## Abstract

Tomographic particle image velocimetry (Tomo-PIV) has become a standard tool for capturing a three-dimensional (3D) velocity fields in nonreacting flows. However, the diagnostic approach can become costly and challenging to implement when extended to applications which require high-speed cameras. This limitation has led to the use of fiber wound bundles to allow for multiple views to be captured on a single camera sensor. Additionally, employing this diagnostic approach on reacting flow-fields becomes more complex as the introduction of the flame causes additional luminosity and optical distortion which impacts the particle field reconstruction. This work seeks to validate and determine the limitations when utilizing a single sensor fiber-coupled approach for capturing Tomo-PIV data on a reacting flow-field. A premixed propane (C3H8) and air Bunsen burner flame is utilized to examine if the single sensor approach can meet the parameters for acceptable reconstruction based on previous research. The resulting velocity fields are then compared to a traditional PIV measurement to assess the deviation of the single sensor approach from a standard velocimetry measurement approach. It is demonstrated that there is strong agreement between the velocity and vorticity for the average flow-fields; however, when comparing the Reynolds shear stresses, a significant deviation is revealed. The deviation is attributed to strong velocity fluctuations occurring within the instantaneous Tomo-PIV data, which creates a significant divergence between the measurement techniques on an instantaneous basis. This demonstrates that while the approach can obtain reliable velocity and vorticity statistics, there are significant limitations in calculating second-order turbulence statistics. Thus, revealing that there is a tradeoff between the ability to extract the full velocity gradient tensor and the extent of the turbulence-related analysis which can be reliably performed.

## 1 Introduction

Developing a further understanding of combustion phenomena has become increasingly important as researchers aim to create more efficient power generation systems [1]. Typical experimental measurements are either planar or line of sight which results in an inability to capture information in the third dimension. Many combustion systems operate under turbulent conditions, which are inherently three‐dimensional (3D), and require improved diagnostics to capture relevant flow-field information across the entire domain [2]. Recent studies have begun implementing tomographic imaging to capture the full three-dimensional flow-field, which renders multiple two-dimensional (2D) projections into a three-dimensional field through a reconstruction technique [35].

Tomographic particle image velocimetry (Tomo-PIV) is an optical diagnostic technique that uses multiple views of illuminated particle fields, taken in synchrony, to extract three-dimensional velocity information using a cross-correlation function applied to reconstructed image pairs [6]. Coupling the tomographic imaging approach with high repetition rate devices renders, it possible to procure the time-resolved velocity gradient tensor which allows for the extraction of relevant quantities important to turbulent flows [7,8]. Previously, many analyses of higher-order turbulence statistics have been restricted to planar measurements, which requirements assumptions to be made and does not capture the three-dimensionality of the flow-field [912].

Since Tomo-PIV is dependent on having accurate reconstructions of the particle field, numerous studies have been performed on nonreacting applications to assess reconstruction techniques and determine standardized parameters to quantify reconstruction acceptability [13]. The first indicator of the accuracy of a Tomo-PIV system is the residual of the curve fitting function determined from the calibration images. This curve fit is used to map the series of two-dimensional calibration images into a three-dimensional volume; however, if the residual is large (greater than 0.5 pixels), then the function does not adequately represent the perspective transformation [13]. The second parameter is the disparity vector which assesses whether the calibration function is implemented accurately to experimental images. This is performed by matching particles in all camera views through a triangulation algorithm, with the residual error being defined as the disparity vector [14]. The final parameter commonly assessed is the signal-to-noise ratio (SNR) which is defined as the intensity of the particles in the flow-field versus the intensity of ghost particles which arise during reconstruction due to the nonunique solution to the multiplicative algebraic reconstruction technique (MART) algorithm [6]. As ghost particles are unavoidable, it is important that their intensities are kept at minimum to not negatively impact velocimetry results. It has been reported that a SNR ≥ 2 is the minimum for acceptable reconstructions [13].

