Open Access
Add to BibSonomy
Abstract
A deep learning method for laser absorption tomography was developed to effectively integrate physical priors related to flow-field thermochemistry and transport. Mid-fidelity reacting flow simulations were coupled with a forward molecular absorption model to train a deep neural network that performs the tomographic inversion of laser absorption images to predict temperature and species fields in flames. The method was evaluated through numerical simulation and experimental testing in benchtop laminar flames. The target flow-fields involved a spatially-convolved laminar ethylene-flame doublet, backlit with tunable radiation from a quantum cascade laser near 4.85 µm probing rovibrational absorption transitions of carbon monoxide. 2D images were collected at 11 different projection angles, yielding an aggregate of 50,688 unique lines of sight capturing the scene with a pixel resolution of approximately 70 µm. A convolutional neural network was introduced to efficiently generate temperature and species profiles and trained with a large dataset of large-eddy simulations of laminar flames at variable conditions. The learning-based approach to the inversion problem was found to more accurately predict species and temperature fields of the flame with fewer projection angles, reduce convergence time, and expand the field domain relative to classical linear tomography.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser absorption tomography (LAT) enables quantitative, spatially-resolved temperature and species measurements in reacting flows via the inversion of spectrally-resolved, line-of-sight integrated data [1]. Unfortunately, in non-axisymmetric, three-dimensional flow-fields, the inversion problem is typically ill-posed [2] due to practical limitations in optical access that yield sparse view angles [3,4]. Solution methods for these ill-posed parameter estimation problems can be assisted by incorporating prior information about the flows [5] to facilitate reconstruction, either explicitly via linear regularization [6–11] or Bayesian inference [12–14], or implicitly by assuming a simplified form of the parameter distribution to reduce the degrees-of-freedom in the problem [15–17]. As considerable physical information is typically known a priori about a reacting flow field of interest, it is desirable to integrate this information in the solution method to improve efficiency and accuracy.
In recent years, deep neural network (DNN) models have emerged as an alternative approach to the inversion problem, as they are efficient in capturing complex non-linear relationships, such as between species thermochemical properties and their associated spectra. A distinction in this approach is that priors are effectively introduced via the neural network training process which reflects a forward model of the projection measurement, introducing input-output pairs to train an implicit functional relationship [18,19]. DNNs have been applied in both emission spectroscopy [20–23] and laser absorption tomography to predict species fields in dynamic flows using multiple beam configurations with varying view angles [24–26]. Previous learning-based absorption tomography studies have employed relatively simple stable distributions (e.g. Gaussian) in training data to represent absorption fields [24–27]. However, such generic field distributions are inadequate representations of many reacting flow-field parameters such as intermediate species or gas temperature. These parameters require commensurate training fields that reflect the physics involved.
In this study, we integrate a mid-fidelity reacting fluid dynamic simulation into the forward model to capture end-to-end physics and generate projected absorption fields to train a deep neural network a priori, independent of flow-field measurements; allowing for incorporation of thermochemistry and transport properties that govern temperature and species distributions. We then apply this neural network to invert multi-angle 2D laser absorption images of a 3D reacting flow (doublet laminar flame) to quantitatively measure temperature and species fields. We first describe the deep learning method as coupled to the unique volumetric laser absorption imaging (LAI) optical setup. We then detail the training process and evaluate predictive capability through numerical simulation and experiment. Reconstructed or predicted fields of CO mole fraction and temperature are compared between the deep learning method and linear tomography with various numbers of projection angles in terms of accuracy, artifacts, and computational efficiency.
2. Methods
2.1 Volumetric laser absorption imaging
Here, we employ laser absorption imaging (LAI), a diagnostic method that produces spatio-temporally rich absorption datasets by capturing flows backlit by tunable laser radiation [28,29] with a high-speed camera (Telops FAST-M3K), as shown in Fig. 1. Volumetric LAI involves the collection of 2D images at multiple angles, readily yielding thousands of unique lines of sight [30] that help constrain the 3D reconstruction of flow-field parameters with steep spatial gradients, as in the reaction zone of small-diameter ($<$1 cm) flames.
Fig. 1. Left: Optical arrangement and transmission images at varying rotation angles of Bunsen-style flames. Right: Flow chart of plane-by-plane processing, including incident ($I_o$) and transmitted ($I_t$) pixel intensity, spectral absorbance, Voigt fits, projected absorbance areas $A_{j,\textrm {proj}}$, and resulting sinograms of $A_{j,\textrm {proj}}$ for every angle.
