\), as well as for PIV.

Experiments are conducted in a flow facility in which a jet flow impinges onto a parallel wall, creating a developing turbulent boundary layer. The flow facility was originally designed to study flame-wall interactions in a side-wall quenching (SWQ) configuration (Jainski et al. 2018; Kosaka et al. 2018, 2020; Zentgraf et al. 2021, 2022). For the purposes of this study, the flow facility operates under non-reacting, cold-flow conditions (i.e., no combustion). This experimental setup was recently presented in (Zentgraf 2022) for characterizing the nozzle exit velocity profiles of the SWQ-burner. A schematic of the facility is shown in Fig. 10a. The central main flow (fully premixed CH₄/air at \(\phi =1.00\); not ignited) was homogenized by meshes as well as a honeycomb structure and subsequently guided through a converging nozzle. At the square nozzle exit (\(\approx 40\times 40\, {\mathrm{mm}}^{2}\)), the Reynolds number was maintained at 5900 and the inflow conditioning yielded a streamwise (\({x}_{1}\)) velocity profile with a nearly top-hat shape (Zentgraf 2022). For turbulent conditions at the nozzle exit, a turbulence grid was used, providing a turbulence intensity of 6–7% (Jainski et al. 2018). The outlet flow impinged the sharp leading edge of a stainless steel wall. The wall’s surface has a mild curvature for improved optical access (radius: 300 mm, see top view in Fig. 10b). The central main flow was shielded from the laboratory environment by a concentric square air co-flow. All flows operated at ambient temperature, which was in agreement with the wall temperature.

A low-speed (10 Hz) PIV setup was used as shown in Fig. 10b. This setup was used previously to characterize the velocity profiles at the nozzle exit as boundary conditions (Zentgraf 2022) and its optical arrangement closely matched the high-resolution, high-speed realization in (Zentgraf et al. 2021). The main flow was seeded with Al₂O₃ particles (Zentgraf et al. 2021) which were illuminated using a dual-cavity Nd:YAG PIV laser (New Wave Research, Gemini PIV, G200, 10 Hz, 532 nm). Laser pulses were separated by \(\Delta t=40\)μs. The laser sheets were guided vertically downward to the wall to minimize reflections at the wall. Measurements were taken in the \({x}_{1}{x}_{2}\)-symmetry plane of the facility, at the wall’s centerline (x₃ = 0 mm). The origin of the coordinate system is defined at the leading edge of the wall along its centerline. Optics exposed to seeding were continuously purged by nitrogen during operation.

The resulting Mie scattering was detected by a sCMOS camera (LaVision GmbH, Imager sCMOS) with an exposure time of 15 μs each frame. The camera was equipped with a 180-mm objective lens (Sigma, APO Macro DG HSM D, f/8) and a bandpass filter (Edmund Optics Inc., #65–216, central wavelength 532 nm, FWHM 10 nm) to suppress ambient light. The field of view (FOV) comprises \(\left(\Delta {x}_{1}, \Delta {x}_{2}\right)\approx (40 \mathrm{mm}, 47.5 \mathrm{mm})\). For velocimetry, the images are cropped to \(\left(\Delta {x}_{1}, \Delta {x}_{2}\right)\approx (38 \mathrm{mm}, 38 \mathrm{mm})\), comprised of 2048 \(\times\) 2048 pixels with the FOV beginning at the wall’s leading edge. At the downstream edge of the FOV, the Reynolds numbers based on the momentum thickness and friction velocity are \({Re}_{\theta }=100\), \({Re}_{\tau }=70\).

For the experimental dataset, wOFV is benchmarked against PIV, as well as PIV + PTV, the latter of which is often used in experiments to improve vector resolution over PIV. It is emphasized that experiments were originally optimized for PIV/PTV. The seeding density was optimized to provide 6–8 particles per final interrogation window and particle displacement was within ¼ of the final interrogation window size in the near-wall region of investigation. Velocity vector fields achieved average cross-correlation values of 0.77. It is therefore emphasized that the PIV quality is not intentionally compromised to exaggerate the advantages of wOFV.

