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

The performance of wavelet-based optical flow velocimetry (wOFV) applied to tracer particle images from hypersonic boundary layer flows is assessed. Two modifications of wOFV to account for the no-slip boundary condition are implemented and assessed to extend the applicability and performance of wOFV to wall-bounded high-speed flows. The first modification (wOFV+) enforces the no-slip boundary condition via constrained optimization, whereas the second modification (wOFV-Ext) performs an antisymmetric extension of the particle images before wOFV processing. The original wOFV and two modified versions are assessed on both (i) synthetic particle images generated from a direct numerical simulation (DNS) of a Mach 4.9 turbulent boundary layer and (ii) actual particle images acquired from a Mach 5 turbulent boundary layer experiment. The velocity estimates using wOFV methods are directly compared to velocity estimates produced by state-of-the-art particle image velocimetry (PIV) software. All three wOFV methodologies were shown to have an increase in accuracy and spatial resolution when compared to PIV. The increase in spatial resolution was shown to be important in the near-wall region (within the viscous sublayer and buffer layer) and for computing shear velocities, u(tau). Overall, both of the no-slip boundary condition modifications were shown to improve accuracy in the near-wall region when compared to both wOFV and PIV.

Laser light sources are commonly used in schlieren-based flow visualization techniques that are in development at NASA Langley Research Center. Of particular interest are high-intensity, low coherence sources that allow for high signal-to-noise schlieren imagery to be acquired without the laser speckle that typically occurs as a result of interference. Results from a self-aligned focusing schlieren system developed at NASA Langley and a background-oriented schlieren system are presented to highlight the effectiveness of these sources.

In this Letter, a digital self-aligned focusing schlieren (D-SAFS) system is introduced. This system uses a digital transparent micro liquid crystal display (μLCD), in combination with a linear polarizer, to act on the linear polarization state of light transmitted in both the forward and reverse directions, essentially acting as both the source and cutoff grids. The use of the μLCD display allows for on-the-fly changes to the cutoff pattern type, spatial frequency, and orientation. This eliminates the need to physically access the source/cutoff grid in order to optimize the instrument's sensitivity, which is necessary with a conventional self-aligned focusing schlieren (SAFS) system.

A high-speed self-aligned focusing schlieren (SAFS) system was used to visualize density gradients in and around the cavity flameholder of the combustor section of the University of Virginia Supersonic Combustion Facility (UVASCF). Images with this system were acquired at a framing rate of 110 kHz with no fuel injection, with fuel injection but no flame, and for fuel injection with combustion corresponding to a global equivalence ratio of phi = 0.18. Images with an air throttle in operation to modify the shock train location with fuel injection and with flame were also acquired. Simultaneous OH planar laser-induced fluorescence (PLIF) images were also acquired at a framing rate of 20 Hz. Results obtained with both visualization techniques are compared to one another to highlight how SAFS can complement more advanced flow visualization techniques and resolve dynamic behavior that may not otherwise be captured. Both proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) analysis techniques are applied to the SAFS image sequences to identify coherent periodic structures for the runs with fuel injection and combustion.

The framework and implementation of a tomographic wavelet-based optical flow velocimetry (TwOFV) technique is demonstrated and assessed. Three-dimensional extensions for both optical flow and the wavelet transform are derived and implemented to extend wavelet-based optical flow velocimetry (wOFV) beyond planar measurements. A quantitative comparison is made for both TwOFV and tomographic particle image velocimetry (tomo-PIV) by applying the algorithms to synthetically generated volumetric particle fields dervied from a direct numerical simulation (DNS) of homogenous isotropic turbulence (HIT). In particular, the techniques were evaluated at four different seed densities that are representative of values used in experiments. TwOFV was shown to have a significant increase in spatial resolution and accuracy at higher seed densities as the TwOFV results show a great deal of similarity between the DNS velocity field ("truth”) and the TwOFV velocity estimate. This work represents a preliminary step towards the realization of high-resolution, volumetric velocimetry using TwOFV and will be utilized to start to guide the selection of a tomographic reconstruction technique along with setting experimental requirements.

The application of the inverse Abel transformation to statistically axisymmetric data is described in this work. The general theory of Abel inversion tomography is discussed, and it is shown that in order to properly apply the Abel inversion, an ensemble operator must be axisymmetric and commute with path integration. For statistically axisymmetric data where the individual realizations are asymmetric (such as turbulent flows issuing from circular nozzles), only the planar mean can be properly recovered. Higher-order moments, such as the standard deviation (root mean square) cannot be recovered because path integration is not commutative with the statistical moment operators. It is further shown that the Abel transformation cannot be used to recover the one-sided Fourier spectrum for the same reason. For the mean, the Abel inversion can be applied to the ensemble quantity, or the ensemble operator can be applied to the Abel inversion of individual asymmetric realizations. These findings are rigorously verified with numerical simulations and demonstrated on experimental data by comparing planar particle image velocimetry data to path-averaged schlieren image velocimetry results from a turbulent jet of CO2 issuing into ambient air.

