Accurate near-wall measurements in wall bounded flows with optical flow velocimetry via an explicit no-slip boundary condition

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.


Introduction
Accurate near-wall velocity measurements in wall bounded turbulent fluid flows remain an extraordinary challenge despite decades of effort by experimentalists.The inherent difficulty in making near-wall velocity measurements is posed by the presence of very small length scales at which the velocity measurements are desired.Commonly used hot wire and hot film anemometry methods are invasive and cannot provide spatially resolved data, and optical methods such as cross correlation-based particle image velocimetry (PIV) cannot provide the spatial resolution needed for near-wall measurements, even if imaging artifacts such as wall reflections are eliminated.Accurate, resolved measurements require a method that avoids spatially averaging the smallest scales while also being non-invasive.One such method, which we shall apply in the present work, is optical flow velocimetry (OFV).
OFV is an alternative to cross correlation-based PIV that uses experimental image data to determine the velocity fields of fluid flows.PIV is a standard technique in fluid mechanics, and the reader is referred to one of several reference texts, e.g.[1], for more information.Briefly, a pair of images of tracer particles are acquired with a known time separation ∆t.The displacements of groups of particles within finite-sized interrogation windows are determined using cross correlation, and the displacements are interpreted as velocity using ∆t.In OFV, on the other hand, instead of using the cross correlations between interrogation windows across a pair of images to determine the averaged displacement of each window, the velocity field is estimated by calculating the displacement at each pixel using optical flow, a technique originating from the computer vision community [2,3].OFV has been demonstrated by the authors and others to yield a higher resolution estimate of the velocity field with increased global accuracy compared to correlation-based PIV [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].A subset of OFV methods are wavelet-based OFV, or wOFV, which is described in section 2.1.
To date, OFV methods have largely been applied to synthetic homogeneous, isotropic turbulence data from simulations in order to compare to conventional PIV.A few recent studies have applied OFV in experiments involving free shear flows as well [19,21].In each of these applications, there are no solid boundaries within the field of view, and so no special considerations need to be taken during the velocity calculation to deal with the sharp gradients imposed by the solid wall(s).The most common boundary condition enforced on the borders of the domain is that the velocity field is periodic, or that the velocity gradients are zero, i.e. a Neumann condition.In a recent study, Nicolas et al [20] applied wOFV to a wall-bounded turbulent flow to study the effects of regularization, described in equation ( 1) in section 2.1, on the near wall velocity structures.They observed that while wOFV increases the fidelity of the velocity measurements compared to cross correlation-based PIV with an appropriately chosen regularization parameter, it still fails to resolve the full turbulent boundary layer profile close to the wall without sacrificing the far-field velocity estimates made by wOFV.This is due to the lack of an explicit no-slip boundary condition, and the authors ultimately concluded that there is a need for a method that enforces the no-slip boundary condition on the wall to resolve the full boundary layer profile while maintaining the gain in accuracy in the far-field region over correlation-based PIV.
In wall bounded turbulent flows such as boundary layers and channel flows, steep gradients exist close to the wall(s) due to the no-slip condition.As noted in the study conducted by Nicolas et al [20], because of the spatial averaging inherent in cross correlation-based PIV methods, these gradients are virtually impossible to resolve with PIV.Hence, large uncertainties in the velocity field measurements persist for y + ≲ 10-20 where the inner maximum of the RMS streamwise velocity (σ u ) is located [22], which is key to measuring the turbulence intensity and estimating the wall shear.As evident from the study conducted by Nicolas et al [20], due to its higher spatial resolution, OFV can potentially overcome this issue by avoiding spatially averaging the small scale motions near the wall.Furthermore, wOFV can make use of known flow physics, i.e. the no-slip condition, which is known a priori and can be used to inform the solution in wOFV.The no-slip boundary condition in fluid flows is based on the observation that the flow velocity must be zero at a stationary fluid-solid interface.In other words the fluid elements on the surface 'stick' to the surface and do not move relative to it [23].Correlation-based PIV cannot exploit the no-slip condition in the same way due to the independence of neighboring velocity vectors.That is, imposing a zero velocity at a solid surface would not affect the velocity vectors computed in interrogation windows away from the surface.
One of the first statistically converged mean velocity profile measurements of a turbulent boundary layer using correlationbased PIV was made in 2001 by Angele and Klingmann [24] in a low speed wind tunnel with the turbulent boundary layer on a flat plate at Re x = 1.76 × 10 6 .The authors were able to resolve the turbulent boundary layer profile in the y + ⩾ 8 region with δ ν = 31 µm, but due to the well known peaklocking phenomenon in PIV [25] the measurements overestimated the RMS by about 8% due to the asymmetric nature of the probability density function in the near-wall region.Progress has been made to resolve these issues and in 2018, Schröder et al [26] reported measurements in an adverse pressure gradient turbulent boundary layer generated on a flat plate in a wind tunnel at Re θ > 10 4 achieving δ ν = 13.5 µm, and the authors were able to resolve a part of the peak in the streamwise RMS at y + ≈ 15 using correlation-based PIV, but the mean profile grossly disagreed with DNS results for the y + < 15 region.The authors ultimately concluded that the cross correlation PIV method is not applicable close to the wall for such high Reynolds number turbulent boundary layer flows which produce small viscous scales, due to the spatial averaging inherent to the method.To date, the full, spatial streamwise RMS profile remains unresolved in the y + < 15 region in laboratory experiments.
