## Abstract

We present a linear model for absorption tomography with velocimetry (LATV) to reconstruct 2D distributions of partial pressure, temperature, and streamwise velocity in a high-speed flow. Synthetic measurements are generated by multi-beam tunable diode laser absorption spectroscopy (TDLAS). The measurement plane is tilted relative to the streamwise direction and absorbance spectra are Doppler-shifted by the gas flow. Reconstruction comprises two stages. First, the thermodynamic state is obtained by reconstructing two or more integrated absorption coefficients and evaluating local Boltzmann plots. Second, the velocity field is directly reconstructed from absorbance-weighted linecenters. Absorbance data are inferred by Voigt fitting and reconstructions are quickly computed by matrix-vector multiplication. Nonlinear parameter combinations, such as the mass flow, are more accurate when computed by LATV than estimates obtained by assuming uniform gas properties along each beam.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Simultaneous 2D measurements of the thermodynamic state and fluid speed in supersonic and hypersonic flows can provide important fundamental and engineering insights [1]. For example, such measurements can be used to determine mass capture and stream thrust through the inlet of a high-speed vehicle, or assess isolator/combustor interactions in a ramjet or scramjet across a range of operating conditions. Established nonintrusive methods for velocimetry do not yield thermodynamic parameters, and may indeed inhibit simultaneous scalar measurements due to the introduction of flow tracers [2,3]. Moreover, these techniques often require considerable optical access and complicated laser/detector systems, which may not be viable in the experimental configurations of interest. Consequently, there is a need for diagnostics that simultaneously measure the speed and thermodynamic state of a flow using robust and easily-configured instrumentation.

Sensors based on tunable diode laser absorption spectroscopy (TDLAS) contain rich information about both the gas state and velocity through the Doppler-shifted absorbance spectrum, and beams (or beam arrays) can be deployed in otherwise optically-inaccessible regions of a test article. Previous applications of TDLAS sensors to high-speed flows have typically (i) used one or more independent beams to measure line-of-sight (LOS) integrated quantities [4] or (ii) synthesized data from multiple beams to resolve 2D distributions through absorption tomography (AT) [5–7]. However, simultaneous 2D measurements of pressure, temperature, and speed have not been achieved to date. We present a novel method for AT that yields the desired measurements. Our approach has a high accuracy and low computational cost, potentially enabling kilohertz-rate monitoring for control purposes through the use of previously-demonstrated TDLAS systems [4,8].

In the conventional approach to TDLAS, intensity data from each beam are evaluated independently, assuming a uniform distribution of gas properties [9]. TDLAS sensors of this variety have been deployed to characterize complex internal flowfields and engine exhausts in support of the design and operation of advanced aircraft [10]. Previous efforts demonstrated the ground test [4,11] and flight test [12–14] viability of TDLAS devices using direct absorption spectroscopy (DAS) or wavelength-modulation spectroscopy (WMS). Target species included O$_2$, H$_2$O, CO$_2$, and CO [15,16], measured to determine inlet mass capture [17] and exit mass fluxes [18], as well as conduct shock train localization [19,20].

The assumption of a homogeneous gas in a dynamic flow can increase measurement uncertainties by up to 50% [21]. AT employs simultaneous DAS or WMS measurements from multiple beams to reconstruct 2D distributions of temperature, the mole fraction of one or more species [22], and pressure [23,24]. By resolving spatial variations in the gas state, AT can potentially improve the accuracy of nonlinear metrics calculated from multi-beam TDLAS data. The technique was originally developed as a laboratory diagnostic, but advances in AT over the past two decades have enabled rapid, in situ imaging of automotive [25] and marine engines [26], aircraft engine exhaust flows [5,6], catalytic reductions [27], 3D combustion structures [28], and more.