The wOFV and PIV velocity fields were processed similarly to that of the synthetic data in Sect. 4. Mie scattering images were first pre-processed with subtraction of the ensemble minimum image followed by a min–max intensity normalization (Adrian and Westerweel 2011). PIV vector processing was performed using a multi-pass correlation with an initial IW size of \(64\times 64\) down to \(16\times 16\) with 75% overlap. The same anisotropic denoising filter used to optimize the PIV results on the synthetic data was applied to the experimental PIV vector fields. PIV + PTV processing was initialized from PIV. PTV was calculated for a particle size range from 1 to 8 pixels and with a correlation window size of 8 pixels. PTV vectors were converted to a structured \(4\times 4\) pixels² grid, as performed in previous boundary layer studies (Ding et al. 2019; Schmidt et al. 2021). This step was performed in DaVis using a “simple averaging / strong filter” scheme in DaVis, which provided the most reliable PTV results. A \(3\times 3\) Gaussian smoothing filter was applied to remove noise in the PTV vector fields. wOFV was performed as described in Sect. 2.

Each velocimetry method provides a different vector spacing and spatial resolution. For PIV, the spatial resolution is 298 μm, as defined by the final IW size of \(16\times 16\), while 75% overlap provides a vector spacing of 74.3 μm (every 4 pixels). The converted \(4\times 4\) pixels² grid used for PIV + PTV provides a vector spacing of 4 pixels or 74.3 μm, which is equivalent to PIV. Since PTV assigns a vector to the centroid of each detected particle, an approximate PIV + PTV spatial resolution is reported as the average particle distance of 5.8 pixels or 107.8 μm. wOFV provides a per-pixel vector spacing of 18.6 μm. A conservative estimate of wOFV’s spatial resolution is reported as the average particle spacing of 107.8 μm. As mentioned in Sect. 4, wOFV’s true spatial resolution is likely to be smaller than the average particle spacing since each particle pixel contains a valid vector, which likely makes the particle centroid spacing an upper limit.

The first vectors from the wall are located 279 μm, 204 μm and 149 μm for PIV, PIV + PTV and wOFV, respectively. These distances are based on geometric masks used to calculate vector fields that are offset from the wall location to avoid light reflections and reduce the frequency of spurious vectors at the wall for both methods. The wall location is approximated using the maximum intensity of the reflection present at the wall. This is then refined using the no-slip pixel position estimate from PIV and wOFV \(\lambda =0.1\) \(\langle {U}_{1}\rangle\) profiles averaged over the downstream distance.

It should be emphasized that the experimental dataset is appreciably different from the synthetic dataset in Sect. 4, and this influences the optimization of wOFV. For example, the near-wall velocity gradient \(\gamma\) is larger and the viscous sublayer is much thinner for the experimental data than the synthetic data. In addition, the image size is \(2048\times 2048\) px compared to \(1024\times 1024\) px in the synthetic data. Therefore, not only is the absolute pixel-wise length halved, but the viscous sublayer comprises a smaller and less significant proportion of the full image FOV. Lastly, while the synthetic data have an average freestream particle displacement of 3 pixels, the experimental freestream flow field has a more substantial displacement of 6 pixels. All of these aspects, in addition to different image characteristics, will influence the regularization weighting for wOFV, such that a suitable range of \(\lambda\) values will be significantly different between the experimental and synthetic datasets. This aspect is common within optical flow literature (Kadri-Harouna et al. 2013). In fact, the experimental data are significantly stricter and less forgiving compared to the synthetic data regarding selection of an acceptable \(\lambda\). The absence of ground truth data means determining the true optimal \(\lambda\) is not possible. For the experimental data, wOFV results from three values of \(\lambda =0.1, 1, 20\) are presented and the most appropriate \(\lambda\) is justified a posteriori based on physical principles as well as general comparison to PIV.

Assessment of wOFV first considers the instantaneous velocity field in comparison with PIV and PIV + PTV. Figure 11 shows an instantaneous velocity magnitude field for PIV, PIV + PTV, and the three wOFV results. An insert is shown for each image, which highlights details of a low-speed streak emerging from the wall.