An experimental campaign was conducted by Klewicki and Flaco [27] in 1990 to measure the wall normal velocity profile of a turbulent boundary layer over the wall of the test section in a low-speed wind tunnel using a four-wire hot wire probe.They reported velocity measurements for the y + > 4 region within an accuracy of 4%, but the y + ⩽ 4 region remained unresolved due to the inability to position the probe close enough to the wall.The study found that the statistics of velocity gradients such as wall shear and vorticity were extremely sensitive to the finite probe size, much more so than the velocity statistics.This is an inherent drawback to pointwise measurement devices such as hot wire probeseven when multiple probes are used to estimate spatial derivative quantities, the flow is averaged over the physical size of the probe.
Recently, the FriDa project, which concerns gathering rigorous experimental data on a flat plate turbulent boundary layer at moderate Reynolds numbers, compiled a comprehensive dataset comparing correlation-based PIV and hot wire anemometry velocity measurements [28].The hot wire velocity measurement closest to the wall was reported at y + = 3.3 at a somewhat moderate friction Reynolds number of Re τ = 624 with an associated error of 3.13%.At high Reynolds numbers, accurate velocity measurements in the y + < 4 region in a laboratory setup is still an active area of need in turbulence research, which is inhibited by the spatial resolution issues inherent to both hot wire measurements and correlation-based PIV.There has been an effort to overcome the spatial resolution issue faced by hot wire anemometry and correlation-based PIV by experimentally studying the large scale boundary layers such as the atmosphere surface layer (ASL) to understand the behavior of the near-wall turbulence, but the similarity in the structure of ASL compared to the typical laboratory turbulent boundary layer has been debated in the light of the experimental data [22,29].The ASL velocity measurements have additional issues that are not described here, and the interested reader is the referred to [30] for more a detailed discussion on the subject.
Despite its potential to improve velocity measurements near walls in optical velocimetry methods, the no-slip condition has not been explicitly enforced in either PIV or OFV.As mentioned above, due to the steep velocity gradients present near walls, the accuracy of these methods suffers when applied to wall bounded flows close to the wall [20], which is a critical measurement location in turbulent boundary layers.This is demonstrated in figure 1(b), where wOFV was applied to synthetic tracer particle images from a direct numerical simulation (DNS) of a turbulent channel flow without the imposition of the no-slip condition.A Neumann condition was used in the wOFV calculation at all of the boundaries.Figure 1 shows a snapshot of the horizontal component of the bottom half of the velocity field from the DNS [31] from the Johns Hopkins Turbulence Database [32,33], with a friction velocity Reynolds number of Re τ ≈ 1000.The velocity fields have been scaled such that the mean centerline velocity is 4 pixels per ∆t, and the full channel width is 1024 pixels.Figure 1(b) shows the profile of the horizontal velocity component at the location marked x ⋆ in (a).Observe that at the lower wall of the channel, the tangential flow velocity is zero, due to the noslip condition.While the velocity profile estimated by wOFV agrees with the true result for the region away from the wall, wOFV is unable to recover the true velocity profile close to the wall due to the steep velocity gradients there, and the computed velocity profile does not satisfy the no-slip condition.
In this work, we aim to improve the near-wall velocity measurements in turbulent flows by explicitly implementing the no-slip boundary condition (i.e. a Dirichlet boundary condition) at solid boundaries in the images in the wOFV method developed by Schmidt and Sutton [16][17][18].We will first develop the method in section 2.2, and then quantitatively evaluate the effectiveness of this modification by comparing the results obtained by estimating the flow field using the simulated tracer particle images from the turbulent channel flow DNS depicted in figure 1.As noted by Saxton-Fox et al [34], near-wall (y + ⩽ 30-40) PIV measurements are difficult not only due to spatial averaging, but also because the tracer particle images are prone to corruption due to reflections of the illuminating laser light off the wall.This would pose a similar challenge for OFV as well, but as this issue is an experimental difficulty rather than one related to the processing algorithm, it will not be considered in the present work.It is noted that several techniques, e.g. using fluorescent particles and blocking the scattering from the wall with optical filters, have been implemented to address this particular experimental imaging challenge [35,36].Similarly, the particles are treated as ideal massless, inertialess tracers which perfectly follow the flow.Therefore effects such as particle lag [37] or slip [38,39] are not considered.Effects such as non-uniform illumination and imaging noise are likewise not evaluated in the present study, but three dimensional out-of-plane motion of the tracer particles is included inherently because the images are created from a 3D simulation.