Absorption tomography has been successfully employed to process multi-beam measurements of the flowfield in a high-speed airbreathing engine. Brown et al. [10,18] conducted AT at the combustor exit of a high-speed engine using eight beams, and Lindstrom et al. [29] probed the isolator of a similar ground test rig with 16 beams. Both studies reported a 2D temperature map, but the velocity field was assumed to be uniform. Busa et al. [30] made simultaneous AT and stereoscopic particle image velocimetry (PIV) measurements to assess combustion progress in a scramjet engine. A large number of projections were measured by rotating the tomography rig around the exit of the combustor. However, AT was not used for the velocity measurements and PIV faces numerous challenges in supersonic flows [31].

Recently, Qu et al. [32] proposed a method to determine the mole fraction and velocity of a target species in a tilted plane by AT, assuming uniform pressure and temperature fields. Gradient-based algorithms were used to solve the mole fraction and velocimetry sub-problems, and the approach was numerically demonstrated with a 48-beam array. However, practical high-speed flows feature large gradients of pressure and temperature. Furthermore, nonlinear solvers are computationally-expensive, and 48 synchronous beams considerably exceeds the number successfully implemented in an engine flow path to-date. As such, there remains a need to generate accurate, spatially-resolved estimates of the full thermodynamic state and velocity field in a high-speed flow using the minimum number of beams required to provide the desired accuracy.

We report a novel, *linear* approach to *AT with velocimetry* (LATV). LATV comprises sequential thermodynamic and velocity reconstructions conducted with integrated absorbance and absorbance-weighted linecenter data. Our method generates 2D estimates of the partial pressure of one or more target species, temperature, and the axial component of velocity in the measurement cross-section. LATV estimates are quickly computed by matrix-vector multiplication. Moreover, nonlinear combinations of the reconstructed parameters are more accurate than estimates computed by a conventional TDLAS evaluation, assuming uniform properties along the beam. In this paper, we derive and numerically validate the LATV measurement model, reconstruct phantoms that have strong spatial gradients using a limited-beam array, and assess the accuracy of nonlinear metrics computed by LATV.

## 2. LATV

Tomographic reconstruction is an inverse problem in which the “forward” measurement model is inverted to determine the spatial distribution of the quantity (or quantities) of interest (QoI). Incorporating velocimetry into AT requires a two-stage reconstruction procedure based on thermodynamic and velocity models, employed in succession. Figure 1 presents a graphical overview of LATV reconstructions. The direction of rays is fixed by the orientation of the pitch and catch optics, which is the defining characteristic of “hard field tomography,” and both measurement models are approximated by an integral equation, which must be discretized.

#### 2.1 Hard field tomography

Hard field LOS-integrated measurements of a spatially-resolved QoI, $x$, generally take the form

where $b$ is the measured value, $L$ is the length of the beam, and $s$ is a parametric variable that indicates a position along the beam. We use a piecewise linear finite element basis to represent $x$, described in [24]. The data are measured along $m$ beams, summarized in the vector $\mathbf {b} = \{b_i\}$, and the QoI are resolved at $n$ nodes, also arranged vector-wise, $\mathbf {x} = \{x_j\}$, where $b_i$ is the measurement from the $i$th beam and $x_j$ is the value of $x$ at the $j$th node. Equation (1) is discretized using the finite element basis, resulting in the so-called “ray-sum,” where $w_{i,j} = \partial b_i/\partial x_j$ is a weight (or sensitivity) determined by integrating along the $i$th LOS over the $j$th basis function. Integrating along each LOS over each basis function yields the weight matrix, $\mathbf {W} \in \mathcal {R}^{m\times n}$, also called a ray-sum matrix or sensitivity matrix.The matrix product $\mathbf {W}\mathbf {x} = \mathbf {b}$ corresponds to the simultaneous measurement of $\mathbf {x}$ along every LOS. Reconstruction consists in the inference of $\mathbf {x}$ given the measured values in $\mathbf {b}$, which is an ill-posed inverse problem. The reconstruction algorithm described in Sect. 3 incorporates prior information into the inversion to obtain accurate estimates of $\mathbf {x}$. Next, we develop a thermodynamic model for LATV that conforms to Eq. (1).

