We describe the design and operation of a high speed optical tomography system for measuring two-dimensional images of a dynamic phase object at a rate of 5 kHz. Data from a set of eight Hartmann wavefront sensors is back-projected to produce phase images showing the details of the inner structure of a heated air flow. Series of animated reconstructions at different downstream locations illustrate the development of flow structure and the effect of acoustic flow forcing.
©1997 Optical Society of America
There is considerable interest within the fluid dynamics community in sensing organized structure in transparent fluid flows at high spatial and temporal resolution with a non-intrusive measurement. Because index of refraction variations within clear air produce distortions that limit the performance of imaging or beam transmission systems, measuring and understanding the structures that produce these distortions can lead to improved methods of adaptive correction. Optical tomography has been used in the past to capture three-dimensional (3-D) images of fluid flows.1 However, in these cases the reconstructions are either snapshots isolated in time or time-averages, with system speed being limited by the large amount of data necessary to acquire and the methods used to acquire it. The tomographic system described in this paper is limited in the number of views, and therefore in spatial resolution. More importantly, each sensor in the system is limited to one spatial dimension so that high data rates can be maintained with inexpensive electronics. We demonstrate tomographic data acquisition at 5 kHz. The entire measurement and reconstruction process is performed electronically so that with more sophisticated acquisition and processing hardware the speed and resolution can be dramatically improved leading to the possibility of high speed 3-D tomography.
2. Optical tomography system
The tomographic system, shown schematically in Fig. 1, is composed of a set of eight one-dimensional (1-D) Shack-Hartmann wavefront sensors2,3 arranged at regularly spaced angular intervals in a semi-circular arc around the flow. The sensors simultaneously measure the path-integrated phase of laser radiation after it has propagated through the near field of a low speed heated jet. These path-integrated phase measurements, called projections, are used as inputs to a standard tomographic reconstruction algorithm to create two-dimensional images of a planar cross section of the flow.
Each Hartmann sensor is simply a 2048-pixel linear CCD camera located at the back focal plane of an 1-D array of 64 small cylindrical lenses that focus light along the axis of the array. The camera and lenslet array are oriented perpendicular to the direction of the flow. Each lenslet is 437.5 microns in diameter and has a focal length of 4.0 cm.
To create a projection, a collimated laser sheet (670 nm) propagates through the flow with the plane of the sheet perpendicular to the flow direction. Temperature variations in the flow produce index of refraction irregularities according to the expression,
where λ is the wavelength [μm], P is the atmospheric pressure [mbar] and T is the temperature [K]. Index of refraction variations along the path induce an optical path delay (OPD) in the beam relative to a path through free air at constant ambient temperature. The OPD of beam path i is given by:
where i refers to the ith lenslet in the array and Ta is the ambient temperature outside of the flow.
The beam is focused by the lenslets into an array of foci that are intersected by the CCD pixels. Temperature induced OPDs within the expanded beam produce shifts in the positions of individual foci. Thus, the foci jitter as temperature variations in the flow move past the sensor. At any instant in time the displacement of the focal spots recorded by the CCD camera is proportional to the local 1-D tilt on the optical wavefront in the same direction,
where Δϕi is a small tilt angle in radians, δi is the focal spot displacement and fL is the lenslet focal length. The tilt can also be expressed as the difference in OPDs from one edge of a lenslet to the other, or,
where d is the width of the lenslet. Equation (4) is approximately the derivative of the OPD. Therefore, once the wavefront tilts are calculated from focal spot displacements at the wavefront sensor, typically by calculating the focal spot’s centroid location, integration of the Δϕ measurements along the length of the sensor yields the OPD:
Tomographic reconstruction is the process of recovering the original two-dimensional index of refraction variations from their integral OPD projections. The OPDs are essentially a Radon transform of the original refractive index data. If the Radon transform were sampled continuously around the flow, the index of refraction image could be obtained by direct transform inversion. However, for most practical systems, data is limited both in number of views and samples per view. Therefore, in practice, direct inversion cannot be applied. Numerous solution methods are available for solving the data-limited imaging problem4 and, recently, these methods have been extended to handle reconstruction from wavefront tilts instead of integrated OPDs.5 The results shown here have been produced using an iterative algorithm employing an algebraic reconstruction technique for solving sets of linear equations.
The 64-lenslet-per-view Hartmann design resulted from a study of focal spot location error as a function of f-number.3 The 8-sensor tomographic system design, shown schematically in Fig. 1, resulted from studies3,6 of reconstructed image quality as a function of the number of projections and number of lenslets per projection.