Wavelet-based optical flow
The details regarding the wOFV method and implementation are described in detail by Schmidt and Sutton [17], and the interested reader is referred to [16][17][18] for further information.Briefly, the velocity field is estimated from a minimization problem given by ( In equation ( 1), θ is a representation of the velocity field u, which differs between various optical flow methods.In socalled 'classical' OFV approaches, θ = u, while in wOFV θ is the wavelet coefficients of u.J D is referred to as the data term, and is a penalty function formed from the displaced frame difference equation (2), which penalizes mismatch between the images I 0 and I 1 at pixel location x subject to displacements given by u (x) ∆t As J D is simply based on an assumption of conservation of brightness between the two images, and hence contains no physical information about the underlying flow, it places no constraints on θ besides minimizing the mismatch between I 0 and I 1 .This can result in high-magnitude spatial gradients that are not physical for real fluids, and also increases the sensitivity to image noise.Hence, physical constraints are introduced via the regularization term J R (θ) to J D , where J R typically minimizes some function of the spatial derivatives of u that are designed to cause the solution to represent a physical flow.Some examples are first-order Tikhonov regularization [40], div-curl regularization [4], and viscosity-based regularization [18], the latter of which is used in the present work.λ is a scalar parameter that balances J D against J R , which is typically manually tuned by the user.The regularization term therefore makes the velocity at a given pixel dependent on the velocities at nearby pixels because of the smoothness enforced on the solution.Due to this, imposition of the no-slip condition at a solid boundary will have a beneficial impact on locations near the wall.