PIV performs a good job resolving the overall velocity field. However, PIV can often struggle to resolve the velocity near the wall as shown by the pockets of unresolved velocity regions near the wall. PIV + PTV resolves closer to the wall than PIV but resembles a noisier velocity field with similarity to the noise seen in \(\lambda =0.1\). It is noted that converting PTV to a larger grid size of \(8\times 8\) pixels² did not reduce the noise level in the PIV + PTV.

All three wOFV results resolve similar general features as PIV, but the quality of the velocity field is determined by the choice of \(\lambda\). wOFV with \(\lambda =0.1\) exhibits the high-frequency, speckle-like noise commonly associated with highly under-regularized findings. While \(\lambda =0.1\) resolves much of the same larger scale features as PIV and PIV + PTV, several artifacts of locally higher and lower velocities exist throughout the image. For wOFV with \(\lambda =1\), the high-frequency noise is removed and the velocity field has strong agreement with PIV and PIV + PTV. The primary differences between \(\lambda =1\) and PIV are that wOFV does not have spurious or missing vectors near the wall and wOFV resolves velocities closer to the wall. In addition, \(\lambda =1\) does not contain the speckle-like noise presented in PIV + PTV. For \(\lambda =20\), the larger flow features are well captured, but the finer-scale features present in PIV, PIV + PTV, and \(\lambda =1\) are mostly removed, likely due to over-smoothing.

The fact that noticeable changes in the velocity field occur over a significantly smaller \(\lambda\) range confirms the challenge in selecting the appropriate \(\lambda\) for the experimental dataset. While it is not possible to determine a \(\lambda\) value that provides the highest accuracy, it would appear that \(\lambda =0.1\) is too under-regularized and \(\lambda =20\) is likely over-regularized. Further analysis of the findings within the turbulent boundary layer is performed to evaluate these aspects and to determine the suitability of \(\lambda =1\).

The near-wall velocity profiles are shown in Fig. 12a, with the normalized profiles shown in Fig. 12b. The \(\langle {U}_{1}\rangle\) values are produced from a 100 image mean and the profiles are spatially averaged over a 2 mm streamwise \({x}_{1}\) distance centered at the location marked by the gray dashed line in Fig. 11. The near-wall gradient \(\gamma\) is calculated from the ensemble average streamwise velocity fields using a linear regression in a similar manner to that described in Sect. 4.2.2. In the viscous sublayer (\({y}^{+}<5\)), there are 15 velocity vectors for wOFV compared to 3 velocity vectors for PIV and 4 for PIV + PTV. The linear regression for wOFV and PIV + PTV uses each of the available vectors, while for PIV, only 2 out of the 3 available vectors are used since the final PIV vector nearest to the wall is frequently spurious. As described in Sect. 4.2, the estimation of \(\gamma\) has a direct effect on normalized wall units through \({u}_{\tau }\). The \(\gamma\) values calculated are \({\gamma }_{PIV}=1919\), \({\gamma }_{PTV}=2402\), \({\gamma }_{\lambda =0.1}=2469\), \({\gamma }_{\lambda =1}=2288\), \({\gamma }_{\lambda =20}=1701\) 1/s, which provide the corresponding \({u}_{\tau }\) values \({u}_{\tau , PIV}=0.1697\), \({u}_{\tau , PTV}=0.1898\) \({u}_{\tau ,\lambda =0.1}=0.1924\), \({u}_{\tau ,\lambda =1}=0.1853\), \({u}_{\tau ,\lambda =20}=0.1597\). Incorrect estimates of \(\gamma\) can result in a strong offset from the exact \({u}^{+}={y}^{+}\) formulation shown by the green dotted line in Fig. 12b. Comparison with this linear relation will be used as an approximate measure to judge the quality of the near-wall vectors in the absence of a ground truth velocity.