The size and weight of high-speed equipment make the final tomographic imaging system large, complex, and easily misaligned. Additionally, the cost associated with such a system makes it difficult for laboratories to utilize the technique. To offset these challenges, a light relay in the form of a fiber wound bundle can be used to record multiple viewing angles on a single high-speed sensor [15]. Individual image resolution is sacrificed for increased benefits such as cost, space, and the flexibility of placement of the final viewing optics. Fiber bundle imaging has been primarily used in biomedical research but has begun to be extended to reacting flow applications to reconstruct chemiluminescence measurements which consist of large (O(10−3 m)) cohesive objects (i.e., flame fronts) [1624]. However, the prior research utilizing the fiber bundles has not focused on reconstructing tracer particle fields which consist of small-scale objects (O(10−6 m)) with a more discontinuous intensity distribution. The fiber-coupled approach has only recently been applied to velocimetry measurements by Reyes et al. under nonreacting conditions [25]. This study focused primarily on average velocity values and did not explore higher-order statistics (i.e., vorticity and Reynolds shear stress) to determine any deviations from traditional velocimetry measurements.

Tomographic particle image velocimetry under reacting conditions has been shown to impact the amount of ghost particles generated in the reconstruction due to optical distortions caused by the reaction [26]. Additionally, these effects are coupled with losses through mediums such as the fiber bundle and optical filters within the single sensor fiber-coupled system [25,27]. Therefore, this work seeks to assess the system's viability and extent to which a fiber-coupled system can be used as a three-dimensional combustion diagnostic tool by capturing Tomo-PIV measurements for a premixed propane (C3H8) and air Bunsen burner flame. The tomographic imaging results are then compared to traditional planar PIV to determine the accuracy and the factors which drive any discrepancies between the two measurements.

## 2 Experiment

### 2.1 Experimental Facility.

Experiments were conducted using a Bunsen burner oriented vertically with an inner (Di) and outer (Do) diameter of 12 and 17 mm, respectively. Nonreacting flow velocities and equivalence ratios are quantified utilizing air and fuel volumetric flow rates. The air flowmeter has a range of 0.25–4.0 SCFM with a resolution of 0.25 SCFM. The fuel flowmeter has a range of 1.0–10.0 LPM with a resolution of 0.25 LPM. This allows for nonreacting flow velocities from 6 to 20 m/s and equivalence ratios ranging from 0.6 to 1.7. A sketch of the resulting reacting flow-field is provided in Fig. 1 for reference. For the analysis performed in this study, stoichiometric conditions are the only equivalence ratio analyzed. Additionally, nonreacting conditions are also implemented to compare the reacting and nonreacting reconstructions. This is done by utilizing same fuel and air conditions for the reacting test case without igniting the flow-field.

Fig. 1
Fig. 1
Close modal

### 2.2 Optical Diagnostics.

Figure 1 displays a schematic of the tomographic imaging system. The imaging system is comprised of a single high-speed camera, a fiber bundle that is split into four legs, and coupling and alignment hardware. The fiber bundle consists of a 1008 × 1008 10 μm diameter array of fibers that are split into four 504 × 504 fiber legs. The fiber array has a size of 10.08 × 10.08 mm on the combined end and 5.04 × 5.04 mm on the split ends. The end of the fiber consisting of the 1008 × 1008 fiber array is denoted as the proximal end, whereas the split 504 × 504 fiber ends are denoted as the distal ends. The proximal end is coupled to the high-speed camera via a relay lens that is sealed from ambient light. The distal ends are coupled to 25 mm f/1.4 objective lenses via a rotational stage that rotates the fiber end with respect to the stationary objective lens to satisfy the Scheimpflug criteria and seal them from ambient light. Images are acquired with the aperture fully closed to allow for the entire region of interest to be in focus. The lenses are rotated 22.5 deg for both the pitch and yaw.