Enforcement of the no-slip condition
The no-slip boundary condition is a Dirichlet boundary condition and can be written as the following where u| ∂D is the flow velocity at the fluid-solid interface ∂D.
In wOFV, as indicated by equation ( 1), the flow field estimation depends on a balance of two terms: J D and J R .While the no-slip condition is satisfied by the flow which produces the input particle images, it will not in general be satisfied by the estimated solution to the OFV problem, even as λ → 0, that is, with no explicit regularization.Explicit regularization (λ > 0) penalizes sharp gradients in the flow, which are nonphysical in most regions but exist near the wall.Resolving these sharp near-wall velocity gradients in the boundary layer without sacrificing the far-field results continues to remain a challenge, and the dependence of the near-wall velocity results on the regularization parameter λ was recently studied by Nicolas et al [20].It was found that while smaller values of λ in wOFV can lead to better near-wall velocity estimates, as then wOFV is able to capture the sharp near-wall velocity gradients, the required values of λ to obtain this increase in accuracy in the near-wall velocity estimates leads to an under-regularized and non-physical 'noisy' velocity field in the far-field region.
It should be noted that in classical OFV, λ = 0 represents an ill-posed inverse problem with no unique solution and is therefore mathematically unstable.wOFV methods resolve this issue by solving for a sparse set of non-zero wavelet coefficients of u, so that the ill-posedness is resolved [16] while introducing some minor implicit regularization.However, as mentioned in section 2.1 and evident from the study conducted by Nicolas et al [20], insufficient or no explicit regularization results in non-physical artifacts in the flow, which sacrifices accuracy in the far-field results in the context of wallbounded flow measurements.Hence the imposition of the noslip condition, just as in real fluid flows, needs to be made at the wall only without reducing the value of λ in the bulk flow away from the wall.The effect of the no-slip condition can then be propagated into the flow by the regularization term, just as the effect of the wall propagates by the action of viscosity in physical flows.The analogue between viscous forces in fluid flows and J R in OFV methods has been established in the literature [18].Finally, it is noted here that explicit mixedtype boundary conditions, i.e.Neumann at some boundaries and Dirichlet at others, is not permitted in wOFV because of the necessity to use the same wavelet functions, which include a boundary treatment, for the entire solution.Furthermore, such a strategy would not be generally applicable in the case of curved surfaces or solid surfaces interior to the image domain.
To achieve no-slip at solid surfaces, a constraint must be constructed that introduces the desired boundary condition at all of the locations in the image domain corresponding to solid surfaces.The strategy is to transform equation (3) to the wavelet domain to obtain a constraint on the wavelet representation of the velocity field, and then to solve equation (1) via constrained optimization subject to the resulting constraint in the wavelet domain.This will lead to a direct interaction of the no-slip constraint with the viscosity-based regularization term J R developed by Schmidt and Sutton [18].Hence the effect of the wall will be propagated into the flow field using physicsbased viscosity regularization combined with the optical flow constraint specified by the data term (J D ).The simplest way to implement such a constraint is to formulate it as a canonical linear equality constraint of the form Ax = 0, where A is a matrix describing the constraint and x is a vector containing the variables over which the minimization is to be performed.In our case, x is the wavelet coefficients of the velocity field.Therefore, we seek a linear operator A that imposes the no slip condition equation (3) on the wavelet coefficients.
The no-slip constraint operator A is found as follows.Let V be an m × n matrix containing a component of the velocity field (i.e.u or v) obtained from wOFV that does not necessarily satisfy the no-slip boundary condition.This velocity field data can be transformed to a two dimensional separable wavelet basis by the following linear operation [17] where F is a matrix factorization of a forward wavelet transform horizontally along each row, F T indicates its transpose, performing the transform vertically along each column, and Ψ is the resulting transformed m × n matrix containing the wavelet coefficients.By reshaping Ψ into a column vector ψ ∈ R mn and reshaping V into a column vector v ∈ R mn , the matrix triple product given in equation ( 4) can be reformulated without any loss of generality into the following matrix vector product where G ∈ R mn×mn is a transform matrix that acts as a linear map translating the 2D velocity field from physical space to the wavelet domain.It should be noted that the act of transforming the matrix triple product given in equation ( 4) to a matrix vector product in equation ( 5) leads to the dimensionality of G being greater than that of F. The dimensionality of the rowspace, i.e. the rank of the linear map, however, is preserved as this act of reformulation of the triple product does not lead to any gain or loss in the information contained in the original transform.Unlike F, then, the matrix G is no longer full rank, which implies that the inverse G −1 cannot be obtained by a simple inversion of the matrix G. Hence the inverse operation of equation ( 5) needs to be obtained in a similar fashion as equation ( 5) itself, i.e. by reformulating the inverse wavelet transform triple product which can be expressed in the same way as equation ( 4), and this can be achieved by the following expression Here, G ∈ R mn×mn is also a rank-deficient transform matrix that maps the wavelet transform of a vector to physical space.
As suggested by equation ( 3), at the boundary points given by ∂D, where no-slip is implemented, the velocity components must be set to zero, which in terms of equation ( 6) can be expressed mathematically as the following constraint: where the a priori information about the flow boundary conditions is contained in the set K, and the cardinality of the set K, denoted by κ, equals the number of boundary points on which the no-slip condition is to be specified.Gi is the ith row of the matrix G corresponding to a point in ∂D.This constraint equation that provides κ equations can also be naturally reformulated in terms of a matrix product by recognizing that all of the κ rows of G corresponding to the constraint given by equation ( 7) can be stacked into a matrix G∂D that by construction would satisfy the following linear system equality: where the linear map G∂D is a κ × mn matrix.Equation (8) implies that for the specified no-slip condition to be satisfied, ψ must be constrained to the null-space of the linear map given by the matrix G∂D .This is precisely the no-slip constraint on the velocity field transformed to the wavelet-domain, expressed via a constraint on the wavelet coefficients of the velocity field.Equation ( 8) is then the required linear equality constraint on the wavelet representation of the velocity field ψ that we seek.The constraint map G∂D is constructed once, in advance, from the wavelet filter matrices and the location of the boundary points on which the constraint is to be specified.To solve this constrained optimization problem, an interior-point method is used as it is efficient at handling problems with a large set of variables, such as the problem at hand, and it satisfies the specified constraint at all iterations [41].The interior-point approach to constrained optimization is to solve a sequence of approximate optimization problems, which can be done by a conjugate gradient using a trust region approach [42].The Hessian of the functional was approximated using Broyden-Fletcher-Goldfarb-Shanno algorithm [43,44] for efficiency and stability.