In Fig. 12a, the profiles above 1 mm from the wall are in excellent agreement. For \({x}_{2}<1\) mm, the \(\lambda =20\) profile shows increasing deviation from all other profiles as the wall is approached. In particular, velocity gradients are weaker leading to a flatter curve and significantly higher velocities at the wall. These features clearly indicate that \(\lambda =20\) is over-regularized; the excessive smoothing washes out the velocity gradient at the wall, creating an underestimate of \({u}_{\tau }\). The resulting normalization creates a strong deviation from \({u}^{+}={y}^{+}\) as shown in Fig. 12b and demonstrates that \(\lambda =20\) is not appropriate since the \(\gamma\) estimation is compromised.

In Fig. 12a, good agreement is shown between \(\lambda =0.1, 1\), PIV, and PIV + PTV until \({x}_{2}<0.4\) mm, where PIV shows a milder gradient for \(0.3\le {x}_{2}\le 0.4\)mm followed by a sharper velocity gradient at the last PIV data point. As will be shown, the last PIV data point is often erroneous, which biases the interpreted flow behavior. In Fig. 12b, PIV is offset from the \({u}^{+}={y}^{+}\), with an abnormal deviation in the curve for the last data point. PTV stays in closer agreement with \(\lambda =0.1, 1\) and is able to resolve closer to the wall than PIV although not to the same extent as wOFV. The resulting normalization to inner variables results in significantly closer alignment with \({u}^{+}={y}^{+}\) for PIV + PTV, although a slight offset remains. The \(\lambda =1\) result, on the other hand, shows perfect alignment with the \({u}^{+}={y}^{+}\) relation and remains in good agreement with a discrepancy of 0.04 \(\delta {u}^{+}\) at the final vector. This suggests that \(\lambda =1\) provides accurate velocity estimates near the wall, as well as an accurate \(\gamma\) estimate. This also indicates that PIV, and to a lesser extent PTV, struggles to correctly estimate \(\gamma\) causing a slight shift in the normalized velocity profile, but not to the same extreme as \(\lambda =20\). The \(\lambda =0.1\) result exhibits the highest \(\gamma\) at the wall, creating a down- and rightward shift in the normalized velocity profile. This shift was not seen for the under-regularized values in the synthetic dataset, which further emphasizes the higher sensitivity of \(\lambda\) for the more challenging experimental dataset compared to the synthetic data.

The streamwise turbulent velocity fluctuations \({u}_{1}={U}_{1}-\langle {U}_{1}\rangle\) are analyzed to further evaluate the capabilities of the velocimetry techniques. Velocity fluctuations provide an assessment of the data quality beyond the ensemble mean and are equally important to evaluate turbulent quantities in the boundary layer. Figure 13 shows the profile of the normalized streamwise velocity fluctuations \({\langle {u}_{1}{u}_{1}\rangle }^{+}\). The fluctuations and wall-normal coordinate in Fig. 13 are normalized by the \({u}_{\tau }\) estimated from \(\lambda =1\) since this \({u}_{\tau }\) value provided the strongest agreement with \({u}^{+}={y}^{+}\). Normalizing each case by \({u}_{\tau ,\lambda =1}\) removes the biased curve shifts as shown in Fig. 12. Similar to \(\langle {U}_{1}\rangle\), the fluctuation profiles are spatially averaged across the 2 mm streamwise distance with the extent of a single standard deviation of these fluctuations illustrated by the shaded area in Fig. 13. The standard deviation of the fluctuations within this 2 mm distance can be considered indicative of the reliability of the velocity estimate and its susceptibility to error.