#### 2.2 Thermodynamic model

Tunable diode laser absorption spectroscopy sensors measure the light produced by a diode laser and attenuated by an absorbing fluid. The attenuation of monochromatic light is described by the Beer–Lambert law [9],

Absorption may take place when the frequency of a photon incident upon a molecule corresponds to the energy of a valid transition between the molecule’s internal energy states, shifting the molecule from its initial energy state, $E^{\prime \prime }$, to an elevated state, $E^{\prime }$. The spectral absorption coefficient, $\kappa _\nu$ describes the fraction of incident spectral light at $\nu$ that is absorbed by the target, given by the sum of all valid transitions at $\nu$. That is, $\kappa _\nu = \sum _{k} \kappa _{\nu ,k}$, where $k$ is the transition index and

Here, $S_k$ is the number density-normalized spectral line intensity, $k_{\mathrm {B}}$ is Boltzmann’s constant, and $\phi$ is the lineshape function. Transitions are characterized by a central wavenumber (at vacuum pressure), $\nu _k = E_k^{\prime } - E_k^{\prime \prime }$, which equals the energy difference between the upper and lower states of the transition. For convenience, the line intensity at $T$ is expressed in terms of the intensity at a reference temperature, $T_{\mathrm {ref}}$ (typically 296 K),Photons at wavenumbers adjacent to $\nu _k$ can be absorbed by a molecule due to line shifting and broadening mechanisms, discussed in Sect. 2.3, which are incorporated into the lineshape function. By construction, $\phi$ integrates to unity, so we can define a local spectrally-integrated absorption coefficient for each transition,

Similarly, the spectrally-integrated absorbance of the $k$th transition isEquations (3) and (7) both have the same form as the generic tomography measurement in Eq. (1). Therefore, the distribution of $\kappa _\nu$ can be reconstructed from spectrally-resolved absorbance data ($\alpha _\nu$) and $K_k$ can be reconstructed from spectrally-integrated absorbance data ($A_k$). In high-speed flows, however, local spectra are Doppler-shifted and the magnitude of the shift depends on the orientation of the beam and fluid velocity. As a result, LOS-integrated measurements of $\alpha _\nu$ from unique perspectives can imply inconsistent local spectra. It is thus necessary to reconstruct spectrally-integrated absorption coefficients in LATV to resolve the thermodynamic state. That is, we take $x$ to be $K_k$ and $b$ to be $A_k$.

For non-reacting flows, two or more transitions must be measured to estimate the temperature and partial pressure of the target molecule; at least one additional transition would be required for each additional species to be measured in a reacting flow. Given reconstructions of $K_k$ for multiple transitions, local Boltzmann plots are used to determine the gas state at a node in the tomography mesh. The Boltzmann plot $y$-axis consists of reconstructed absorption coefficients, normalized by the reference line intensity and Boltzmann population factor, and the $x$-axis consists of lower state temperatures [33],

In summary, coupled measurements of $A_k$ for multiple transitions are used to reconstruct $K_k$ by inverting Eq. (7), and local values of $K_k$ are used to determine the temperature and partial pressure of target molecules via Eq. (9).

#### 2.3 Velocity model

Line broadening and shifting mechanisms alter the probability that a molecule will absorb light at a given wavenumber. Broadening distributes the strength of each transition over a finite spectral range, resulting in a spectral line, and shifting alters the location of the line. The shape and position of lines are observed in absorbance spectra, which can thus be used to infer the factors responsible for broadening and shifting along a beam. These factors include pressure, temperature, the incidence of electrical perturbations, the abundance and variety of collision partners, and the fluid velocity, the latter of which is the basis of LATV.

