# Error Analysis of Integrated Absorbance for TDLAS in a Nonuniform Flow Field
## 4 Discussion and Application  
### Effect of the Absorption Line Overlap  
As discussed in Section 2, the above analysis is based on isolated absorption lines. In most practical applications, there may also be an absorption line overlap. The target absorption line overlaps with the weaker surrounding absorption line, distorting the absorption signal. This problem arises regardless of whether the flow field is uniform. To analyze the coupling effect of nonuniformity and absorption line overlap, this section takes two typical applications (i.e., the swirling flame and inlet oblique shock wave [10; 11]) as examples and uses the method described in Section 2. Figure 7 is a schematic diagram of the working conditions. Both examples can be simplified as a combination of two quasi-uniform flow field areas. In the swirling flame, the red circle divides the high-temperature zone from the low-temperature zone, the static pressure is constant, and the component concentration is linearly related to the temperature. In the inlet oblique shock wave, the shock wave divides the high-temperature zone from the low-temperature zone. The component concentration is constant, and the pressure changes drastically. \(L\) is the total absorption optical path length of the laser in the flow field, \(\delta\) and \(c\) are the absorption optical path lengths in the hot zone and the cold zone, respectively. The Mach number (Ma) is equal to 2.5, and the component concentration (X) is equal to 0.1, in the case of the inlet oblique shock wave. The physical parameters of the two flow fields are given in Table 1.  
Figure 7: Schematic of the case working conditions: **(a)** swirl flame and **(b)** inlet oblique shock. Four widely used absorption lines of water vapor [10; 43; 44] at 6807.8, 7185.6, 7444.3, and 7466.3 cm\({}^{-1}\) are analyzed. According to the HITRAN database [29], there are more than a dozen absorption lines of water within 1 cm\({}^{-1}\) around the four target lines. These adjacent absorption lines interfere with the target lines to varying degrees. They are thus used to illustrate the effect of line-shape distortion generated by absorption line overlap on the IA. Suppose that there is a laser beam that includes the four wavenumber ranges shown in Figure 8. As shown in Figure 7, the laser moves \(n_{1}\to n_{2}\) in the case of the swirling flame and \(n_{3}\to n_{4}\) in the case of the inlet oblique shock wave. As \(\delta/L\) (which is defined as the ratio of the length of the high temperature zone to the total absorption optical path) changes, the IA errors in the isolated (target absorption line) and blended (all absorption lines within the wave number range) spectra are calculated. Figure 8 is a schematic diagram of the absorption signal simulation and fitting when the laser passes through any of the flow fields in Figure 7. The red line at the bottom is the simulated blended spectrum, and the black solid line is the result of the multi-peak fitting of the simulated blended spectrum. The blue line is the simulated isolated spectrum, and the black dashed line is the result of the single-peak fitting of the simulated isolated spectrum. Here, the isolated spectrum is assumed to exist. This is to compare with the blended spectrum to analyze the effect of the absorption line overlap. The red and blue lines at the top are the fitting residuals of the two absorption signals. The difference between the two residuals shows the effect of the absorption line overlap on the line-shape fitting. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Work Condition**} & \multirow{2}{*}{**Flow Field**} & \multicolumn{2}{c}{**Temperature**} & \multirow{2}{*}{**Water Partial Pressure (Atm)**} \\  & & & **(K)** & \\ \hline \hline  & & & **Static Pressure** & \\  & & & **(Atm)** & **Concentration** \\ \hline Swirling Flames & 11 & 500 & 1.0 & 0.015 \\  & 22 & 1500 & 1.0 & 0.045 \\ Oblique shock wave & 11 & 900 & 1.0 & 0.1 \\  & 22 & 1440 & 4.0 & 0.1 \\ \hline \hline \end{tabular}
\end{table}