Synthetic images from channel flow
The no-slip constraint derived in section 2.2 was implemented in the wOFV method developed by Schmidt and Sutton [17].The resulting constrained optimization algorithm (denoted here as wOFV+) was tested on the particle images obtained from the experiment described in section 3.2, and simulated tracer particle images generated from the bottom half of the 3D turbulent channel flow DNS described in section 1 and represented in figure 1, using inertia-less Lagrangian particles and an artificial laser sheet, following [45,46].About 1000 pairs of 512 × 512-pixel synthetic particle images with an average of 80 × 10 3 particles per image were generated at randomly selected streamwise and spanwise locations in the channel at random time instants to minimize correlation bias in the data and allow accurate computation of converged velocity statistics [47].The dimensions of the data acquisition window were chosen such that the spatial resolution of the flow closely matches the highly resolved laboratory PIV experiments conducted by Scarano et al [28] in terms of the wall viscous units.Therefore, the half channel width was set to h = 1536 pixels, which at a friction Reynolds number of Re τ ≈ 1000 leads to one viscous unit δ ν = 1.536 pixels and a simulated 1/e 2 laser sheet width of about a millimeter, corresponding to 10 pixels in image units.By comparison, the experiments of Scarano et al had a resolution of 35.55 pixels per millimeter, achieving δ ν ≈ 1.4 pixels at Re τ ≈ 922.
The synthetic particle image pairs were then processed with a state of the art commercial correlation-based PIV code (Insight4G), wOFV without the no-slip boundary condition, and wOFV with the no-slip constraint applied at the bottom wall (denoted as wOFV+).For processing the synthetic particle images with correlation-based PIV, a final interrogation window size of 16 × 16 pixels with 50% overlap was used after pre-processing the synthetic particle images through a min-max filter [48].The PIV algorithm uses window deformation, multiple passes, and recursive outlier removal and replacement.A min-max filter was also used to pre-process the synthetic particle images for wOFV and wOFV+.Although imaging noise and illumination non-uniformity were not considered in this study when testing on synthetic images, outof-plane motion of the tracer particles can cause errors in OFV methods just as it does for correlation-based PIV, which motivates the use of the min-max filtering for all velocimetry algorithms [1,49].
Figure 2 shows the instantaneous (a) streamwise and (b) wall-normal velocity components computed using wOFV, wOFV+, and the correlation-based PIV algorithm plotted as the abscissa, normalized by the friction velocity of U τ = 0.179 pixels per ∆t, where ∆t is the time between consecutive tracer particle images in a pair.The ordinate is the coordinate spanning from the bottom channel wall to the channel centerline in the viscous units, defined as y + = Re τ y/h, where h = 1536 pixels is the half channel width and Re τ = 1000 is the friction velocity Reynolds number.The true DNS results are also plotted for comparison.Data from the correlation algorithm is shown as individual circles, where each circle represents a velocity vector.These vectors are spaced 8 pixels apart, due to the size and overlap of the interrogation windows, and the first vector in the wall-normal direction is 7 pixels from the wall, i.e. as close to the wall as possible.From figure 2(a), it is evident that the implementation of the no-slip boundary condition as described in section 2.2 results in wOFV+ being able to successfully resolve the near-wall streamwise velocity field more accurately than wOFV without the boundary condition.In the region of y + ⩽ 15, wOFV+ nearly perfectly traces the DNS ground truth whereas wOFV does not, while correlationbased PIV fails to make any estimates in that region due to its finite-sized interrogation windows, and shows significant inaccuracy in the lower portions of the log-law region (y + ≲ 50).As previously noted, making accurate velocity measurements in this near-wall region is extremely important, yet very challenging for state-of-the-art optical methods.It is also evident from figure 2 that the far field velocity profile from the wOFV+ algorithm is not significantly altered compared to wOFV, so the near-wall accuracy is improved without sacrificing fidelity in the far-field.The wall-normal velocity has a much smaller magnitude and exhibits less steep gradients than the streamwise velocity, and so the improvements of wOFV+ are not as significant compared to the other velocimetry methods.
Figure 3 shows a snapshot of the full-field streamwise velocity estimates obtained by wOFV, wOFV+ and the correlation-based PIV method, compared to the true DNS  results.The velocity field away from the wall computed by wOFV+ compares favorably to the true DNS results, and does not suffer from any adverse effects due to the introduction of the no-slip condition compared to wOFV.The correlationbased results, on the other hand, fail to capture the near-wall field as it estimates flow field vectors over finite sized windows.Close inspection of panel (d) further reveals smoothing of fine flow features, which has been previously documented in the literature [16,17].
This observation that the wOFV+ preserves the far field fidelity is further solidified by figure 4(a), which shows the root mean square error (RMSE) of the calculated velocity fields normalized by the RMS of the turbulent fluctuations over all 1000 image pairs.The wOFV+ results show improved accuracy compared to wOFV and are comparable to correlationbased PIV for the full field.Figure 4(b) shows the normalized local RMSE in the near-wall region of y + ⩽ 13.7 for the same images as (a).In this region, wOFV+ produces superior results compared to both wOFV and correlation, although the differences in mean error are rather small.The large fluctuations observed in the RMSE for the correlation results are due to the small number of sample points available in the near-wall region.