The aggregate effect of collisional and Doppler broadening is frequently modeled with a Voigt lineshape, $\phi _{\mathrm {V}}$, characterized by a linecenter, $\nu _0$, and Lorentzian and Gaussian half-widths at half-maximum. The linecenter of a transition is located near $\nu _k$, but the precise center depends on the local pressure and velocity as well as the orientation of the beam. Alternate lineshapes may be necessary to account for correlations between collisional and Doppler broadening or speed-dependent effects, subject to the application and desired accuracy.

The Voigt profile integrates to unity and is symmetrical about $\nu _0$ such that, for a line of integrated area $A$,

This “absorbance-weighted” measure of $\nu _0$, i.e., the first moment of $\alpha _{\nu ,k}$, is a key feature of lineshapes for the purpose of LATV [34,35]. Consider the superposition of two lines, A and B, with areas $A_{\mathrm {A}}$ and $A_{\mathrm {B}}$ and linecenters $\nu _{0,\mathrm {A}}$ and $\nu _{0,\mathrm {B}}$. The absorbance-weighted center of the combined profile is*absorbance-weighted linecenter*and denote $\bar {\nu }_{0,k}$. Note that the contribution of each line to $\nu _{0,\mathrm {A}+\mathrm {B}}$ in Eq. (11) is the product of $A$ and $\nu _0$ divided by the total line area, which is analogous to the spectrally-integrated absorbance, $A_k$. Therefore, taking the individual lines in Eq. (11) to be homogeneous segments along a beam, as the segment size goes to zero, the differential contribution of the local spectrum at $s$ to the absorbance-weighted linecenter at $L$ is Integrating Eq. (12) along a beam yields the absorbance-weighted linecenter, Crucially, $\bar {\nu }_{0,k}$ can be determined from DAS or WMS data by multipeak-Voigt fitting, or by numerically evaluating Eq. (10) in the case of DAS data. Moreover, $A_k$ is measured and the distribution of $K_k$ along the beam is determined by thermodynamic reconstruction. Therefore, the only unknown in Eq. (13) is the spatial distribution of $\nu _{0,k}$, which is a function of the unknown velocity field.

Two line shifting mechanisms are of direct relevance to LATV: pressure shifting and Doppler shifting. Pressure affects the distribution of intermolecular distances in a gas, perturbing $E_k^{\prime \prime }$ and $E_k^{\prime }$ and therefore $\nu _k$ (i.e., energy levels are characterized by a potential function that depends on the intermolecular distance between collision partners). The resulting shift is

where $\delta _{\mathrm {p},k,l}$ is the pressure shift coefficient of the $k$th transition for the $l$th species and $X_l$ is the corresponding mole fraction. Fundamentally, $\delta _{\mathrm {p},k,l}$ is a function of temperature, typically modeled using a power-law. While these effects can be incorporated into our model, we employ a constant mole fraction-weighted pressure shift coefficient, $\delta _{\mathrm {p},k}$, in this paper for convenience. As a result, $\Delta \nu _{\mathrm {p},k} = \delta _{\mathrm {p},k}p$.Doppler shifts occur when there is a substantial component of fluid velocity along the beam. The frequency of laser light is shifted in the fluid’s frame of reference,

where $\nu ^{\prime }$ is the “apparent” frequency; $u$ is the dominant, streamwise component of the fluid velocity; $\theta$ is the angle between the beam and the streamwise direction; and $c$ is the speed of light. Note that $\nu -\nu ^{\prime }$ is effectively constant across a spectral line. Therefore, the Doppler-shifted line position in the reference wavenumber axis is and the velocity shift coefficient for the $k$th transition is The final pressure- and velocity-shifted linecenter is Substituting Eq. (18) into Eq. (13) and isolating the Doppler-shifted component yields our velocity model,If significant chemical reactions take place in a system, multiple species must be reconstructed and/or a thermochemical model must be introduced to determine $p$. Moreover, if the temperature- or composition-dependence of $\delta _{\mathrm {p},k}$ are known then the spatial distribution of $\delta _{\mathrm {p},k}$ can be estimated from reconstructed quantities and included in the pressure integral in Eq. (19).