In this configuration, we target carbon monoxide mole fraction and temperature with a distributed feedback (DFB) quantum cascade laser (QCL) near 4.85 $\mu$m that is spectrally scanned across the P(0,20) and P(1,14) rovibrational transitions in the fundamental band of CO in the mid-wave infrared [31]. The beam is horizontally expanded with a cylindrical lens and re-collimated with a concave mirror, then pitched through the flow-field comprising two Bunsen-style flames, each with flame brushes approximately 3 mm in diameter. The beam is spectrally isolated with a bandpass filter (4860 $\pm$ 96 nm), and a plano-convex lens focuses the expanded beam onto the detector array of the camera. The camera detector subwindow of size 128$\times$24 captures the beam with a frame rate of 40 kHz and integration time of 5 $\mu$s. The QCL is injection-current tuned with a 400 Hz sawtooth waveform, resulting in 100 points per scan for subsequent spectral fitting, shown in the center of Fig. 1. The dual flame assembly is mounted on a rotational and vertical translation stage to capture multiple projection angles and heights for the tomography. 2D $A_{j,\textrm {proj}}$ images were collected at up to 11 different projection angles, yielding an aggregate of up to 50,688 unique lines of sight capturing the scene with a pixel resolution of approximately 70 $\mu$m, evaluated by imaging a wire mesh backlit with laser radiation [29]. Mass flow controllers (MKS MFC GE50A) supply reactants with overall flow rates of 128 sccm C$_2$H$_4$, 79 sccm N$_2$, and 101 sccm O$_2$, resulting in a fuel/oxidizer equivalence ratio of $\phi = 3.80 \pm 0.07$. After the tubing is split to the two burners, one flow is measured with a rotameter to ensure equal flow through each burner. The exit velocity of each flow is 0.41 m/s and the jet exits of the stainless steel burners are 1.6 mm in diameter, providing a laminar jet Reynolds number of $\sim$44.
For each projection angle measurement, incident ($I_0$) and transmitted ($I_t$) intensity data is collected and averaged over 1 s (400 scans), an interval over which the flames are assumed steady. For a non-uniform gas medium, the Beer-Lambert law integrated over wavenumber $\nu$ [cm$^{-1}$]—or the projected absorbance area $A_{j,\textrm {proj}}$ [cm$^{-1}$]—can be expressed for each line-of-sight with pathlength $L$ [cm] and related to thermodynamic gas properties in Eq. (1) [32],
2.2 Tomographic inversion
We employ two approaches to the reconstruction process of LAI data, as depicted in Fig. 2. First, we apply Tikhonov-regularized linear tomography, which uses an analytical prior (smoothness) to assist the reconstruction of absorption coefficient ($K_j$) fields, from which we extract temperature and mole fraction via two-line thermometry [32]. We subsequently evaluate a novel physics-trained deep learning-assisted tomography method, which uses reacting flow simulations as training-based priors to assist in the prediction of temperature and mole fraction fields directly. While linear tomography is a more established laser absorption technique for spatially resolved measurements in non-uniform reacting flows [1], the deep learning approach (and associated training process) is hypothesized to enable convenient integration of prior knowledge related to combustion physics, the experimental setup, and spectroscopic properties, which are captured in the forward projection process. We detail both approaches in this section, with specific application to the volumetric LAI setup described in Fig. 1.
As a basis for reference, the forward projection process is modeled as a linear parallel-beam tomography problem and the flow field is discretized into a $100\times 100$ rectangular grid probed by $128$ parallel lines of sight from $N$ projection angles (nominally $N=6$). Writing Eq. (1) for all $N\times 128$ lines of sight yields a system of linear equations:
where $\mathbf {A_{j,proj}}$ and $\mathbf {K_{j}}$ represent the projection sinogram and 2D field of the absorption coefficients, respectively, both in vector form. $\mathbf {A}$ is the projection matrix, where $A_{ij}$ represents the absorption length for the $i$th beam passing through the $j$th pixel. Due to limited-angle measurements, matrix $\mathbf {A}$ is rank-deficient (and ill-posed), which is addressed here with Tikhonov regularization [5]. This linear tomography approach is well-established [6–8,10,11], and was previously used to perform initial volumetric laser absorption imaging [9].In the learning-based approach, a deep neural network performs the inversion by predicting 2D temperature and species fields from the projected absorption measurements provided as inputs. In this work, we train the neural network by modeling the physics associated with the forward process to create an implicit function between input and output pairs. The forward model includes reacting fluid dynamics simulations (detailed in Section 2.3) to generate temperature and concentration field data, shown in the top of Fig. 3. Spectral simulations using line-by-line parameters from the HITEMP database [33] are performed in each grid cell to simulate $K_j$ fields, after which a forward projection using a Radon transform is applied to calculate path-integrated line-of-sight projections $A_{j,\textrm {proj}}$ that reflect the VLAI setup depicted as sinograms in the top right of Fig. 3. The physics-governed data are used as a labeled dataset to train the neural network.
Fig. 3. Convolutional neural network architecture for temperature and mole fraction field predictions provided six different angles of projection data.