A Photron SA-Z high-speed camera (Japan) is used to record the Tomo-PIV images. The camera has a 1024 × 1024-pixel CMOS sensor that can capture 20,000 frames per second at full resolution and has an interframe time of 150 ns. The coupling between the high-speed camera and fiber bundle has been previously described by Reyes et al. [25]. The result is one image that contains four 512 × 512-pixel views with spatial resolutions of 7.85 pixel/mm after volumetric reconstruction utilizing the MART algorithm. PIV images are captured using a Photron SA-1 high-speed camera, at a resolution of 1024 × 1024 pixels, which is equipped with a 50 mm f/1.2 focal length lens. The PIV camera is placed on the opposite side of the Tomo-PIV system to allow for simultaneous measurements as depicted in Fig. 1. The resulting field of view is 60 × 60 mm providing a spatial resolution of approximately 17 pixel/mm. A timing box that allows for picosecond accuracy acts as the external clock for both the laser and camera system to ensure synchronization of the PIV measurement.

Planar and tomographic PIV is performed by seeding flow-field with 5 μm aluminum oxide particles which are illuminated using a dual pulsed Nd:YAG Evergreen laser capable of emitting 200 mJ per pulse. An 8 mm laser sheet is utilized to illuminate a suitably large enough volume for the three-dimensional measurement. A test was performed comparing the thick (8 mm) and thin (0.8 mm) laser sheet to confirm that there is minimal deviation in the velocity field of the PIV due to the increased laser sheet thickness. The interpulse time is 20 μs and image pairs are captured at 10 Hz with an exposure time of 1 ms to ensure enough light is captured on the sensor. Previous research has indicated uniformity in light collection across all distal ends and minimal distortion effects to the captured image for this optical setup [25]. However, when comparing intensity measurements between a tomographic system utilizing a fiber bundle and one consisting of full resolution cameras, the measured intensity is approximately –10× less for the fiber-coupled system [27]. Therefore, it is important to quantify the light loss through the fiber-coupled system when imaging the 532 nm wavelength to ensure that there is enough light throughput to allow for accurate particle field reconstructions. The light loss is computed by imaging a backlit panel producing a uniform light field with only the high-speed camera and comparing its mean intensity value to the mean of images captured when the additional components are added. The results are presented in Table 1 and show that there is an overall light loss of 23% across the entire system when accounting for the spectral responses of the individual components of the 532 nm wavelength for both nonreacting and reacting conditions.

Table 1

Light loss across tomographic imaging system as components are added with respective spectral responses

Component532 nm sensitivityMean intensity
High-speed camera72%215
Fiber bundle90%204
532 nm filter94%196
Component532 nm sensitivityMean intensity
High-speed camera72%215
Fiber bundle90%204
532 nm filter94%196

The tomographic reconstruction is calculated using the calibration tool and MART built into LaVision's imaging software, davis (version 10.0). Calibration is performed by capturing an image of a dot pattern that is traversed a known distance from the origin. Five calibration images are acquired within the region of interest and uploaded into davis 10.0 for calculation of the projected volume. The calibration target consists of 3 mm dots with 5 mm spacing from center to center for a total of 121 dots. A sample set of calibration images is presented in Fig. 2. After inputting the target's dimensional parameters, a common origin is selected in each calibration image, a shape finding algorithm locates the dots and records their pixel location, and a third-order polynomial is used to compute the projected volume. This produces a 646 × 696 × 159 voxel volume that represents a physical space of 82 × 89 × 20 mm. Only 66% of the volume is used to avoid inaccuracies on the edges of the reconstructed volume limiting the final interrogation region to 425 × 460 × 105 voxels which provides a physical space of 54 × 58 × 13.5 mm. The single sensor fiber-coupled system is reported to have spatial calibration errors of 0.07 of a pixel or 5 μm and has a three-dimensional location uncertainty of 0.1% [25]. The calibration is further optimized by using davis 10.0's self-calibration tool to self-correct for any minor discrepancies in the projected volume by utilizing 825 realizations of a recording as an input.