#### 2.4 Matrix systems

Quantities of interest and intermediate parameters in LATV are represented in terms of a vector of coefficients. Individual coefficients are the value of QoI at nodes in the finite element mesh. In LATV, the QoI are the absorption coefficient vectors, $\mathbf {k}_k = \{K_{k,j}\}$ (one vector per measured transition); temperature vector, $\mathbf {t} = \{T_j\}$; pressure vector $\mathbf {p} = \{p_j\}$; mole fraction vectors $\mathbf {x}_l = \{X_{l,j}\}$ (one vector per species); and streamwise velocity vector $\mathbf {u} = \{u_j\}$. Data vectors for the thermodynamic reconstruction are $\mathbf {a}_k = \{A_{k,i}\}$ (one vector per measured transition); and the absorbance-weighted linecenter data for the velocity reconstruction are $\mathbf {v}_k = \{\bar {\nu }_{0,k,i}-\nu _k\}$ (one vector per measured transition), which must be adjusted to account for the pressure shift per the left-hand side of Eq. (19).

For the thermodynamic reconstruction, integrated absorbance data correspond to the linear system from Sect. 2.1,

which is inverted to reconstruct $\mathbf {k}_k$ from $\mathbf {a}_k$. Thermodynamic vectors $\mathbf {t}$ and $\mathbf {p}$ are computed from the reconstructed vectors $\mathbf {k}_k$, node-by-node, using Eq. (9).Integral equations in the velocity model (Eq. (13)) feature a kernel that depends on $\mathbf {a}_k$ and $\mathbf {k}_k$. The operator that embeds this kernel is

where $\circ$ is the Hadamard product and $\mathbf {a}_k^{\circ -1}$ indicates the Hadamard inverse of $\mathbf {a}_k$, applied to each element. One more operator is required for the velocity integral because $\delta _{\mathrm {u},k}$ depends on the orientation of each beam relative to the streamwise direction. We define the diagonal matrix $\mathbf {R} \in \mathcal {R}^{m\times m}$ with elements $r_{i,i} = \mathrm {cos}\,\theta _i$ for the $i$th beam, and compute a velocity operator, The pressure-shifted data vector for velocity reconstructions, based on the left-hand side of Eq. (19), is These quantities can be averaged,#### 2.5 Mass flow measurement

An advantage of the simultaneous measurement of the thermodynamic state and velocity in a plane is the ability to compute nonlinear quantities arising from the measured variables. We demonstrate the calculation of nonlinear performance metrics using LATV by computing mass flow through the measurement plane. Mass fluxes at the nodes are calculated assuming an ideal gas,

Overall mass capture is computed by integrating $\dot {m}^{\prime \prime }$ over the measurement plane. For the linear finite element mesh used in this paper, the mass flow across each element is given by the average of $\dot {m}_j^{\prime \prime }$ at the element nodes multiplied by the element area, and $\dot {m}$ is the sum of individual element mass flows across the mesh.

## 3. Reconstruction

Tomographic imaging is an ill-posed inverse problem and supplemental information must be included in the reconstruction procedure to obtain reasonable estimates of the gas distribution [36]. Numerous algorithms have been developed to solve linear tomography systems like Eqs. (20) and (25), including the standard, multiplicative, and simultaneous algebraic reconstruction techniques [37,38]; Tikhonov regularization [39]; Bayesian methods [40]; entropy-regularized functionals [41]; dimension-reduction techniques [42]; and many more. The LATV model is compatible with all of these techniques and we do not attempt to review them here. Readers are directed to Cai and Kaminsky [22] for a recent overview of AT reconstruction algorithms.