The neural network architecture is adapted from an implementation previously demonstrated to efficiently reconstruct species distributions directly from limited-angle spectroscopy data [26]. The number of convolutional layers and filter sizes are determined following this work, based on the convergence and the prediction accuracy of the test data set. The network comprises two convolutional layers, two pooling layers and one fully connected layer. As illustrated in the bottom of Fig. 3, the input to the neural network comprises $2$ sinograms ($N \times 128$), one from each CO spectral transition. The first convolutional layer C1 convolves 8 filters of $3\times 3$ with stride 1 followed by a rectifier nonlinearity, a batch normalization layer BN1, and a max pooling layer with filters of $2\times 2$. The second convolutional layer C2 similarly convolves 14 filters of $2\times 2$ with stride 1 followed by a rectifier nonlinearity, a batch normalization layer BN2, and a max pooling layer with filters of $2\times 2$. After a flatten layer and a fully connected layer FC, an output vector of size $20000\times 1$ is obtained, which can easily be reshaped as two $100\times 100$ 2D profiles of temperature and CO mole fraction, respectively. The loss function minimized during training was a simple mean squared loss between the network output and target normalized temperature and CO mole fraction values. The RMSprop algorithm is used with minibatches of size 16, learning rate 0.001, momentum 0.0, and decay 0.9. The network was trained for 100 epochs (typically 20–30 mins) on the Tensorflow deep learning framework using an 8 GB NVIDIA RTX 2080 graphics card. After the training process finishes, an effective inversion operator with implicit physical priors on combustion thermochemistry, transport, and flame symmetry is learned, and can be applied to new path-integrated measurements to reconstruct the temperature and concentration fields. This effectively bypasses the inversion of $A_{j,\textrm {proj}}$ to $K_j$ obtained through linear tomography [9], as well as two-line thermometry [32], and directly results in thermochemical profiles of the flow-field. Although DNN requires a 20–30 min training process as mentioned, once the networks are established the subsequent reconstructions are computationally efficient. When implemented on an Intel Core i7-9700K 3.60 GHz CPU, DNN completed reconstructions of all pixel rows in $\sim$1 s while linear tomography took $\sim$30 s, largely due to the nonlinear operations required for two-line thermometry.
2.3 Neural network training
To provide a priori information for the DNN based on combustion physics, thermochemical fields are generated using NIST’s Fire Dynamics Simulator (FDS), a large-eddy simulation (LES) code for thermally-driven, low-speed reacting flows [34]. The training dataset comprises 14,700 2D flow-field cross-sections of temperature and CO mole fraction, generated from 139 time-averaged ($\sim$1 s) LES predictions of a partially-premixed Bunsen-style flame in an axisymmetric 2D domain, as shown in the top of Fig. 3. To obtain top-down cross sections from the axisymmetric flowfields, the 2D results are azimuthally projected to generate 3D flow-fields representative of the doublet flame experimental geometry, as shown in Fig. 4(a). In these simulations, the input parameters of fuel/oxidizer ratio, reactant gas temperature and flow rate, burner geometry, ambient oxygen concentration, and co-flow rate are systematically varied so as to produce a wide range of possible thermochemical fields, shown in Fig. 4(c). For data encompassing flame interactions, the training set is supplemented by a smaller number of 2D flow-field cross-sections (2,040) from 51 simulations of two partially-premixed Bunsen-style flames in a 3D domain. The inclusion of 3D doublet simulations was shown to improve reconstructions at higher planes where there is greater interaction between the two flames. A simplified diffusion-limited two-step combustion model is employed in the simulations, wherein all fuel is converted to CO prior to final oxidation to CO$_2$. For each of the 2D simulations, the domain comprises a 30$\times$100 Cartesian mesh corresponding to 15 mm $\times$ 50 mm and encompassing burner diameters of 5.0–7.5 mm, while the domain for each of the 3D simulations comprises a 50$\times$50$\times$40 mesh corresponding to 40 mm $\times$ 40 mm $\times$ 32 mm and encompassing burner diameters of 8 mm each. These dimensions are larger than the corresponding experiments and were chosen to reduce computational effort in generating multiple distinct flowfields. The time-steps of the simulations were automatically varied to satisfy the Courant-Friedrichs-Lewy condition [34] (CFL $\leq$ 1). To avoid over-constraining flowfield priors and account for real parameter distributions that differ from the mid-fidelity LES model, we supplement the training dataset with mixed temperature and mole fraction distributions randomly selected from the results. This step is intended to build flexibility into the neural network (reduce stiffness) while learning the implicit functional relationship of temperature and species with absorbance fields. By mixing temperature and mole fraction distributions, prediction accuracy on a test set of 1000 samples (not included in the training set) is improved by a factor of 2 as measured by mean percentage error across all samples. Additionally, the two flames are allowed to be distinct to account for any potential differences in the flow rates encountered in the measurements. A representative 2D histogram of the thermochemical state-space captured in the training dataset is shown in Fig. 4(d).
Fig. 4. Reacting CFD-based training dataset; (a) 3D flow-field in experimental geometry; (b) 2D fields for various flow conditions; (c) selected 1D profiles of CO mole fraction and temperature; (d) probability distribution of simulated thermochemical state space.