Fig. 2
Fig. 2
Close modal

LaVision davis is used for processing of both the PIV and Tomo-PIV measurements. All images are preprocessed using a nonlinear sliding minimum filter. The PIV images are processed using a multipass technique utilizing a 64 × 64 pixel window which is decreased down to 16 × 16-pixel interrogation region with a 75% overlap for the reacting conditions. The resulting interrogation area is 0.92 × 0.92 mm. Tomographic PIV images are processed by first performing a volumetric reconstruction utilizing the MART algorithm. The resulting volume is then processed using a multipass 3D cross-correlation for each image pair, the interrogation volume begins at 96 voxels with an 8-voxel search radius down to a final size of 16 voxels with 1-voxel search radius. Each pass of the 3D cross-correlation is implemented using a 75% overlap. The final interrogation region is a 2.04 × 2.04 × 0.51 mm volume. For both measurement techniques, a universal outlier detection scheme which employs a 3× median filter is used to detect and remove any spurious vectors which could negatively impact the flow-field statistics [28]. This technique has been applied successfully to previous reacting flow experiments [29,30]. The uncertainty for the u and v velocity components is reported from davis software to be 0.41 and 0.52 m/s for the planar data and 0.8 and 1.3 m/s for the tomographic data. This uncertainty in each component primarily accounts for the uncertainty in the correlation statistics between image pairs [31]. This approach for uncertainty quantification is based on symmetry (or asymmetry) of the pixelwise intensity contributions for each image pair. Low uncertainty is linked with symmetric contributions, while high uncertainty is linked with asymmetric contributions in the correlation between image pairs. Additionally, previous research explored other experimental parameters such as particle image diameter and seed density to determine optimal values for examining higher-order statistics such as vorticity and Reynolds shear stress [32]. To further minimize the uncertainty when experimentally measuring these higher-order statistics, a particle image diameter of approximately 2 pixels and a densely seeded flow are utilized to align with previous research investigations performed by Wilson and Smith.

## 3 Results

Particle image velocimetry and Tomo-PIV measurements are captured on a laboratory Bunsen burner and repeated at a single equivalence ratio to obtain a suitable sample size to for averaging and higher-order statistic measurements. An example image of the Tomo-PIV and PIV is presented in Fig. 3. To validate the tomographic PIV results, it is important to characterize several parameters which have been presented in prior research in nonreacting and full resolution applications by Scarano such as the SNR, residual of the curve fitting function, and disparity vector [13]. A fundamental problem in Tomo-PIV is the presence of ghost particles in the reconstruction which can skew the resulting velocity field. It has been shown in previous results that a SNR, which is defined as the intensity of the reconstructed particles versus the ghost particles, of greater than 2.0 is the lower limit for acceptable reconstructions. The SNR is measured for the current experiment is determined by reconstructing a volume which is larger than the illuminated region caused by the laser sheet. The average intensity of the image within the laser sheet (intensity of actual particles) can then be compared to the average intensity outside of this region (intensity of ghost particles). The average intensity of the reconstruction is extracted in each XY plane and presented in Fig. 4 as a function of Z-voxels. The signal-to-noise ratio is then determined by comparing the intensity of the region where the flame resides (approximately 50–120 voxels) to the intensity of the regions on the edge of the reconstruction which should correspond to ghost particle intensity. Comparing these values gives a SNR of 8.7 and 4.0 for nonreacting and reacting conditions indicating that the single sensor fiber bundle system can provide acceptable particle field reconstructions [13], although it is worth noting that the introduction of the flame decreases the SNR by approximately half. Combining this with information from the calibration process and use of the volume self-calibration tool indicates that all parameters listed in prior literature have been met. The quantities examined thus far regarding the Tomo-PIV are all digital quantities which does not guarantee the physical quantities extracted (i.e., velocity) are accurate. Therefore, the velocity results for the Tomo-PIV and PIV are compared to assess the measurement technique.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

