Evaluation of nitric oxide laser-induced fluorescence thermometry techniques in a hypersonic boundary layer

Abstract

Nitric oxide planar laser-induced fluorescence was performed to measure the wall-normal distribution of static temperature through a hypersonic boundary layer. A 10-degree half-angle wedge model was oriented at a 5-degree angle of attack in the NASA Langley 31-in Mach 10 facility, resulting in a 5-degree flow turning angle and an edge Mach number of 7.6. Nitric oxide was seeded through a spanwise slot into the boundary layer upstream of the imaging region and was excited with a pulsed ultraviolet planar laser sheet. The laser was spectrally scanned across six fluorescence transitions in the (0, 0) band of the $A^2\Sigma ^+$–$X^2 \Pi$ system. Eighteen thermometry methods were assessed through comparison to predictions of the temperature field from computational fluid dynamics simulations. The effect of spectral resolution and laser linewidth on measurement uncertainty was also investigated. The most accurate technique was spectral peak thermometry, which achieved an accuracy of ± 31.6 K ($12.6\%$ error relative to CFD temperature). The spectral peak thermometry technique required a minimum spectral resolution between 0.074 and 0.102 ${\mathrm {cm}}^{-1}$ and a maximum laser linewidth of 0.49 ${\mathrm {cm}}^{-1}$ to extract meaningful temperature information from the spectra.

Graphic abstract

1 Introduction

The properties of the hypersonic boundary layer (BL) on the surface of high-speed vehicles significantly affect both safety and performance (Cheng 1961; Spalding and Chi 1964; Tirtey et al. 2011). Quantifying heat flux through the surface of the vehicle is critical, as it affects decisions made on the composition and sizing of the thermal protection system (TPS) and limits flight operating conditions (Narayana et al. 2013; Narayana and Sato 2012; Van Wie et al. 2004; Lees 1956; Squire and Marschall 2010). A major motivation of current research is predicting the heat transfer to the vehicle during transition from laminar to turbulent flow. One major study investigated the complex thermodynamic and flow properties of hypersonic BL transition by placing a flow protuberance on the wing of the Space Shuttle Orbiter Discovery, while thermocouple and thermal imaging data were collected during atmospheric re-entry (Berger et al. 2015). During re-entry of STS-119, a region of unexpected turbulent flow was observed, resulting in additional heating on the surface of the vehicle, as determined by long-range thermal imaging. These observations demonstrate the complexity of hypersonic BL transition and the importance of developing accurate temperature measurement techniques in hypersonic flows for advancing numerical model development and improving our understanding of the hypersonic BL environment.

Laser-induced fluorescence (LIF) has been used in past studies to obtain a wide range of quantitative flow (e.g., velocity) and thermodynamic (e.g., pressure, temperature, and concentration) measurements in hypersonics (Danehy et al. 2015). In particular, LIF thermometry has been used to measure time-averaged and instantaneous rotational, vibrational, and translational temperatures in various flows (Palma et al. 2000, 2003; Combs et al. 2016; Seitzman et al. 1985). As a non-intrusive technique, LIF measurements do not disturb the flow in contrast to what occurs with physical-based probes (Chapman 1949). A recent paper by Chaudhry and Candler (2017) details the large influence a shock ahead of a pitot probe can have on dynamic pressure measurements in a hypersonic environment. The combination of sheet-forming optics and modern digital camera detectors results in planar-LIF (PLIF), which offers high spatial resolution and short exposure times (Kohse-Höinghaus 1990; Freegarde and Hancock 1997; Sánchez-González and North 2018). Since LIF occurs at a different wavelength than the laser, scatter near surfaces can be minimized using spectral filters. As a result, PLIF can offer higher signal–noise ratio (SNR) near surfaces than other laser-based measurements, such as particle image velocimetry (PIV) (Kinsey 1977; Bridges and Wernet 2003). The sensitivity of LIF to the fluid flow and thermodynamic state of the tracer species allows for various quantitative measurements, including temperature, pressure, mole fraction, and velocity (Danehy et al. 2015).

Nitric oxide (NO) is often used as a PLIF tracer species in high-speed flows due to its similar properties to air, which minimizes affecting natural flow properties and also results in relatively high mass diffusion rates (Arisman et al. 2013, 2015). The molecule is also relatively stable, allowing it to be purchased and stored in a conventional compressed gas bottle. Additionally, since NO occurs naturally in many high enthalpy flow facilities, LIF techniques developed for NO have a wide range of applications (Inman et al. 2013). Specifically, NO-LIF thermometry techniques have been thoroughly investigated in a wide variety of flows (Tamura et al. 1998; Gross et al. 1987; Vyrodov et al. 1995; Lee et al. 1993).