Despite the numerical simplifications and adjustments, the mid-fidelity model provides key physical information that helps guide (via training) flow-field parameter solutions to the ill-posed inversion problem. First, the training data reflects a physical range of temperatures and mole fraction values governed by thermochemistry—e.g., the temperature range is bound by the mixture’s adiabatic flame temperature, and CO mole fraction is limited by the amount of carbon in the reactant inlet stream. Second, the training data provide a general correlation among temperature and mole fraction; although the distribution of thermochemical states shown in Fig. 4 has significant spread (which is somewhat deliberate), field solutions in which temperature is high and CO mole fraction is low (and vice versa) are improbable, thereby further informing more likely solutions. Lastly, the mid-fidelity model provides a relative spatial distribution of mole fraction and temperature by creating distinct regions of temperature increase associated with the oxidation of the fuel to CO, the oxidation of the CO to CO$_2$, and diffusive mixing with ambient temperature air. The flame geometry, including convoluted circular ring-like distributions of the mean CO and temperature properties, is also captured. The training process enables the neural network to recognize these physical features.
3. Results
In this section, we evaluate reconstructions produced by the deep learning approach in comparison to those calculated using linear tomography with respect to three key metrics: (1) overall flow-field parameter reconstruction accuracy given sparse view angles, (2) artifact reduction and sharp gradient resolution, and (3) geometric and thermochemical similarity to physical flames. These quantitative evaluations are performed via both simulated and experimental LAI-obtained non-axisymmetric flow fields using varying numbers of projection angles.
3.1 Reconstruction accuracy
To assess and compare the reconstruction accuracy of the inversion methods, both approaches (deep learning and 2D linear tomography) were applied to reconstruct “known” CO and temperature fields—independent of the training dataset—representative of measured flames. These known thermochemical fields were produced by first applying a Tikhonov-regularized Abel inversion [6] to projection measurements of isolated single flames assuming steady, axially-symmetric conditions, as demonstrated in prior work on LAI-obtained projection data [28]. Different pairs of these reference profiles of varying intensity were then combined to represent a non-axisymmetric doublet flame configuration, and serve as “ground truth” for the simulation study. These reference profiles were compared with reconstructions produced by both 2D linear tomography and the deep neural network. A mean percentage error within the domain of interest was used as a comparative metric, and representative results alongside simulated “ground truth” fields are shown in Fig. 5. As is typically observed in ill-posed inversion problems [5], utilizing a greater number of projection angles was shown to increase the overall accuracy of the reconstructed fields. For example, in the CO mole fraction fields shown in Fig. 5, the mean percentage error reduced from 5.4% to 2.7% when using six instead of three angles in the linear tomography reconstruction; the corresponding error reduction for the neural network reconstruction was from 1.4% to 1.1%. For the corresponding temperature fields, the increased number of angles reduced the error from 5.8% to 4.9% using linear tomography; this error reduction using the neural network reconstruction was from 1.6% to 0.7%. Notably, even when trained with only three projection angles, the neural network was able to predict the high spatial gradients in the flow-field with greater accuracy than 2D linear tomographic methods using 11 angles, significantly reducing the required number of projections for a reconstruction. For a given number of projection angles, the deep learning approach, on average, reduces mean percentage errors relative to linear tomography by a factor of at least three.
Fig. 5. Comparison of linear tomography (LT) and deep learning (DL) reconstruction methods for mole fractions (top) and temperatures (bottom) with representative case.
To further evaluate the interpolation and extrapolation capability of the trained DNN, additional LES simulations inside (500 cases) and outside (500 cases) the envelope of LES input parameter spaces are used as testing cases. These parameter spaces include ranges of mixture ratio, inlet temperature, burner inlet velocity, burner diameter, co-flow velocity, ambient oxygen concentration, and fuel type (ethane or ethylene). The reconstruction accuracy of the testing cases in temperature and mole fraction is evaluated as a function of their distance to the nearest training sinogram as shown in Fig. 6. The distance is quantified by the $L_2$ distance between the sinogram of the testing case and its nearest training sample:
Fig. 6. Reconstruction accuracy of testing cases in mole fraction (left) and temperature (right) as a function of the $L_2$ distance to their nearest training case.
Within training parameter space, most testing cases are found to remain high reconstruction accuracy ($e_X<2\%, e_T<5\%$) even when they are different from the training cases (large $\mathrm {D_{sinogram}}$), indicating good interpolation capability of the DNN. However, for cases outside training parameter space, the prediction errors are usually 3 times higher than cases within training parameter space. Additionally, the DNN predictions are less stable with out-of-sample inputs, with prediction errors increasing with distance to the training set. Such limitations on the extrapolative capability must be considered when developing and applying comprehensive physics-trained DNNs to measurements of experimental reacting flows.
With the deep learning approach validated via simulation, the neural network inversion was applied to experimental LAI-obtained projection measurements of various flame doublets such as those shown in the left of Fig. 1. Reconstructed 2D fields of temperature and mole fraction are shown in the left of Fig. 7 for both the 2D linear tomographic and deep learning-based inversion methods. Since a "ground truth" for the thermochemical fields determined from experimental measurements of the non-axisymmetric flowfield is unknown, the quality of the reconstructions was evaluated by re-projecting the predicted temperature and CO fields using the forward model described in Section 2.2 and comparing the result with the measured projected absorption areas, as shown in the right of Fig. 7. An aggregate root mean square error (RSME) was used as a comparative metric, calculated as the square root of the averaged squared differences between the re-projected absorbance image and the corresponding measurement image all over pixels. Despite the lack of an analytical relation, the re-projected absorbance fields using the deep learning method were shown to have comparable RSME, within 10%, to linear tomography results (which are analytically constrained) using experimental VLAI data from the same number of projection angles, demonstrating the neural network’s ability to well-capture the non-linear relationships associated with the end-to-end physics of the experiment.