### 3.1 Average Results.

First, the velocity information is compared using the average vector fields which are calculated using a total of 825 image pairs. To ensure that 825 vector fields are sufficient for statistical analysis, a convergence test is performed by analyzing the average velocity and vorticity as a function of the number of vector fields. It is determined that each statistic reaches a nominally constant value after approximately 300 frames, ensuring that a sufficient number of frames are captured for analysis. For accurate comparison, the coordinate systems from each diagnostic are normalized to begin at the top left corner of the Bunsen burner. Since the PIV field has a higher vector resolution, it is interpolated down to match the coarser vector field of the Tomo-PIV. Additionally, since traditional PIV uses a thin laser sheet to illuminate the particle field, the Tomo-PIV vector field is averaged using planes in front and behind the center of the burner to match the laser sheet thickness. The results are presented in Fig. 5 with contours of the axial velocity (v) and vorticity (ωz) fields. The bulk axial velocity features are present in both the planar and tomographic data; however, notable discrepancies lie in the lack of velocity information below 0.5 Di in the Tomo-PIV data due to the visual obstruction from the base of the burner in the lower two views (lens 3 and 4 from Fig. 3). Similarly, the contours for ωz also presented in Fig. 5 demonstrate the Tomo-PIV's ability to capture the relevant flow features as seen by the shear layer regions depicted by the high vorticity magnitude regions.

Fig. 5
Fig. 5
Close modal
To examine the differences more closely, two velocity profiles are extracted at 1.5 and 3.0 Y/Di above the lip of the burner and are presented in Fig. 6(a). The results indicate that the correct trends are extracted from the Tomo-PIV data but it underpredicts the velocity magnitude by an average of 0.6 m/s with a maximum of 2.5 m/s for both heights. This underprediction in the velocity field is attributed to the presence of ghost particles which can lead to the cross-correlation incorrectly locking onto intensity peaks which can result in an underpredicted velocity magnitude. To assess the deviation between the two measurements, the absolute and relative errors are examined within the flow-field. The absolute error and relative error are defined as
$ϵabs=VTPIV−VPIV$
(1)
$ϵrel=VTPIV−VPIVVPIV×100$
(2)
Fig. 6
Fig. 6
Close modal

The PIV velocity field is treated as the ground truth to quantify how the Tomo-PIV deviates from a standard measurement technique commonly utilized in combustion research. Extracting the velocity within the region between these two profiles within the flame region and comparing the results provide a relative error in the measurement of 3.8%. The u-velocity component is similarly presented in Fig. 6(b), although the velocity component is an order of magnitude smaller than the axial velocity resulting in the relative error percentage increasing to 25%. It is noted that the absolute error is approximately the same for both velocity components, but the relative error is larger for the u component due to being an order of magnitude smaller.

Higher-order statistics, which rely on accurate velocity measurements, need to be validated for the single sensor approach due to their importance for developing a deeper understanding of turbulent reacting flows. The simplest of these first-order statistics is vorticity. Profiles are extracted at the same heights as the axial velocity is presented in Fig. 6(c). Within these profiles, the Tomo-PIV measurement tends to underpredict, similarly to velocity, by an average of 86 1/s and a maximum of 457 1/s. In the central region where the vorticity is nominally zero, fluctuations are experienced, but the Tomo-PIV is still capable of accurately predicting the vorticity. The relative error of ωz in the peak regions is approximately 10%, while there is an increase in the central region due to the lower order of magnitude.

Since both direct comparison and first-order statistics agree, the Reynolds shear stress is evaluated to further assess the order at which statistics begin to diverge. The Reynolds shear stresses Rxx and Ryy for both tomographic and planar results extracted from a height of 1.5 Y/Di above the burner and are presented in Fig. 7. The profiles indicate that the tomographic measurement has a strong tendency to overpredict the planar measurement, although it is worth noting that common peaks arise in the same locations and overall trends remain similar with the primary difference lying in the magnitude. Extracting relative error from these profiles indicates a relative error of 95% and 70% for Rxx and Ryy, respectively. Expanding beyond the single profile and examining the region occupied by the flame, the relative error for Rxx is 70% and Ryy is 51%. The driving factor in this error is the difference between the velocity fluctuations measured in both the Tomo-PIV and PIV. A convergence test was performed for the Reynolds shear stress to confirm that the sample size was adequate for second-order statistic measurements. Each approach converges to a nominally constant value; however, the Tomo-PIV value is considerably larger. This indicates that the error is a result of the measurement technique and not due to a lack of convergence for the turbulent statistical analysis.