Simulations of noise and measurement errors in individual Hartmann sensors and reconstruction errors from the combined set of sensor measurements4 produced a system model summarized in Fig. 2. In the system model, an optical wavefront is propagated through a model of the flow and detected by a model of the wavefront sensor. The simulated intensity is input to the spot location, tilt, and OPD calculation algorithms and the resulting OPD is compared to the true OPD of each projection to arrive at the error. A unique error is calculated for each view by applying random electronic noise and non-axisymmetric models of the flow. Reconstructions of the flow created from the simulated projections are compared to the flow models, which are used as truth images, in order to determine the errors of the 8-view tomographic system. A set of non-axisymmetric flow models were created from computational fluid dynamics (CFD) data of a round jet. The CFD models were supplemented with actual cross-sectional flow visualizations of the jet. Although the visualizations were not originally a quantitative representation of the flow, they were scaled to match the size and (centerline) temperature of the flow.
In the comparison of the reconstructed and truth images a set of performance metrics were used including single-valued metrics such as average, maximum, and rms error over the entire image. Because tomographic reconstruction often results in high spatial frequency artifacts, these single-valued metrics can be inconsistent with the information content actually present in the reconstructions. To extract information in the presence of high frequency noise we employed a correlation metric6 to identify the spatial frequency cutoff for low-pass filtering. The performance of the system after filtering shows that there is a trade-off between resolution in a reconstruction defined by a particular filter cutoff and rms temperature error. For a reconstruction resulting from the 8-view system, a higher spatial frequency filter leads to slightly higher rms error.3 We believe this effect is due mostly to the restricted number of views. At 2-mm resolution the spatial rms error in temperature is calculated as 0.7 C, or about 7% of the maximum temperature variation we expect to measure. A discussion of error measurements in reconstructed experimental data is given in section 4.
3. Reconstructions of dynamic fluid flow
The utility of the high speed system is demonstrated by imaging a dynamic air flow. In all of the experiments described here, the flow is a low-speed round heated-air jet created using a blower that forces room temperature air across heaters, into a plenum chamber, and out through a nozzle with an entrance diameter of 19.7 cm and an exit diameter (D) of 1.27 cm. The centerline temperature of the flow at the nozzle exit is about 10 C above ambient room temperature, the centerline velocity is 8 m/s. After the air exits the nozzle, its laminar flow becomes unstable due to Kelvin-Helmholtz instability waves7, excited acoustically at the nozzle exit plane. These instabilities evolve into periodic vortices in which cool ambient air is rolled up into the heated jet air creating well defined ring-shaped interfaces between hot and cold fluid in the shear layer around the core of the flow. A streamwise cross-sectional flow visualization of the vortices in the shear layer is shown in Fig 3. It is these vortices that provide the spatially and temporally evolving index of refraction features that are detected by the optical tomography system.
Several distinct modes governing formation of these vortices can be excited using frequencies from small acoustic speakers.8 A uniform sound pressure level from a small speaker located in the plenum directly beneath the nozzle excites axisymmetric modes of the shear layer at the nozzle exit plane. Helical modes, that spiral around the flow, are excited by varying the sound pressure level around the jet perimeter using a set of speakers.9 Tomographic reconstructions depicting several of these modes are shown in Fig. 4 – 6. Each sequence is composed of 256 individual tomographic reconstructions. The data for each sequence was acquired at 5 kHz. These images are 8-bit digital representations of temperature variations across one plane of the flow orthogonal to the flow direction with color brightness representing temperature. In the sequences shown the actual temperatures represented range from 24 to 42 C.
In Fig. 4, tomographic reconstructions depict the downstream development of shear layer structure at three downstream locations in the jet. The position of each sequence is given in units of nozzle exit diameter, D, and the locations are at 1D, 2.5D, and 3.5D. The sequences in Fig. 4(a)–(c) show the periodic passage of the vortices under the influence of a forcing frequency of 470 Hz. The sequence at 1.0D in Fig. 4(a) clearly shows the ring-like rolled vortex surrounding the core of the flow as it passes through the tomographic plane. Further downstream, at 2.5D and 3.5D, the vortex pattern become more complex as the structures develop, grow and combine. Eventually the shear layer grows large enough that the core of the flow disappears.
The sequences in Fig. 4(d)–(f) were recorded using a 590 Hz forcing signal and display a persistent pentagonal vortex structure where side jets eject heated air from each corner.10 These jets are related to streamwise vortex structures and therefore persist in the sequence over many snapshots. The only difference between the sequences shown in Figs. 4(a)–(c) and 4(d)–(f) is the forcing frequency, 470 and 590 Hz, respectively. Comparison of the images shows distinct differences in the flow structure due to forcing. Such structural differences can only be inferred from anemometry probe measurements that do not possess the spatial resolution of the tomography data.