Fig. 7. Left: Two-dimensional tomographic reconstructions of CO mole fraction (top) and temperature (bottom) generated from linear tomography and deep learning methods; Right: Reprojection of reconstructed fields at an angle of 120$^\circ$.
3.2 Blurring and artifact reduction
In linear tomographic reconstruction, the ill-posed nature of the inversion problem in conjunction with the smoothness constraint imposed by Tikohonov regularization will often result in blurring of reconstructed flow-field parameters as well as the presence of polygonal petal-like artifacts associated with the limited view angles [5]. These phenomena are seen in both the simulation results shown in Fig. 5 and the experimental results shown in Fig. 7 produced by linear tomography. Specifically, at 3 view angles, the peak values in the species field distributions are approximately 50% lower than the 11-angle LT solution, while the minimum values associated with the core of the flow are higher by approximately 30%. This reflects an apparent blurring and lower spatial resolution, and has the effect of increasing the size of the reconstructed flame brush—in Fig. 5, using three instead of six angles increases the flame brush width (here marked by 1% CO concentration) from 1.6 mm to 2.0 mm when using linear tomography, while this increase is imperceptible in the reconstructions produced by the neural network. Further, it can be noted that the polygonal petals of the field distributions for linear tomography are an apparent function of the number of view angles, where the number of polygonal corners or petals in circular or ring-like distributions equate to twice the number of view angles. Additionally, we note that in reconstructed regions in which the absorption coefficients $K_j$ of both spectral transitions approach zero, the temperature sensitivity of the spectral transition pair can generate physically improbable temperature and mole fraction values via two-line thermometry (using the LT method) if the signal-to-noise ratio (SNR) in the projected absorbance area $A_{j,\textrm {proj}}$ data is below $\sim 5$. By contrast, since the neural network need not satisfy Eq. (2) explicitly as is the case with linear tomography; this method mitigates artifacts in the periphery of the flames (which would not have shown up in the training data), reduces artificial polygonal bias in the reconstructed field, and ultimately yields sharper features in the regions of higher intensity, as seen in the left of Fig. 7.
Additionally, due to physical features and trends associated with the neural network training data, it is feasible to predict a larger domain of the species and temperature fields. Notably, the domain representing final oxidation of CO to CO$_2$ and diffusion into the ambient is difficult to resolve with linear tomography due to lower absorbance caused by both lower CO and declining temperature in the regions where ambient oxidizing air is entrained into the flow. Thus, in Figs. 5 and 7, neither the temperature nor the mole fraction are resolved in regions with very low values of absorption coefficient ($K_{\textrm {P(1,14)}} < 0.0005$) for the 2D linear tomography reconstructions, and so these regions are not plotted for clarity (due to extensive non-physical results and artifacts). With six projection angles, this limits the resolved peripheries of the flames to $r/D\approx$ 0.94 from the flame centers ($D=$ 1.6 mm), where the temperature is 2400 K. By contrast, the physics-trained neural network was able to predict thermochemical profiles beyond the domain constrained by the analytical solutions, as the method bypasses two-line thermometry of the $K_j$ fields. Assuming a temperature cutoff of 850 K (the lowest temperature predicted in the flame cores by the deep learning method), this extends the resolved regions to $r/D\approx$ 1.25 from the flame centers, representing a domain expansion of 33% compared to LT.
These 2D fields corresponding to different rows of pixels are assembled into 3D images of mole fraction and temperature for the C$_2$H$_4$ flames, shown in Fig. 8 for both the 2D linear tomographic and deep learning-based inversion methods. Both linear tomography and the neural network results show that the cooler cores narrow as flame height increases and more of the unburned fuel is oxidized in the reaction layers of the flames, which is consistent with the general evolution of partially-premixed Bunsen-style flames [35]. Higher spatial gradients are resolved within the flames when using the deep learning-based method, revealing the flame cores of the flows containing lower temperatures and concentrations of CO, and sharper rings with higher temperatures and concentrations of CO. For example, when using six projection angles, the lowest temperature resolved by linear tomography in the core of the flames at the lowest measured flame height is 1500 K, while the neural network reconstruction produced 850 K. The corresponding CO mole fraction values were 0.034 for linear tomography and 0.012 for the deep learning-based method. This is most evident in the lower planes of the flames, where CO concentration and temperature are both lower, providing for weaker absorption and higher susceptibility to measurement noise.