Fig. 7
Fig. 7
Close modal

To assess the causes for the inaccuracy for each statistic, the relative error of each is calculated as a function of the height above the burner. The absolute and relative errors are averaged at each height for only points within the region consumed by the flame. The results for the axial velocity (v), vorticity (ωz), and Reynolds shear stress Ryy are presented in Figs. 8(a) and 8(b). It is evident that there is a height at which each statistic becomes nominally constant, which is 1.2, 1.7, and 2.3 Y/Di for v, ωz, and Ryy, respectively. The axial velocity reaches a nominally constant value first which indicates that this error is primarily a function of viewing angles. As the height above the burner is increased, there is increased overlap between the images which results in a higher confidence reconstruction. As the reconstruction quality is increased, the probability of the 3D cross-correlation accurately capturing the flow direction and magnitude increases as shown in Fig. 8.

Fig. 8
Fig. 8
Close modal

The error for the vorticity (ωz) becomes constant further downstream demonstrating that there is another error source present. This additional source error comes from the velocity gradient measurement, which is increased in regions of high vorticity production (i.e., burner shear layer). The error decreases further downstream as the shear layer expands in width and the vorticity begins to dissipate. This indicates that the Tomo-PIV measurement has difficulty extracting sharp gradients experienced in the flow-field. The impacts of low-resolution imaging are experienced here as the vector resolution becomes limited, reducing the measurement technique's ability captures spatial gradients.

The final statistic Ryy is shown to consistently have the highest relative error and does not become nominally constant until 2.3 Y/Di downstream. Prior research has demonstrated and confirmed that velocity fluctuations continuously decrease downstream of the burner [33,34]. This indicates that as the fluctuations become less intense, the Tomo-PIV produces more similar results to the PIV measurement. Since the Reynolds shear stress is a function of the velocity fluctuations, which are reliant on instantaneous measurements, this provides an indication that there is a significant deviation between the instantaneous measurements. To quantify the deviation between the two measurements, the correlation confidence value quantified by davis software is examined. It is noted that there is a 37% decrease in the correlation value in the Tomo-PIV which is also coupled with an increase in the standard deviation. This indicates that the fluctuations in correlation confidence are resulting in an increased number of spurious vectors within the domain. This could be a result of the MART algorithm not effectively reconstructing the volume, the presence of ghost particles resulting in incorrect locking of the cross-correlation, or fiber bundle grid lock. These causes in addition to previous errors measured result in the overprediction of the velocity fluctuations.

Determining the limitations of this optical setup as a function of height above the burner is of interest because many research studies have been performed using planar diagnostics on domains of interest which span from the near field to downstream of the flame stabilization device. In the near field, several studies have been performed which rely on determining the vorticity, flame strain, and turbulence levels to understand the physics of the reacting flow-field [3538]. These measurements rely on accurate quantification of the velocity field, spatial gradients, and the fluctuations. To further expand on the knowledge presented in prior research, experiments need to be performed which resolve the full three-dimensional flow-field with high accuracy. Understanding the potential error sources which can arise based on the domain of interest shows that careful consideration of view orientation and image resolution are required when determining the proper experimental Tomo-PIV setup for specific sets of research objectives.