In Fig. 13, each curve follows a relatively similar trend from \({y}^{+}=200\) to \({y}^{+}=10\); \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) values increase from the freestream region and exhibit a local maximum in the buffer layer at \({y}^{+}\approx 10\) as seen in other boundary layer studies (e.g., (Spalart 1988)). From \({y}^{+}=10\) toward the wall, each curve exhibits different trends. For PIV, \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) values continue to increase quite substantially into the viscous sublayer. This trend is non-physical as the turbulent fluctuations are expected to decrease in the viscous sublayer as the wall is approached. PIV also exhibits a very large standard deviation below \({y}^{+}=10\), which is primarily caused by spurious vectors within the last 2–3 PIV vectors. This feature illustrates the challenges of PIV to accurately resolve small-scale fluctuations in the presence of strong velocity gradients. Reliable PIV measurements are often challenging directly near surfaces. While ensemble average PIV quantities can be represented with sufficient accuracy, higher-order velocity statistics and instantaneous velocity fields more clearly reveal challenges with PIV. PIV + PTV shows improvement from PIV; PTV resolves a greater extent of the buffer layer peak and initially shows the expected decrease in \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) toward the wall. However, PIV + PTV still shows the non-physical increase in \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) within the final 2–3 vectors at the wall and contains a large standard deviation. Although these artifacts are less severe compared to PIV, they demonstrate that PIV + PTV can still struggle to accurately resolve the flow nearest the wall.

wOFV findings, on the other hand, do not exhibit such large deviations in \({\langle {u}_{1}{u}_{1}\rangle }^{+}\), indicating that wOFV is less susceptible to the same errors as PIV and PIV + PTV near the wall. Indeed, \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) values are large for \(\lambda =0.1\) due to the results being under-regularized; however, \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) values and their deviation are significantly lower than those for PIV or PIV + PTV near the wall. Below \({y}^{+}=10\), all wOFV findings show the expected decrease in \({\langle {u}_{1}{u}_{1}\rangle }^{+}\). The profile \(\lambda =20\) shows a milder peak near \({y}^{+}=10\) and a milder decrease near the wall compared to the other wOFV findings. The \(\lambda =20\) velocity is over-regularized for which excessive smoothing reduces the variation between peak and trough in the curve from \({y}^{+}=10\) to \({y}^{+}=1\). \({\langle {u}_{1}{u}_{1}\rangle }^{+}\) values for \(\lambda =1\) show the greatest decrease as the wall is approached, which follows the expected trend of turbulent fluctuations being suppressed within the viscous sublayer in close proximity to the wall. In addition, the extent of the shaded region for wOFV remains constant near the wall suggesting that velocity errors are not being influenced by the proximity of the wall. Overall, wOFV with \(\lambda =1\) shows the most promising findings in terms of ensemble average values, as well as behavior of the velocity fluctuations.

An example is presented which highlights the advantages of wOFV in resolving turbulent flow phenomena within a boundary layer. This example is demonstrated for an instantaneous velocity field comparing the optimized wOFV with \(\lambda =1\), PIV and PIV + PTV.

One of the added benefits of wOFV over PIV or PIV + PTV is the improved vector spacing together with physically sound smoothing, and with that, the ability to better resolve velocity gradient quantities. Figure 14 shows the instantaneous vorticity field \(\omega\) for PIV, PIV + PTV, and wOFV. The vorticity is calculated using the 8-point circulation approach described in (Raffel et al. 2018). Individual turbulent structures with relatively high vorticity magnitude are generated near the wall’s leading edge and are advected downstream within the developing boundary layer. The inlays shown in Fig. 14 highlight a region that captures a prograde vortex that was generated from the wall’s leading edge. This is a particularly challenging region because of the vortex’ proximity to the wall, where small pixel displacements coupled with the spatially varying sharp velocity gradients present difficulties for velocimetry techniques. Indeed, PIV has been used to resolve small-scale vortex structures near walls, but this is often accomplished by using high image magnifications yielding FOVs smaller than 5 × 5 mm² (Jainski et al. 2013), rather than a large FOV that is present in the current work.