The effects of azimuthal forcing9,11 were also studied using a set of eight small speakers flush mounted into the nozzle exit plane at 45 degree intervals in a circle around the jet. Helical modes were excited by driving the speakers at 530 Hz and varying the sound pressure level with azimuthal location γ around the jet perimeter as cos(mγ), where m is an integer. The m= ± 1 case shown in Fig. 5, establishes a shear layer vortex mode structure in the shape of a stretched helical spring wrapped around the core of the flow. This vortex structure appears to “flap” from one side of the jet to the other as the helix travels through the plane of observation at 1.5D. More details of the aero-optical studies performed on the jet flow using the tomographic technique will be outlined in a forth-coming publication.12
In addition to viewing the tomographic reconstructions as an animated sequence, the snapshots can also be stacked to form a two dimensional time history of the structures passing through a plane of the flow. When plotted as a surface of constant temperature, as in Fig. 6(a), such a stack shows at a glance the passage of ring vortex features. As shown in Fig. 6(b), a cut through the stack of 64 slices reveals the inner structure of the periodic rolled vortices. Since a cut can be made through the tomographic stack in any direction, we may examine the flow data at any angle.
It is possible to produce true 3-D reconstructions with this system under the restrictive condition of strictly periodic dynamics using conditional sampling techniques.13 A more general solution is to replace the 1-D Hartmann sensors with 2-D wavefront sensors. Because of the digital data acquisition and processing format of the Hartmann sensors, the sampling rate, number of samples per view, and number of views can grow with improving detector array readout rate, acquisition hardware, and processing power.
4. Experimental reconstruction accuracy
Quantitative comparison of experimental tomographic reconstruction to simulated results is made difficult by the lack of knowledge about the true spatial distribution of the temperature in any particular realization of the flow. We can attempt to infer the accuracy of the reconstructions by comparing the temporal features of the reconstructions to the temporal behavior of the flow. Because the core of the flow remains essentially laminar we can assume that the temperature of the central region of the flow remains constant at a level elevated from ambient. The temperature should also remain constant at ambient outside the bounds of the flow shear layer. Constant temperatures in these two regions have been measured experimentally using temperature probes.
Comparison of mean probe and mean tomographic temperature measurements shows agreement at a point at the flow center and at a point outside the flow. The rms of tomographic temperature at a point outside the flow also shows agreement with the rms of probe measurements at that point. However, the rms of temperature measurements in the center of the tomographic reconstructions varied considerably from the probe-measured central rms due to fluctuations of up to 4 C in the tomographic temperature measurement. Because these fluctuations were periodic with the same period as the shear layer vortices, we surmised that temperature error at the center of the tomographic reconstructions is sensitive to the surrounding flow structure. In support of this assertion we found that these fluctuations occurred in noise-free simulations and that the temperature error was maximum when the vortex ring structure was most fully developed and minimum when the structure was least developed. We also found that different reconstruction algorithms responded differently to the structure. These results show that a quantitative study of tomographic reconstructions should consider that the results of simulations are strongly dependent on the structure in chosen model set.
An optical tomography system made from off-the-shelf components that is capable of operating at data acquisition frame rates of up to 5 kHz has been used to obtain spatially resolved high speed cross-sectional temperature images of a round heated jet. These tomographic images show dynamic details in the evolving vortical flow structures found in the shear layer of the jet. Reconstructions produced by the system are quantitative temperature distributions of a planar cross section of the jet whose accuracy at a spatial resolution of 2.0 mm was calculated at 0.7 C in simulation for the 8-view system. Actual temperature measurement accuracy was found to vary with the complexity of the structure present in the flow.
We have used the tomographic reconstructions and other optical techniques to measure the effect of acoustic forcing on axisymmetric and helical modes in the near nozzle region of the jet. Investigations performed to date have identified the dominant modes in the turbulent axisymmetric jet. Our data acquisition system does not push any fundamental limitations in resolution or speed, both of which may be enhanced through custom electronics design. Tomographic reconstruction of the data is performed separately and in the data limited case described here reconstruction of a sequence of 256 images takes a few seconds. Increases in the amount of data necessary for higher resolution measurements may eventually limit the speed of the tomographic reconstruction process. High spatial and temporal resolution in a non-intrusive optical flow measurement system has potential application in wind tunnels, internal combustion studies, and in flows containing complex structure where intrusive probes may be inappropriate. High speed tomography may also be suitable for the experimental verification of numerical flow models.
We gratefully acknowledge the contributions of Ellen Chen and Kenneth Bishop (ATA, Inc.) to the content of this paper, and Mervin Kellum and Matthew Fetrow (ATA, Inc.) in the preparation of the animated figures. This work was supported in part by the Air Force Office of Scientific Research.
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