### 3.2 Instantaneous Results.

Average flow-fields can provide valuable insight into the reacting flow physics, but instantaneous results are key to further developing understanding of the temporal dynamics of turbulent reacting flow-fields. Additionally, the examination of the Reynolds shear stress within the average data provided some insight into potential fluctuations occurring within the instantaneous Tomo-PIV data which are further explored. Similar to the average results, instantaneous velocity profiles are presented at the same heights above the burner as before in Fig. 9(a). The profiles show the Tomo-PIV's ability to capture the flow-field, with variations occurring on average of 2.0 m/s within the profiles and maximum of 12 m/s. It is beneficial to examine the absolute error in measurement throughout the domain of interest to determine the effectiveness of the Tomo-PIV system. A contour of the absolute error, which is calculated in the same manner as described in Sec. 3.1, is shown in Fig. 9(b). The error contour begins at approximately 1.0 Y/Di above the burner as Fig. 8 illustrated that error prior to this height is significant. The contour indicates that the Tomo-PIV measurement corresponds well with the PIV measurement within the domain above burner; however, there are pockets within the vector field which experience large absolute error. The instantaneous data presented represent the time instance where the lowest average error occurred within the domain of interest; in the case, the average error between the two measurements was approximately 7%. However, there are pockets of high error which arise due to the previously mentioned sources of error. It is important to note that although the average relative error is low at the time instance depicted, the values fluctuate and have a maximum of approximately 20%. The average instantaneous error taken across all sample sets within the same flame region is determined to be 13% depicting reasonable agreement between the measurement techniques, although nearly four times as large as the relative error demonstrated for the average results.

Fig. 9
Fig. 9
Close modal

The instantaneous vorticity fields are presented in Fig. 10 for the same time instance as Fig. 9. While the same general flow features are present, the fields are less similar than the average results previously presented in Fig. 5. When comparing average error spatially within the shear layer regions, an average error of 30% occurs at the time instance depicted, while the mean error for all samples increases to approximately 51%. An instantaneous profile at Y/Di = 1.5 is also presented in Fig. 11 which further demonstrates the increased noise and error present in the Tomo-PIV measurement. On an instantaneous basis, higher-order statistical analysis of reacting flow-fields becomes limited as calculations involving gradients become increasingly noisy and uncertain due to limitations in image resolution and accuracy in the reconstruction and cross-correlation.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

### 3.3 Flow-Field Three Dimensionality.

Although there is a loss of accuracy when performing single sensor tomographic diagnostics, the tradeoff becomes the acquisition of the full velocity gradient tensor allowing for more complete analysis. These results have indicated that despite discrepancies in the magnitude of velocity calculated, the general flow features extracted from both measurements remains similar. Figure 12(a) demonstrates the full three-dimensional axial velocity (v) field above the burner which could not be captured using traditional measurement techniques which illustrates the strong flow acceleration due to the density decrease as reactants are converted into products. Furthermore, the Tomo-PIV measurement allows for the spanwise velocity (w) to be analyzed and visualized across the entire measurement domain as depicted in Fig. 12(b). It is seen that the areas impacted most by this velocity component are in the shear layer region where the vorticity generation is the highest and the greatest three-dimensional motion is anticipated. Since the planar PIV only allows for the measurement of two velocity components (u,v), the continuity equation can be used to approximate the gradient dw/dz by
$dudx+dvdy+dwdz=0$
(3)
Fig. 12
Fig. 12
Close modal

This allows for a comparison between a two-dimensional approximation of dw/dz and the results extracted from the Tomo-PIV measurement. The gradients are calculated from PIV and extracted directly from the Tomo-PIV from the products' region and presented within Fig. 13. The results indicate that the Tomo-PIV measurement is approximately 1.7 times greater than the PIV and spans a smaller distribution of values. To further quantify this measurement, an examination of the substantial derivative of velocity is performed. The substantial derivative is calculated for each within the products' region for both. The substantial derivative of the axial velocity Dv/Dt is examined, and the difference between the two measurement techniques is 6.6% which provides further insight into the three dimensionalities of the flow-field. Although this flow-field lacks significant contributions from the z-component of velocity, it does contribute to terms that can only be calculated with the full velocity gradient tensor available.

Fig. 13
Fig. 13
Close modal

## 4 Conclusion

Tomographic particle image velocimetry is performed on a premixed propane–air flame utilizing a single sensor fiber optic approach. The system uses a single high-speed camera coupled to a fiber bundle to acquire four viewpoints simultaneously on one image sensor. The optical setup can reconstruct a volume of 82 × 89 × 25 mm volume equating to 646 × 696 × 159 voxels, but only values in a 425 × 460 × 109 voxel range are used to avoid inaccuracies on the edge of the reconstruction domain. It was found that a total light loss of 23% through the entire system is experienced when imaging the 532 μm wavelength band to isolate the illumination caused by the laser pulse.