Fig. 8. Three-dimensional tomographic reconstructions of CO mole fraction (top) and temperature (bottom) generated from linear tomography and deep learning methods
3.3 Thermo-physical state space
In this subsection, we discuss the thermo-physical state space associated with the tomography results, including the range and correlations associated with species and temperature fields that relate to thermochemistry and transport, as well as geometric features of the flame configuration. Flames possess steep spatial gradients associated with balancing diffusive heat and mass transfer, and intermediate species like CO usually exist in a very thin reaction zone ($<$1 mm) within a corresponding large temperature rise ($>$2000 K) during oxidation from reactants to products [35]. Correlations of temperature and concentration are often used to quantify overall flame thermochemistry and evolution [36], and blurring associated with limited angle tomography can bias these correlations by muting the steep gradients in flow-field parameter distributions in the reconstruction. Thus, the ability to quantitatively assess the combustion physics in these flames is directly coupled to the inversion method’s ability to resolve sharp features in the flow-field parameters. Additionally, the Bunsen-style flames possess inherent symmetries associated with coupled mass, momentum, and heat transport phenomena. Except for planes with significant interactions between the two flames, the temperature and mole fraction distributions are expected to be approximately axially symmetric in each flame; that the distributions are not preferentially grouped on one side of the flame or the other. In reconstructions using linear tomography, however, such as in Figs. 5 and 7, axial asymmetries are present and worse when employing fewer projection angles, manifesting as the petal-like artifacts and azimuthal variation discussed in Section 3.2.
To quantitatively assess the individual flame symmetry, a Tikhonov-regularized Abel inversion [28] was applied to a projection measurement of an isolated single flame to reconstruct the radial temperature and mole fraction profiles assuming steady, axisymmetric conditions. It should be noted that this reference comparison is only valid at low flame heights, where the two flames minimally interacted. The left of Fig. 9 compares averaged 1D radial profiles of temperature and CO mole fraction for flames in the flow, obtained from both linear tomography and deep learning with respect to the reference Abel-inverted profiles. These profiles were generated by averaging different 1D profiles marked by the white dashed lines in Fig. 7, and the error bars in Fig. 9 quantify the absolute deviations from this average in the azimuthal coordinate. When using three view angles, the mean deviations in temperature and mole fraction produced by linear tomography are 6% and 17%, respectively, while the corresponding deviations produced by the neural network are 2% and 4%, demonstrating that reconstructions performed with the neural network better capture the axial symmetry of the flames. With linear tomography, reducing the number of projection angles from six to three causes large absolute errors in temperature and mole fraction compared to the reference Abel results, while the deep learning approach was found to be less sensitive to the number of projections used. Even with only three projection angles, the neural network predicts temperature and CO profiles closer to the reference Abel profiles than linear tomography with six angles, thereby effectively reducing the required number of unique projections for a parameter field reconstruction. Notably, peak mole fractions are reduced more significantly than peak temperatures ($-50$% vs. $-16$% when using three projection angles, respectively) in the linear tomography reconstructions, which has the effect of producing physically improbable temperature and mole fraction flow-fields for the flames.
Fig. 9. Left: One-dimensional reconstructions of temperature and CO mole fraction from linear tomography and deep learning; Right: Thermochemical state-spaces of the deep-learning-based and linear tomography solutions alongside training data. Training samples with closest sinograms shown in green.
An analysis of the thermochemical state-space representation of the linear tomography and deep learning results is conducted by examining correlations between CO and temperature based on six projection angles, as shown in the right of Fig. 9. Scatter data from all measurement planes using both methods are plotted alongside the training data, and the training case with closest sinogram to the measurements is highlighted (green). Notably, linear tomography is found to produce many solutions that are likely non-physical—the thermochemical state space produced by linear tomography contains points with temperatures greater than 2000 K and CO mole fractions lower than 0.05, which are unlikely to simultaneously occur in physical flames in the captured domain. Conversely, the thermochemical state-spaces of the reconstructions predicted by the deep learning-based method largely stay within the reacting CFD-generated state space; however, the neural network predictions do not strictly follow the most probable regions of the 2D histogram produced by the CFD or their nearest samples in the training set. Notably, the temperature / CO state-spaces of the deep learning-predicted reconstructions are generally consistent with those measured in analogous laboratory flames by coherent anti-Stokes Raman spectroscopy and two-photon laser induced fluorescence [37], though peak CO concentrations are much higher in the present study ($\sim$12% vs. $\sim$5%) due to the fuel-rich mixtures used. Given the sparse number of view angles, these results highlight the ability of the neural network to assist in capturing complex relationships associated with combustion physics when trained with mid-fidelity reacting CFD simulations. Moreover, these simulations do not appear to overly bias the DNN results given that even the closest simulation cases do not match the resulting predictions, suggesting that the variation in the training data provides for flexibility in the implicit functional relationships of the network.
4. Conclusions
A deep neural network trained with mid-fidelity reacting flow simulations and a forward absorbance model was applied to enhance the measurements of CO and temperature fields in steady flames using a relatively limited number of viewing angles. Notably, the DNN training was performed a priori—without inclusion of measurement results—while effectively introducing end-to-end physical priors that inform the implicit functional relationships. When paired with the spatially-rich projection data provided by the volumetric laser absorption imaging optical configuration, the neural network helps resolve steep thermochemical gradients associated with the intermediate CO in the thin reaction zones ($<$1 mm) of small diameter flames. Resulting 3D images of thermochemical structure suggest that this physics-trained neural network inversion has potential to more accurately predict complex temperature and concentration fields of intermediate species in flames with less blurring and artifacts than linear tomographic methods, while concurrently reducing the required number of projections and computational load.
