A numerical investigation of high-resolution multispectral absorption tomography for flow thermometry
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Multispectral absorption tomography (MAT) is now a well-established technique that can be applied for the simultaneous imaging of temperature, species concentration, and pressure of reactive flows. However, only intermediate spatial resolution, on order of 15 × 15 grid points, has so far been achievable in previous demonstrations. The aim of the present work is to provide a numerical validation of our MAT algorithm for thermometry of combusting flows, but with greatly improved spatial resolution to motivate its experimental realization in practical environments. We demonstrate a grid resolution that is comparable to that of classical absorption tomography (CAT) containing 80 × 80 elements from only two orthogonal projections, which is impractical to realize with CAT but especially desirable for applications where optical access is limited. This is achieved using the smoothness assumption, which holds true under most combustion conditions. The study shows that better spatial resolution can be obtained through a simple increase in the spatial sampling frequency for the two available projections, as the smoothness condition becomes more reliable on smaller spatial scales. Our work also demonstrates the first application of MAT for full volumetric reconstructions. The studies thus provide robust guidelines for the implementation of MAT over large spatial scales and lay solid foundations for its development and application in complex technical combustion scenarios, where spatial resolution is crucial to investigate the interaction of flow phenomena with chemical reactions.
KeywordsReconstruction Quality Smoothness Condition True Profile Spectral Sampling Wavelength Modulation Spectroscopy
Optical imaging techniques are indispensable for the resolution of non-uniformities in technical flow fields [1, 2]. Generally the techniques can be divided into two categories, which are planar imaging techniques on the one hand and tomography, on the other. As evident from its name, the former category, which includes planar laser-induced fluorescence/phosphorescence [3, 4] and Raman/Rayleigh imaging [5, 6, 7], is two-dimensional in nature and requires a pulsed laser source for the selective illumination of the plane of interest. The signal stems from the interaction of the illumination light field with the gas molecules, and the generated light emission is imaged directly onto a two-dimensional detector array, typically a camera. Mathematically speaking, planar imaging techniques can be considered as a straightforward linear field mapping operation. On the other hand, tomography relies on the mapping of integrals of the target field along the line of sight, LOS, which in what follows we refer to as integral mapping . As a consequence, the target field has to be integratable along the LOS and the corresponding integrals have to be physically meaningful. Since emission along the LOS is accumulative and hence integratable along its path toward the detection plane, essentially all planar imaging methods can be upgraded into 3D tomographic modalities. In practice, limitations are only set by the available optical access to the system under study and the excitation power available for volumetric sample illumination. In contrast to planar imaging, which solely targets on emission fields, tomography can also recover the fields for absorption coefficients, which can be further processed to retrieve other fundamental gas properties, such as temperature, species concentration, and pressure [9, 10, 11, 12, 13, 14]. Compared with emission tomography, the absorption counterpart enjoys further advantages such as being calibration free, species selective, and highly sensitive [15, 16, 17, 18, 19, 20].
However, since only two projections were used in previous implementations of MAT and the spectral sampling could only partially compensate (in a mathematical sense) for spatial sampling deficiencies, the spatial resolution is inevitably undermined. Nevertheless, we point out that there is no limitation per se in the number of projections that can be accommodated by the MAT algorithm, however, at a commensurate loss in temporal resolution due to beam scanning requirements. But in this case, MAT enjoys a much improved reconstruction fidelity due to better immunity against experimental noise such as originating from beam steering, window fouling, and etalon fringing . Moreover, since MAT can be combined with advanced detection techniques, such as wavelength modulation spectroscopy (WMS), it can be used for high- and/or varying pressure scenarios, for which CAT is not optimally suited. In summary, the nonlinear MAT technique offers full flexibility, to either be deployed with high temporal but somewhat limited spatial resolution, as demonstrated in an earlier article, or, as we demonstrate here, at very high spatial resolution at the cost of increased data acquisition requirement.
We demonstrate this effect by increasing the spatial sampling frequency along each projection direction. This finer meshing of the flow filed is permissible under the assumption of the smoothness condition, which is valid for most combustion environments practically encountered. We demonstrate a resolution spatially for grid sizes containing 80 × 80 elements to achieve a resolution that is comparable to that of CAT although requiring only a fraction of projections. For high-resolution MAT, the temporal resolution is furthermore only limited by the bandwidth of the data acquisition system. High-resolution MAT has thus the practical potential for deployment in situations where both spatial and temporal resolutions are crucial, for example to provide full resolution of complicated flow fields such as supersonic combustion systems within ramjets/scramjets .
So far, both CAT and MAT have been limited to applications in two dimensions, simply because of prohibitive experimental costs for the enabling technology. However, the potential for the implementation of MAT with inexpensive tunable diode lasers with coarse wavelength-division multiplexing (CWDM)  has made volumetric absorption tomography a more practical proposition. This provides us with a strong motivation to further develop this technique theoretically and perform numerical validation studies in preparation for experimental demonstrations in the near future.
The remainder of this paper is organized as follows: Sect. 2 briefly introduces the mathematical formulation for the MAT algorithm; Sects. 3 and 4 present studies for large-scale planar and volumetric implementations of the technique, and the final section provides a summary of our findings.
2 Mathematical formulation of MAT
The mathematical formulation of MAT using both direct absorption spectroscopy (DAS) and WMS has been detailed in previous publications [15, 22]. To facilitate the discussion here, we focus on MAT implementations based on DAS as an example and briefly summarize the formulation to use MAT for flow thermometry.
3 Large-scale planar MAT
4 Volumetric MAT
So far, MAT, and indeed CAT, has been limited to 2D situations simply due to prohibitive experimental costs. Fortunately, recent advances in MAT with inexpensive tunable diode lasers have made the experimental realization of volumetric absorption tomography a more realistic proposition. This is what encourages us to perform preliminary numerical studies here in preparation for later experimental demonstrations. We note that volumetric MAT implementation is a worthwhile endeavor not only from an application point of view (i.e., providing full 3D information for the system under study), but it brings advantages also for tomographic inversion process, because extra information gained along directions between adjacent planes offers further constraints (e.g., smoothness between the layers) that make the method even more robust.
In summary, we present the numerical studies of nonlinear MAT for large computational mesh sizes. We demonstrate a spatial resolution with meshes containing up to 80 × 80 grid points for planar MAT requiring just two orthogonal projections. The spatial resolution obtained is comparable with that of CAT, which needs many more projections. Even better spatial resolution is in principle possible at the expense of increased computational cost. We show that reconstruction fidelity is improved simply by increasing the spatial sampling frequency along the available orthogonal projections before the effect of increasing N/I (the ratio between the number of variables and nonlinear equations) outweigh the benefit of smoothness condition. However, it has to be pointed out that for more complicated turbulent flow fields featuring sharper gradients and more intense fluctuations, more projections are necessary to resolve all information on relevant spatial scales. In theory, resolution can be improved by using more projections, but in practice, the achievable resolution is dictated by the maximum number of projections available with limited optical access and the optimal signal noise ratios that are possible. An advantage of the MAT algorithm is that it features a better noise immunity when the same number of projections was used as CAT against experimental noise originating from beam steering, window fouling, and etalon fringing . Finally, we demonstrate that full volumetric MAT is a feasible and realistic proposition for experimental realization. The availability of cost efficient laser and detector technologies mean that full 3D reconstructions of dynamic combusting flows will soon become a reality.
This work was funded by the European Commission under Grant No. ASHTCSC 330840 and was partly performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service. Clemens F. Kaminski also wishes to acknowledge EPSRC for funding (Grant EP/L015889/1).
Supplementary material 1 (MP4 2998 kb)
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