We demonstrate LATV using a regularization functional inspired by Bayesian reconstruction. Future work will fully explore the implications of Bayesian tomography for a LATV diagnostic. Reconstructions presented in this paper are computed by minimizing an objective function,

The mean and covariance of QoI are generally unknown and must be estimated. A typical minimally-informative “prior” features a zero mean, $\boldsymbol \mu _{\mathrm {x}} = \mathbf {0}_{\mathrm {n}}$, where $\mathbf {0}_{\mathrm {n}}$ is a $n\times 1$ vector of zeros, and an exponential covariance matrix with unit variances, $\boldsymbol \Gamma _{\mathrm {x}} = \boldsymbol \Gamma _{\mathrm {exp}}$ [24]. This matrix describes the correlation between fluctuations at any two nodes, $i$ and $j$,

where $d_{i,j}$ is the Euclidean distance between the nodes and $\beta$ is a correlation length-scale.Given estimates of $\boldsymbol \mu _{\mathrm {x}}$ and $\boldsymbol \Gamma _{\mathrm {x}}$, the objective function is minimized by computing the least squares solution of the corresponding augmented system,

Thermodynamic reconstructions for each transition are conducted using the integrated absorbance measurements, $\mathbf {a}_k$, with unique regularization elements for each transition, i.e., $\lambda _k$, $\mathbf {L}_k$, and $\boldsymbol \mu _k$. Velocity reconstructions substitute the matrix $\mathbf {C}$ for $\mathbf {W}$, and the measurement vector ($\mathbf {d}$) is calculated using Eqs. (23) and (24). An approximate absorption kernel-weighted operator may be constructed, a priori, assuming uniform thermodynamic fields,

## 4. Numerical demonstration

#### 4.1 Synthetic measurements

High-speed flows of interest generally exhibit considerable spatial variations in thermodynamic properties and velocity due to, for example, shock waves and boundary layers. We demonstrate LATV using synthetic phantoms that exhibit appropriate spatial gradients but are not otherwise meant to replicate a particular physical configuration.

Figure 2 depicts the measurement scenario. We model the phsyical domain as a cylindrical channel with a radius of 5 cm. By convention, the $x$, $y$, and $z$ directions are referred to as the streamwise, normal, and spanwise direction, respectively. TDLAS beams are aligned with a 2D measurement plane. We rotate a $y$-$z$ plane $\pi /8$ rad about the normal and spanwise axes, resulting in an elliptical cross-section of the channel with an eccentricity of 0.521. By rotating the plane, a component of streamwise velocity is aligned with the beams, enabling velocity reconstructions via Doppler-shifted linecenters. The plane is discretized using a finite element mesh, having 484 nodes and linear support in each element. Figure 2(b) shows the mesh, the streamwise direction projected onto the measurement plane, and an arrangement of 15 beams. The beam arrangement was selected using the optimization procedure described in [43], which utilizes a Bayesian algorithm with a uniform prior.

We assume a standard composition of dry air throughout the flow path, with an O$_2$ mole fraction of 0.21. Pressure, temperature, and velocity phantoms are constructed from random draws of a multivariate normal distribution. The mean pressure field is uniform at 1 atm, and the mean temperature and velocity fields are parabolic. Unique vectors are drawn for each field such that there are no correlations between $p$, $T$, or $u$. Naturally occurring correlations between these parameters, e.g., due to constant stagnation properties, are not considered here, but could be incorporated into both the phantoms and the priors. Pressure bounds of 0.5 and 1.5 atm were enforced, as well as a minimum temperature of 300 K and minimum velocity of 0 m/s. We created 250 phantoms from 750 random vectors. Sample distributions are shown in Fig. 3.