The use of a deep learning-based approach and associated reduction in optical hardware requirements expands the possible future developments of the LAI method; 3D flame measurements presented here represent 1-s averaged data from steady flames—to expand the 3D tomographic LAI method to time-resolved capability, as has been demonstrated previously in two dimensions [30], multiple projection angles must be imaged simultaneously. The use of deep learning assistance may reduce the number of high-speed IR cameras required and increases the practicality of utilizing time-resolved 3D tomographic LAI for investigations of unsteady, convoluted flame structures.
Funding
Air Force Office of Scientific Research (FA9550-19-1-0062).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. W. Cai and C. F. Kaminski, “Tomographic Absorption Spectroscopy for the study of Gas Dynamics and Reactive Flows,” Prog. Energy Combust. Sci. 59, 1–31 (2017). [CrossRef]
2. J. R. Howell, M. P. Mengüç, K. Daun, and R. Siegel, “Inverse Problems in Radiative Transfer,” in Thermal Radiation Heat Transfer, (CRC Press, Boca Raton, FL, 2021), chap. 18, pp. 825–863, 7th ed.
3. L. Ma, X. Li, S. T. Sanders, A. W. Caswell, S. Roy, D. H. Plemmons, and J. R. Gord, “50-kHz-rate 2D imaging of temperature and H2O concentration at the exhaust plane of a J85 engine using hyperspectral tomography,” Opt. Express 21(1), 1152–1162 (2013). [CrossRef]
4. F. Stritzke, S. van der Kley, A. Feiling, A. Dreizler, and S. Wagner, “Ammonia concentration distribution measurements in the exhaust of a heavy duty diesel engine based on limited data absorption tomography,” Opt. Express 25(7), 8180 (2017). [CrossRef]
5. K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transfer 172, 58–74 (2016). [CrossRef]
6. K. J. Daun, K. A. Thomson, F. Liu, and G. J. Smallwood, “Deconvolution of axisymmetric flame properties using Tikhonov regularization,” Appl. Opt. 45(19), 4638 (2006). [CrossRef]
7. J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013). [CrossRef]
8. C. Liu, L. Xu, J. Chen, Z. Cao, Y. Lin, and W. Cai, “Development of a fan-beam TDLAS-based tomographic sensor for rapid imaging of temperature and gas concentration,” Opt. Express 23(17), 22494 (2015). [CrossRef]
9. C. Wei, K. K. Schwarm, D. I. Pineda, and R. M. Spearrin, “Volumetric laser absorption imaging of temperature, CO and CO2 in laminar flames using 3D masked Tikhonov regularization,” Combust. Flame 224, 239–247 (2021). [CrossRef]
10. Y. Bao, R. Zhang, G. Enemali, Z. Cao, B. Zhou, H. McCann, and C. Liu, “Relative Entropy Regularized TDLAS Tomography for Robust Temperature Imaging,” IEEE Trans. Instrum. Meas. 70, 1–9 (2021). [CrossRef]
11. N. A. Malarich and G. B. Rieker, “Resolving nonuniform temperature distributions with single-beam absorption spectroscopy. Part I: Theoretical capabilities and limitations,” J. Quant. Spectrosc. Radiat. Transfer 260, 107455 (2021). [CrossRef]
12. K. J. Daun, S. L. Waslander, and B. B. Tulloch, “Infrared species tomography of a transient flow field using Kalman filtering,” Appl. Opt. 50(6), 891–900 (2011). [CrossRef]
13. S. J. Grauer, J. Emmert, S. T. Sanders, S. Wagner, and K. J. Daun, “Multiparameter gas sensing with linear hyperspectral absorption tomography,” Meas. Sci. Technol. 30(10), 105401 (2019). [CrossRef]
14. J. D. Christopher, O. A. Doronina, D. Petrykowski, T. R. Hayden, C. Lapointe, N. T. Wimer, I. Grooms, G. B. Rieker, and P. E. Hamlington, “Flow parameter estimation using laser absorption spectroscopy and approximate Bayesian computation,” Exp. Fluids 62(2), 43 (2021). [CrossRef]
15. X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of nonuniform temperature distributions using line-of-sight absorption spectroscopy,” AIAA J. 45(2), 411–419 (2007). [CrossRef]
16. L. H. Ma, L. Y. Lau, and W. Ren, “Non-uniform temperature and species concentration measurements in a laminar flame using multi-band infrared absorption spectroscopy,” Appl. Phys. B 123(3), 83 (2017). [CrossRef]
17. D. Wen and Y. Wang, “Spatially and temporally resolved temperature measurements in counterflow flames using a single interband cascade laser,” Opt. Express 28(25), 37879 (2020). [CrossRef]
18. D. Wu, K. Kim, G. El Fakhri, and Q. Li, “Iterative low-dose CT reconstruction with priors trained by artificial neural network,” IEEE Trans. on Med. Imaging 36(12), 2479–2486 (2017). [CrossRef]
19. X. Zheng, I. Y. Chun, Z. Li, Y. Long, and J. A. Fessler, “Sparse-View X-Ray CT Reconstruction Using l1 Prior with Learned Transform,” arXiv (2017).