As noted earlier and in many previous studies in the literature, correlation-based PIV suffers from an inherent loss of spatial resolution, which becomes apparent when derivative quantities and statistics are computed from the flow field estimates (see e.g.[16,17]).Figure 5 shows the near-wall vorticity fields computed using wOFV, wOFV+, and correlation compared to the DNS truth.Clearly, correlation-based PIV does a comparatively poor job at estimating derivative quantities, and substantially underestimates the vorticity magnitudes.The vorticity field computed using the wOFV flow field agrees quite well with the true vorticity except very close to the wall.The introduction of the no-slip constraint allows wOFV+ to improve upon the wOFV results in the near-wall region, again without sacrificing fidelity in the flow away from the wall.
Figure 6 presents profiles of statistical quantities from the 1000 uncorrelated realizations of the turbulent channel flow.Figure 6(a) shows the mean of the streamwise velocity field, and (b) shows the RMS of the streamwise turbulent fluctuations.While all three velocimetry methods faithfully reproduce the true profiles for y + ≳ 100, i.e. in the upper part of the log-law region and in the outer layer, the profiles show that wOFV+ captures the statistics in the near-wall log-law region, buffer layer, and viscous sublayer much more accurately than either wOFV or correlation.Correlation-based PIV underestimates the magnitude of the peak in the RMS near 10 < y + < 13 due to the very strong velocity gradients in this region, but wOFV and wOFV+ are able to capture the peak in   the RMS accurately.Furthermore, wOFV+ captures the correct mean and RMS profiles down to the wall while wOFV fails to capture the profiles in the viscous sublayer (y + ⩽ 20) accurately due to a lack of an explicit no-slip boundary constraint.
Another commonly encountered issue when performing velocimetry in shear flows using optical techniques stems from limited dynamic range [1,50].Essentially, optical velocimetry methods use a time interval ∆t, which is optimized for capturing the velocity component along the principal flow direction (streamwise in this case).This comes at an expense of reliably measuring the orthogonal velocity component (wallnormal) if its magnitude is substantially smaller than that of the principal flow direction.It is unknown whether OFV methods have improved dynamic range compared to correlation-based PIV, but the wall-normal velocity results shown in figure 7  indicate that this may be the case.Figure 7(a) shows the mean wall-normal velocity and (b) shows the RMS of the vertical velocity fluctuations.The correlation-based method significantly overestimates the mean wall-normal velocity, even in the outer layer, whereas both wOFV and wOFV+ captures it with reasonable accuracy away from the wall.This is due to the very small magnitude of the wall-normal velocity, and a small but consistent bias towards positive values in the correlationbased PIV results.Near the wall, the estimations of the mean become poor for all methods due to dynamic range limitations impacting wOFV and wOFV+ as well as correlation.It should be noted, however, that wOFV+ captures the wall-normal RMS nearly perfectly through the viscous sublayer while both correlation-based PIV and wOFV fail to do so as illustrated by figure 7(b).
An additional statistical measure important in turbulent flows is the Reynolds stress, profiles of which are shown in figure 8. wOFV+ accurately captures the Reynolds stress profile all the way to the wall, through the viscous sublayer.
Note also that while correlation-based PIV slightly underestimates the Reynolds stress magnitude in the region 40 < y + < 100 in the log-law layer, both wOFV and wOFV+ capture the profile in this region accurately.Figure 9 shows the full Reynolds stress field computed using wOFV, wOFV+, and correlation compared to the DNS ground truth.Similar to the observations from the vorticity fields in figure 5, both wOFV and wOFV+ are able to reproduce the fine-scale details in the Reynolds stress field compared to correlation, which smooths out these features.
This improvement in the capture of the near-wall velocity profiles results in increased accuracy in the wall shear stress estimates, which is notoriously difficult to measure, even when using optical methods.Figure 10 shows histograms of the error in the wall shear stress computed from wOFV, wOFV+, and correlation compared to the DNS plotted over 1000 realizations of the velocity field data at all 512 wall points in the domain.The magnitude of the mean error in the estimated wall shear for wOFV is 0.2438/∆t and for the correlationbased PIV method the mean error compared to the true DNS is 0.2739/∆t, while for wOFV+ the mean error in the wall shear estimates is 0.0168/∆t, which is about fifteen times lower than of wOFV and correlations.The standard deviation in the error for wOFV+ is 0.0122/∆t, which is significantly lower than the standard deviations for wOFV and correlation, which are 0.1134/∆t and 0.0648/∆t, respectively, so wOFV+ not only estimates the wall shear more accurately than the other methods, but it also does so more precisely.As anticipated, the implementation of the no-slip boundary condition results in significant improvements in the wall shear stress estimates compared to wOFV and correlation-based PIV.