TDLAS measurements of internal air flows are often conducted using an O$_2$ absorption band in the visible range [44]. Figure 4 shows $R$ branch lines from the O$_2$ $A$-band near 759 nm, which comprises rovibronic transitions between the $X^{3}\Sigma _g^{-}$ electronic ground state and $b^{1}\Sigma _g^{+}$ excited state of O$_2$ [45]. We employ the RQ(23,34) and RQ(33,34) transitions for our LATV tests, modeled using line parameters and total internal partition sums from HITRAN2012 [46]. These lines exhibit a unique sensitivity to temperature at flight-relevant conditions, which can be seen in Fig. 4, and are thus well-suited to Boltzmann plot postprocessing. We assume a conservative, 1 cm$^{-1}$ direct scan from 13,163.4 to 13,164.4 cm$^{-1}$ with 500 spectral resolution elements. Our synthetic data consist of absorbance estimates corrupted by independent and identically-distributed (IID) normal errors. The errors have a standard deviation equal to 3.5% of $\max (\alpha _\nu )$, which is akin to the average of 50 high-speed scans with 25% noise. IID errors are consistent with intensity noise in a short scan with limited intensity modulation. Absorbance data are generated by Doppler-shifting local spectra based on the local velocity and beam orientation and integrating along the beam.

#### 4.2 LATV model

First, we tested the velocity model using our noisy synthetic absorbance data. Modeled absorbance-weighted linecenters were computed using Eq. (25) and estimated linecenters for both transitions were determined by simultaneous Voigt fitting. Noise in the data corrupts estimates of $A_k$ and $\bar {\nu }_{0,k}$, limiting the resolution of these quantities (see Sect. 4.3). A sample Voigt fit is shown in Fig. 5. We tested the model using all 250 phantoms, measured along the 15 beams shown in Fig. 2(b), resulting in 3750 comparisons per transition.

Figure 6 shows comparisons of modeled and fitted linecenters for both measured transitions along with the corresponding error probability density functions (PDFs). The relationship is linear with a correlation greater than 0.99 for RQ(23,34) and 0.976 for RQ(33,34), which is a weaker, more noise-affected absorption feature. The standard deviations of modeled linecenter-shifts for RQ(23,34) and RQ(33,34) were $5.2\cdot 10^{-3}$ cm$^{-1}$ and $5.3\cdot 10^{-3}$ cm$^{-1}$ (the discrepancy between these shifts is due to differential pressure shift coefficients). The corresponding errors had standard deviations of $3.5\cdot 10^{-4}$ cm$^{-1}$ and $1.2\cdot 10^{-3}$ cm$^{-1}$, resulting in signal-to-noise ratios of 11.7 and 6.6, respectively. Notably, the PDFs in Fig. 6(c) and (d) are centered and bell-shaped, and the linecenter errors can be approximated as IID normal variables.

Line-of-sight-integrated absorbance data aggregates local spectra along a beam. Net velocity gradients along the beam can produce a lopsided absorbance profile when coupled with pressure and temperature gradients, even if the local lineshapes are symmetric. Therefore, the assumption of a Voigt profile in the fitting algorithm introduces errors into estimates of the integrated absorbance and absorbance-weighted linecenter for each transition. Nevertheless, our results demonstrate that the linear velocity model developed in Sect. 2.3 is a valid approximation of line shifting, and Voigt fitting is an appropriate method to estimate $\bar {\nu }_{0,k}$ when local spectra conform to the Voigt lineshape.

#### 4.3 LATV reconstructions

Second, we tested the feasibility of reconstructing velocity fields by LATV. The velocity model requires estimates of the integrated absorption coefficient and pressure fields, which must be obtained by thermodynamic reconstruction. As a result, it is necessary to determine whether errors in the thermodynamic reconstruction prevent accurate estimates of the velocity field. We reconstructed all 250 phantoms using Voigt-fitted estimates of $A_k$ and $\bar {\nu }_{0,k}$. Absorption coefficient and pressure reconstructions were employed to create the velocity operator, and mass capture estimates were obtained by summing the mass fluxes computed with Eq. (26). Tomographic estimates of $\dot {m}$ were compared to conventional, “individual beam” TDLAS estimates, described in [4,8]. These are obtained by evaluating data from each beam, separately, assuming uniform properties throughout the cross-section, and averaging the results.