20. T. Ren, M. F. Modest, A. Fateev, G. Sutton, W. Zhao, and F. Rusu, “Machine learning applied to retrieval of temperature and concentration distributions from infrared emission measurements,” Appl. Energy 252, 113448 (2019). [CrossRef]
21. J. Huang, H. Liu, and W. Cai, “Online in situ prediction of 3-D flame evolution from its history 2-D projections via deep learning,” J. Fluid Mech. 875, R2 (2019). [CrossRef]
22. Y. Jin, W. Zhang, Y. Song, X. Qu, Z. Li, Y. Ji, and A. He, “Three-dimensional rapid flame chemiluminescence tomography via deep learning,” Opt. Express 27(19), 27308 (2019). [CrossRef]
23. T. Ren, H. Li, M. F. Modest, and C. Zhao, “Efficient two-dimensional scalar fields reconstruction of laminar flames from infrared hyperspectral measurements with a machine learning approach,” J. Quant. Spectrosc. Radiat. Transfer 271, 107724 (2021). [CrossRef]
24. C. Wei, K. K. Schwarm, D. I. Pineda, and R. M. Spearrin, “Deep neural network inversion for 3D laser absorption imaging of methane in reacting flows,” Opt. Lett. 45(8), 2447 (2020). [CrossRef]
25. Y. Jiang, J. Si, R. Zhang, G. Enemali, B. Zhou, H. McCann, and C. Liu, “CSTNet: A dual-branch convolutional network for imaging of reactive flows using chemical species tomography,” arXiv pp. 1–9 (2020).
26. J. Huang, H. Liu, J. Dai, and W. Cai, “Reconstruction for limited-data nonlinear tomographic absorption spectroscopy via deep learning,” J. Quant. Spectrosc. Radiat. Transfer 218, 187–193 (2018). [CrossRef]
27. J. Si, G. Li, Y. Cheng, R. Zhang, G. Enemali, and C. Liu, “Hierarchical Temperature Imaging Using Pseudo-Inversed Convolutional Neural Network Aided TDLAS Tomography,” arXiv (2021).
28. C. Wei, D. I. Pineda, C. S. Goldenstein, and R. M. Spearrin, “Tomographic laser absorption imaging of combustion species and temperature in the mid-wave infrared,” Opt. Express 26(16), 20944 (2018). [CrossRef]
29. R. J. Tancin, R. M. Spearrin, and C. S. Goldenstein, “2D mid-infrared laser-absorption imaging for tomographic reconstruction of temperature and carbon monoxide in laminar flames,” Opt. Express 27(10), 14184 (2019). [CrossRef]
30. K. K. Schwarm, C. Wei, D. I. Pineda, and R. M. Spearrin, “Time-resolved laser absorption imaging of ethane at 2 kHz in unsteady partially premixed flames,” Appl. Opt. 58(21), 5656 (2019). [CrossRef]
31. R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014). [CrossRef]
32. C. S. Goldenstein, R. M. Spearrin, J. B. Jeffries, and R. K. Hanson, “Infrared laser-absorption sensing for combustion gases,” Prog. Energy Combust. Sci. 60, 132–176 (2017). [CrossRef]
33. L. Rothman, I. Gordon, R. Barber, H. Dothe, R. Gamache, A. Goldman, V. Perevalov, S. Tashkun, and J. Tennyson, “HITEMP, the High-Temperature Molecular Spectroscopic Database,” J. Quant. Spectrosc. Radiat. Transfer 111(15), 2139–2150 (2010). [CrossRef]
34. K. B. McGrattan, R. J. McDermott, C. G. Weinschenk, and G. P. Forney, “Fire Dynamics Simulator, Technical Reference Guide, Sixth Edition,” Tech. Rep. Special Publication (NIST SP) - 1018, National Institute of Standards and Technology, Gaithersburg, MD (2016).
35. C. K. Law, Combustion Physics (Cambridge University Press, New York, 2006).
36. D. I. Pineda, L. Paxton, N. Perakis, C. Wei, S. Luna, H. Kahouli, M. Ihme, F. N. Egolfopoulos, and R. M. Spearrin, “Carbon oxidation in turbulent premixed jet flames: A comparative experimental and numerical study of ethylene, n-heptane, and toluene,” Combust. Flame 221, 371–383 (2020). [CrossRef]
37. M. Mann, C. Jainski, M. Euler, B. Böhm, and A. Dreizler, “Transient flame-wall interactions: Experimental analysis using spectroscopic temperature and CO concentration measurements,” Combust. Flame 161(9), 2371–2386 (2014). [CrossRef]