Planar velocity measurements were performed in a non-premixed turbulent jet flame using double-shot OH planar laser-induced fluorescence (PLIF) imaging and a wavelet-based optical flow velocimetry (wOFV) method. This study represents an initial experimental evaluation at the feasibility of ``seedless'' velocimetry in reacting flows using wOFV. Results were compared against simultaneous PIV measurements to assess the accuracy of the OH PLIF-wOFV measurements. The accuracy of OH PLIF-wOFV is found to be highly sensitive to the inter-frame displacement of the flow features in the OH PLIF images, as well as the magnitude of the local OH PLIF image intensity gradients. When these two factors are controlled for OH PLIF-wOFV (i.e., rejection of vectors where the image intensity gradient is too small), the velocity results from the two techniques agree in the mean for the streamwise component of the velocity field, although more images are required for OH PLIF-wOFV to converge statistics. However, the transverse velocities estimated with OH PLIF-wOFV have considerably larger error than the PIV results, likely due to a combination of smaller magnitude inter-frame displacements than the streamwise component, and the effect of out-of-plane scalar transport.

An implementation of stereoscopic wavelet-based optical flow velocimetry (swOFV) using a general mapping approach for stereoscopic reconstruction is presented that is suitable for experiments. The methodology is evaluated quantitatively using synthetic tracer particle images generated from a direct numerical simulation (DNS) of homogeneous isotropic turbulence (HIT) and experimental data from a laminar axisymmetric vortex ring. Statistical comparisons between velocity components and gradient quantities are made between swOFV and stereoscopic PIV (sPIV) results. Overall, swOFV shows higher accuracy (approximately 35\% less error near the optimal off-axis angle) and increased spatial resolution as evident through field quantity visualization. The increases in spatial resolution also are observed in the energy and dissipation spectra, as swOFV is capable of resolving the finest dissipative scales. Experimental particle images from a laminar axisymmetric vortex ring were were evaluated and showed a strong topological agreement between swOFV and sPIV as expected for laminar flows. However, the swOFV results exhibited much less noise in the velocity field estimates, which was especially evident in the reconstruction of the out-of-plane velocity ($w$ )component and the out-of-plane vorticity, $\omega_z$. Overall, the updated and generalized implementation of swOFV, which can be implemented experimentally, has been shown to provide higher accuracy and spatial resolution capabilities when compared to sPIV.

The influence of several non-ideal experimental effects on the accuracy of a wavelet-based optical flow (wOFV) method is evaluated using data from a series of simulated flows. Out-of-plane particle displacements and image intensity non-uniformity are shown to decrease the accuracy of wOFV in a similar manner to correlation-based PIV, but with an increased sensitivity compared to correlation-based PIV. However, wOFV is still found to be more accurate than PIV when the magnitude of these effects remain within expected experimental bounds. It is noted that for some cases, the results are significantly improved by using image pre-processing filters. It is observed that flow divergence does not have an appreciable effect on the accuracy of wOFV velocity estimation, even though the fluid transport equation on which wOFV is based implicitly assumes that the motion is divergence-free. This is a significant finding for the broader applicability of planar velocimetry measurements using wOFV. Finally, it is noted that the accuracy of wOFV is not significantly reduced in regions of the image that do not contain particles, as long as the seeding density is sufficiently high. This explicitly demonstrates that wOFV (when applied to particle images) yields an accurate whole field measurement, and not just at the particle locations.

Experimental assessment of a wavelet-based optical flow velocimetry (wOFV) approach for the estimation of two-dimensional velocity fields from tracer particle images is performed. Assessment is performed within a series of free-shear flows and compared to correlation-based particle image velocimetry (PIV). Particle field images were acquired in a series of turbulent jets and in laminar and turbulent wakes behind a circular cylinder. Broad variations in operating conditions were considered, including Reynolds numbers, measurement locations, and image resolution (i.e., magnification or sampling frequency). wOFV is a novel technique that produces a high-resolution velocity estimation with the potential for significant increases in spatial resolution and accuracy when compared to traditional correlation-based PIV. Velocity fields, vorticity fields, and turbulent energy spectra are derived from the wOFV and PIV results for direct comparison. While qualitative agreement is observed between wOFV and PIV results, wOFV is shown to significantly outperform PIV in terms of noise for experiments in which the flow is overly resolved spatially (vector resolution << smallest flow scales), and in terms of effective spatial resolution where the flow is not overly resolved.

Velocity measurements were made in a turbulent jet of carbon dioxide issuing into air using a wavelet-based optical flow velocimetry (wOFV) method applied to schlieren image pairs. Schlieren image pairs were acquired with a two-camera setup using a single conventional Z-type schlieren system. The mean axial velocity shows good agreement with PIV data in the same jet after application of an Abel transformation. The applicability of Abel transforms for schlieren image velocimetry (SIV) in turbulent flows is discussed in general, and it is demonstrated that it is only appropriate to apply the Abel transform to the mean velocity field rather than instantaneous velocity fields, schlieren images, or other higher-order statistics. These restrictions apply regardless of the velocimetry approach (i.e., correlation-based analysis or wOFV) due to the path-integrated nature of the current Schlieren measurement. Due to these limitations, it is not clear whether a high-resolution technique such as wOFV provides additional benefits compared to correlation-based methods for the current Schlieren measurements.