The overall vorticity fields calculated from the velocimetry techniques are in good agreement; all methods show similar overall features such as the high vorticity regions extending from the wall’s leading edge. PIV + PTV shows higher fluctuations in the vorticity field than PIV and wOFV. This attribute is no doubt due to the higher degree of speckle-like noise present for PTV as shown in Fig. 11. The inserts in Fig. 14 highlight the capabilities of resolving the finer vorticity structures near the wall for each method. Overall, the same spatial distribution of positive/negative vorticity structures is captured by each method, however, the effect of greater vector resolution is immediately seen; in particular, PIV and PIV + PTV images are substantially more pixelated compared to wOFV. PIV can exhibit larger discontinuities in the vorticity field (i.e., larger changes from pixel-to-pixel), which are absent in the PIV + PTV and wOFV results, with wOFV achieving a highly resolved and more continuous vorticity field. The PIV + PTV vorticity field deviates more substantially from PIV and wOFV with several strands of high vorticity extending from the larger vorticity structures. These elevated vorticity strands are likely due to elevated noise levels present in PIV + PTV as discussed in Figs. 11 and 13. wOFV is able to resolve the vorticity much closer to the wall and without troublesome unresolved regions from erroneous vector calculation as in the PIV and PIV + PTV fields. wOFV faithfully preserves the features shown in both the PIV and PIV + PTV results, but achieves a much finer-detailed vorticity field.

Figure 15 shows the corresponding vector field within the green rectangle shown in Fig. 14. The vector field shows all available vectors for PIV, PIV + PTV and wOFV shown in red, blue and black, respectively. wOFV is capable of resolving the prograde vortex in much more detail than the other methods. While the vortex is visible in PIV and PIV + PTV, the vortex structure is more difficult to interpret due to sparser vector spacing and the presence of quasi-erroneous vectors that deviate from a vortical flow pattern. In Fig. 15, all vector fields show good agreement above \({x}_{2}=0.6\) mm; most vectors are in alignment and are of the same magnitude. However, closer to the wall, there are larger disagreements between wOFV and PIV. In many locations, PIV vectors are aligned orthogonally to wOFV vectors. Some PIV vectors are clearly erroneous as they differ significantly from their neighboring PIV vectors. Additionally, PIV vectors are absent in the upper left corner where spurious vectors are detected and removed during post-processing. Closest to the wall, PIV vectors point inwards toward the wall with a relatively large velocity magnitude, which strongly disagree with the wOFV vectors directed parallel or outward from wall with a velocity magnitude more consistent with the neighboring vectors. PIV + PTV improves on PIV in this regard with suitable quality vectors at the wall and calculates vectors in all regions. However, PIV + PTV exhibits select vectors that disagree with PIV and wOFV. In addition, PIV + PTV vectors near its vortex core center are misaligned with its circulation and struggle to resemble a coherent vortex core. It is likely that PIV + PTV struggles to successfully resolve the strong gradients present in this region. The velocity field features shown in Fig. 15 reveal some challenges cross-correlation-based PIV and combined PIV + PTV experience in resolving small-scale intricate flow dynamics with high velocity gradients in the vicinity of physical boundaries. Assuming a suitable \(\lambda\) is selected, these findings positively indicate that wOFV is better suited to resolve these turbulent flow structures in the boundary layer region.

Lastly, to assess the potential of resolving fine-scale turbulent velocity fluctuations using wOFV, the normalized streamwise turbulent kinetic energy spectrum (\({E}_{11}^{*}\left({\kappa }_{1}\right)\)) is analyzed. This is calculated using the Fourier transform of the streamwise velocity fluctuations (\({u}_{1}\)) across the entire field of view. The 1D turbulent kinetic energy spectrum, normalized by its peak value, is presented in Fig. 16 for PIV, PIV + PTV and the optimized wOFV with \(\lambda =1\). Due to the moderately low turbulence level, there is insufficient separation of scales to produce a significant inertial subrange (− 5/3 region). The spectra reveal a high-frequency noise present for PIV at increasing wavenumbers. The PIV spectra do not show the classical energy decay at increasing wavenumbers, indicating the velocity measurement noise floor and spectral resolution limit have already been reached. The PIV + PTV spectra do not show the high-frequency noise present in the PIV profile. However, PIV + PTV spectra show elevated energy at all wavenumbers compared to PIV together with a non-physical modulation after \({\kappa }_{1}>2\times {10}^{4}\). wOFV is in close agreement with PIV and PIV + PTV at the low wavenumbers, but shows an energy decay at higher wavenumbers and resolves a significantly greater proportion of the energy spectrum without obvious indications that the measurement is being corrupted by noise or accuracy issues at high wavenumbers.