Experimental data from turbulent boundary layer
The experimental data used in this work was obtained from the PIV experiments conducted by de Silva et al in the high Reynolds number boundary layer wind tunnel at University of Melbourne [51].Images were acquired in a zero pressure gradient turbulent boundary layer at Re τ ≈ 8000 with a free stream velocity of 10 m s −1 .Polyamide particles of  size ≈ 1 µ m were seeded into air and imaged with a high magnification camera.The reader is referred to [51] for a detailed exposition of the experimental setup.Briefly, the test section of the facility has a development length of approximately 27 m, enabling the achievement of high Re τ at relatively low free stream velocity.PIV measurements were made at a location 21.5 m downstream of a boundary layer trip with a PCO4000 camera with resolution 4008 × 2672 pixels to obtain the high spatial resolution necessary to capture the fine turbulent motion in the boundary layer.The high magnification imaging setup was achieved via a combination of bellows, a tele-converter, and a series of extension tubes to utilize the full sensor, resulting in a field of view of approximately 30 mm × 20 mm, which roughly translates to 700 × 500 viscous wall units in the streamwise and wall-normal directions, respectively, and a pixel resolution of 7.5 µ m per pixel.A schematic of the experimental setup is shown in figure 11.
The particles were illuminated by two overlapping laser sheets generated using two Spectra Physics Quanta-Ray PIV 400 Nd:YAG double-pulse lasers that deliver 400 mJ pulse −1 each.The camera captured 1680 statistically independent particle image pairs with a separation of 15 µ s between each frame.To obtain high-fidelity near-wall particle images, the scattering of the laser light at the wall surface was weakened by replacing the bottom wall of the test section with transparent glass, which has been shown to mitigate the near-wall scattering [35], and additional seeding was injected into the near-wall region far upstream of the measurement location to account for the lower seeding concentration typically observed close to the wall.It should be noted that the particle density and inter-frame displacement in the captured images were not optimized for processing with wOFV/wOFV+, but for correlation.wOFV requires a higher seeding density and smaller inter-frame displacement than correlation-based PIV for optimal performance [16].
The images were processed with wOFV, wOFV+, and Insight4G.For PIV processing, a final interrogation window size of 32 × 32 pixels with 50% overlap was used without any pre-processing of the images besides intensity normalization.A sliding background filter was used prior to processing with wOFV and wOFV+ to mitigate the errors due to nonuniformity in illumination, as suggested in [1,49].Figure 12 shows instantaneous streamwise and wall-normal velocity profiles obtained using wOFV, wOFV+, and correlation normalized by the friction velocity U τ = 0.334 m s −1 .The velocity vectors obtained from correlation are spaced 16 pixels apart, due to the size and overlap of the interrogation windows, and the first vector in the wall-normal direction is 12 pixels away from the wall.It is evident that all three methods agree in the far-field region (y + ⩾ 100), but the results begin to deviate in the log-law region, and diverge further in the y + ≲ 5 near-wall region.Correlation fails to capture the near-wall region of y+ < 5, while the rapid changes in the velocity field present in the near wall region lead to inaccurate results in wOFV.The imposition of the no-slip condition in wOFV+, however, successfully increases the accuracy of the velocity measurements in a real-world experimental setting, despite the seeding density and inter-frame particle displacement not being optimized for wOFV/wOFV+ computations.
The increase in accuracy of the near-wall velocity field in wOFV+ becomes more apparent when the statistical results are analyzed.The mean streamwise velocity is shown in figure 13 for the various velocimetry algorithms, along with the results obtained by de Silva et al [51] from an in-house PIV algorithm, and the DNS results obtained by Schlatter and Örlü [52].All methods and data agree well away from the wall, but only wOFV+ is able to capture the full mean streamwise velocity profile near the wall, whereas both the correlation results, from de Silva et al's in-house correlation based PIV code and Insight4G, fail to extract the full near-wall velocity profile from the particle images due to the spatial averaging inherent to cross-correlation.Furthermore, wOFV struggles to capture the near-wall profile due to the presence of strong velocity gradients near the wall as observed in the study conducted by Nicolas et al [20].The streamwise mean velocity profile obtained by wOFV+ traces the DNS results obtained by Schlatter and Örlü [52] accurately all the way to the wall.This includes the viscous sublayer, which is notoriously difficult to measure, even in the experimentally untouched y + < 1 region, establishing a new state of the art result in the near-wall turbulent boundary layer measurements and pushing the boundaries of the capability of optical methods for near-wall velocimetry.