Linear AT with velocimetry produced accurate estimates of the pressure, temperature, and velocity phantoms from error-laden absorbance data using a limited number of beams. Combined thermodynamic and velocity reconstructions were computed in 47 ms, on average, by MATLAB R2019b running on a computer with a dual-core processor (2.1 GHz clock speed) and 8 GB of memory. This time can be considerably reduced by running a compiled solver on dedicated hardware, potentially enabling kilohertz-rate evaluations of various important quantities.

Figure 7 shows three sample phantoms and reconstructions of $p$, $T$, and $u$. The position, shape, and magnitude of integral-scale structures were faithfully recovered by LATV using minimal measurement and prior information. Average normalized root-mean-square errors (NRMSEs) for $p$, $T$, and $u$ were 0.044, 0.066, and 0.129, respectively. By construction, normalized variation of the phantoms is roughly equal for all three quantities, with normalized standard deviations ranging from 0.101 for $T$ to 0.132 for $p$. Moreover, we demonstrated that model errors in Eq. (25) are minimal. There are two potential reasons for moderately-larger NRMSEs in the velocity estimates: (i) errors in $\mathbf {a}_k$ and $\mathbf {k}_k$ that affect $\mathbf {K}_k$ and (ii) decreased stability in the velocity operator due to the beam array orientation matrix, $\mathbf {R}$. That is, the condition number of $\mathbf {C}$ was typically five times greater than the condition number of $\mathbf {W}$ in our tests, which implies a greater sensitivity to errors in the spectroscopic evaluations. Nevertheless, the reconstruction accuracy of all three QoI was high, despite the limited number of beams and simple reconstruction procedure.

A comparison of mass capture errors from LATV and individual beam estimates is shown in Fig. 8. The average mass flow for the phantom set was 1.34 kg/s. LATV produced a time-averaged estimate of 1.33 kg/s, accurate to within 1%; the individual beam estimate was 1.24 kg/s, representing a 7% error. Standard deviations of LATV and individual beam errors were 0.11 and 0.26 kg/s, respectively; moreover, individual beam estimates were correlated with $\dot {m}$. Greater errors in the individual beam measurements are produced by the naïve spatial model. Conventional measurements of $\dot {m}$ are much more sensitive to the position of beams and distribution of flow structures. Tomographic estimates increase both the overall accuracy and the temporal resolution of mass capture statistics. Further improvements to nonlinear metrics can be realized by refining the prior.

## 5. Conclusion

We have derived a novel, linear model for AT with velocimetry: LATV. Our approach reconstructs 2D distributions of the partial pressure of one or more target species, temperature, and streamwise velocity using integrated absorbance and absorbance-weighted linecenter data. Sequential thermodynamic and velocity reconstructions are carried out by matrix-vector multiplication.

Our measurement model was verified by comparing modeled and Voigt-fitted linecenters for two O$_2$ transitions using noisy absorbance data. The correlation between modeled and fitted data exceeded 0.975 in both cases. Moreover LATV model errors were roughly centered and adequately modeled as IID normal variables. In a phantom test, LATV reconstructions captured large-scale features of the phantoms using a sparse beam array. Nonlinear combinations of $p$, $T$, and $u$ (in this case, the target mass flow rate) were more accurate when obtained by LATV than estimates based on uniform-beam TDLAS evaluations.

Linear AT with velocimetry has several attractive features. The linear formulation enables rapid reconstructions, and the simultaneous, non-intrusive, spatially-resolved reconstructions of partial pressure, temperature, and velocity have considerable potential to be used in scientific and engineering analyses of high-speed flowfields.

## Funding

Innovative Scientific Solutions Incorporated (SB20247); Air Force Research Laboratory (FA-8650-14-D-2317, FA8650-19-F-2416).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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