Tomographic and planar PIV measurements are performed on a Bunsen burner flame at a constant equivalence ratio to compare velocity field measurements. The signal-to-noise ratio of the Tomo-PIV reconstructed particle field is first calculated and compared for nonreacting and reacting conditions. It is determined that the single sensor facility can provide SNR's which are necessary for acceptable reconstructions, although it is worth noting that the introduction of a flame reduces the SNR by approximately half. First, the average velocity fields are compared, and it is seen that the Tomo-PIV system provides accurate results in the bulk flow direction. Similarly, the vorticity is examined and shown to produce accurate results as well. The average Tomo-PIV data begin to diverge significantly when comparing the higher-order statistics such as Reynolds stress where it has a strong tendency to overpredict compared to the 2D results. This indicates that the fiber-coupled Tomo-PIV approach is a feasible diagnostic approach for reacting flow-fields when analyzing average velocity and vorticity but has significant limitations when measuring second-order turbulence statistics.

To isolate potential sources of error for each statistic, the relative error is plotted as a function of height above the burner. It is shown that each statistic reaches a nominally constant error at a specified height above the burner. The axial velocity occurs first with the vorticity and Reynolds's shear stress occurring further downstream. The relative error of the axial velocity component is attributed to the viewing orientation. The vorticity measurement is then linked to the viewing orientation and image resolution which limits the spatial gradients that can be accurately calculated. Finally, the Reynolds shear stress error is attributed to reconstruction inaccuracies as the overprediction of velocity fluctuations drives the high magnitude of error. These reconstruction inaccuracies result in a decreased cross-correlation confidence coupled with an increased standard deviation which drives the overprediction in Reynolds shear stresses.

Since the Reynolds shear stress indicated that there were fluctuations occurring within the instantaneous Tomo-PIV velocity fields, they are examined to determine the significance in the deviation between the measurement techniques. The axial velocity was determined to have an average relative error of 13% when comparing each time instance. Although it can have a minimum relative error of 7% on an instantaneous basis, there are time instances which reach 20%. While the Tomo-PIV can capture instantaneous velocity fields accurately, the relative error increases significantly when extended to higher-order statistics such as vorticity. Analysis on first-order statistics is prone to increased error due to the low resolution of the images, resulting in decreased resolution of the vector field which contributes to increased noise in spatial gradient measurements on an instantaneous basis. The optical diagnostic approach becomes limited to zeroth-order statistics, inhibiting its usefulness as combustion diagnostic tool on an instantaneous basis.

The single sensor fiber-coupled approach to Tomo-PIV is shown to be capable of capturing accurate velocity and vorticity fields for reacting flows on an average basis; however, the accuracy begins to diminish on an instantaneous basis. In particular, the system does not reliably capture second-order turbulence statistics, which warrants further exploration of the accuracy of the particle field reconstructions utilizing a fiber-coupled system. The impacts of the viewing angle, image resolution, and the fiber bundle grid on the cross-correlation procedure in PIV applications should be further explored to minimize any errors which arise in instantaneous velocity and higher-order statistic measurements. Further research examining these topics will allow for the optical diagnostic technique to be extended to increasingly turbulent flows and allow for more complete turbulent analysis to be performed.

## Funding Data

• Air Force Research Laboratory (FA8650-17-C-2029).

• Air Force Office of Scientific Research (19RT0258/FA9550-19-0322). Public release (88ABW-2018-6179).

## Nomenclature

• Di =

inner diameter of the Bunsen burner

•
• Do =

outer diameter of the Bunsen burner

•
• MART =

multiplicative algebraic reconstruction technique

•
• PIV =

particle image velocimetry

•
• Rxx =

Reynolds shear stress x-component

•
• Ryy =

Reynolds shear stress y-component

•
• Tomo-PIV =

tomographic particle image velocimetry

•
• V =

axial velocity

•
• ϵabs =

absolute error measurement

•
• ϵrel =

relative error measurement

•
• ωz =

spanwise vorticity component

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