Conclusions
A technique to enforce the no-slip condition in wOFV has been developed to improve velocity measurements in the nearwall region in wall-bounded turbulent flows.This technique involves reformulating the no-slip boundary condition as a constraint on the wavelet representation of the velocity field, and then making use of a constrained optimization algorithm to minimize the functional of the optical flow equation subjected to the resulting constraint in the wavelet domain.This technique was successfully implemented in wOFV and was found to significantly improve the near-wall flow field results, leading to substantial gains in the accuracy of the estimates of the near-wall velocity profiles, flow statistics, wall shear stress, and Reynolds stresses, without sacrificing the accuracy and spatial resolution benefits of wOFV in the far field.wOFV+ was also demonstrated to be capable of resolving the near-wall flow field and fine-scale flow features more accurately than correlation-based PIV, which is inherently limited by finite sized interrogation windows.The test of the method on experimental particle images lead to state of the art results with accurate mean velocity results in the y + < 1 region, demonstrating that the new method provides an increase in the accuracy of the near-wall velocity measurements even when non-ideal experimental effects are present in the tracer images such as non-uniform illumination, imaging noise, and wall reflections.

Figure 1 .
Figure 1.(a) Horizontal velocity magnitude in a channel flow simulation [31].(b) Horizontal velocity profiles at location x ⋆ along the channel.The wOFV profile was calculated using synthetic tracer particle images generated from the DNS.

Figure 2 .
Figure 2. Instantaneous (a) streamwise and (b) wall-normal velocity profiles obtained from wOFV, wOFV+, correlation-based PIV, and the DNS ground truth.Data markers for the PIV algorithm represent individual velocity vector locations.

Figure 4 .
Figure 4. Root mean square error (RMSE) of (a) the full velocity field and (b) the velocity field in the y + ⩽ 13.7 region computed by wOFV, wOFV+ and correlation, normalized by the RMS of the turbulent fluctuations over 1000 uncorrelated image pairs.

Figure 5 .
Figure 5. Vorticity of the velocity field computed using (b) wOFV, (c) wOFV+ and (d) correlation, compared to the true DNS velocity field (a).

Figure 6 .
Figure 6.(a) Mean and (b) RMS velocity profiles of the computed streamwise velocity compared to the ground truth from the DNS.

Figure 7 .
Figure 7. (a) Mean and (b) RMS velocity profiles of the computed wall-normal velocity compared to the ground truth from the DNS.

Figure 8 .
Figure 8. Normalized Reynolds stress profile computed using wOFV, wOFV+ and correlation compared to DNS truth.

Figure 10 .
Figure 10.Error in the computed wall shear stress computed by wOFV, wOFV+ and correlation compared to the DNS.

Figure 11 .
Figure 11.Schematic of the experimental setup showing the location of the field of view and the camera.
Friction velocity Reτ = Uτ h/ν Friction velocity Reynolds number Re θ = U∞θ/ν Momentum thickness Reynolds number Rex = U∞x/ν Local Reynolds number along the plate δν = h/Reτ One viscous unit y + = y/δν Wall normal distance in viscous units u + = u/Uτ Streamwise velocity in viscous units v + = v/Uτ Wall-normal velocity in viscous units