Sánchez-González et al. (2012) measured spatially resolved, instantaneous temperature fields in a Mach 4.6 flow using two-line PLIF with NO used as a tracer. This study achieved good agreement with computational fluid dynamics (CFD) simulations (differences of < 10% in most areas of the imaging region). By exciting the transitions with two simultaneous laser pulses, it was possible to measure transient temperature information in the flow field. Inman et al. (2013) measured the Doppler broadening effect from NO-PLIF spectra to determine an upper limit for free-stream translational temperature in a non-equilibrium, Mach 5 flow in an arc-jet facility. However, this study did not account for saturation broadening effects, which could result in an over-prediction in temperature. Palma et al. (2003) performed two-line rotational and vibrational thermometry in a Mach 7 shock-tunnel flow. The rotational temperature measurement uncertainty was minimized by choosing transitions that varied widely in rotational quantum number ($J^{\prime \prime }$). Using a similar method, Lachney and Clemens (1998) characterized two-line thermometry error in a supersonic bluff body LIF experiment. Bessler and Schulz (2004) later developed a multi-line approach to measuring temperature using NO-LIF. The approach was validated on a bunsen burner flame, where the method achieves less than $10\%$ disagreement with theory. The method modeled both the Doppler broadening and Boltzmann fraction temperature dependencies of the NO spectra. A later study reported agreement within $9\%$ error when compared to pyrometer measurements in high-pressure flames up to 40 bar (Lee et al. 2005a). Other groups have validated this technique using different tracer species (NO, SiO, OH) in a variety of experiments (Kirschner et al. 2014; Denisov et al. 2014; Chrystie et al. 2017; Lee et al. 2005b). The multi-line method has also been demonstrated in high-speed flows (Kirschner et al. 2014; Kaseman et al. 2017). The method used in Bessler and Schulz (2004) assumes that saturation effects are negligible, which holds in high-density flows.

The present study aims to build upon the work of previous LIF thermometry studies, particularly in validating a wide range of techniques to (1) measure temperature (2) correct for saturation effects (3) correct for absorption effects and (4) assess the applicability of these techniques for data sets of lower spectral resolution. A PLIF experiment was performed in NASA Langley Research Center’s 31-in. Mach 10 air tunnel with a 10-degree half-angle wedge model. Detailed descriptions of the facilities capabilities are given in Micol (1998) and Berger et al. (2015). Gas-flow properties of this facility were provided by Hollis (1996). Approximately 1.5 mJ/pulse of ultraviolet (UV) laser light was formed into a laser sheet and used to partially saturate the NO transitions. A 3D rendering of the wedge model and PLIF set-up is shown in Fig. 1, which shows free-stream and laser conditions. The CFD simulations completed previously showed good agreement with molecular tagging velocimetry (MTV) data (Bathel et al. 2012) and surface pressure data (Arisman et al. 2015) for these set-up and flow conditions. Using a simple geometry and laminar flow conditions, it can be assumed that CFD predictions of the temperature, pressure, and velocity fields are accurate, and therefore can be used to assess the thermometry techniques developed. The assessment of NO-PLIF thermometry techniques on a canonical flow will provide the foundation for their application on more complex flows around high-speed vehicles.

Fig. 1

Reproduced from Danehy et al. (2015)

Wedge model schematic showing gas seeding.

2 NO fluorescence model

A brief description of the underlying NO-LIF theory required for thermometry is presented here. The reader is directed to Palma et al. (1999) and McDougall (2018) for more complete descriptions of the two-level NO fluorescence signal ($S_{\mathrm{f}}$) model:

$$\begin{aligned} S_{\mathrm{f}}=\frac{1}{1+s}\chi _{_{\mathrm{NO}}} N_{\mathrm{T}} B_{12} F_{\mathrm{B}} I_{\mathrm{o}} G \varPhi t_{\mathrm{det}} V \frac{\varOmega }{4 \pi }\eta , \end{aligned}$$

(1)

where s, $B_{12}$, $F_{\mathrm{B}}$, and $\varPhi$ represent the saturation parameter, Einstein $B_{12}$ absorption coefficient, Boltzmann fraction and fluorescence yield, respectively, which are transition-dependent terms. $\chi _{\mathrm{NO}}$ and $N_{\mathrm{T}}$ represent fluorescence species mole fraction and total population of molecules per unit volume, respectively, and are flow-dependent terms. $I_{\mathrm{o}}$ and G represent the laser irradiance and spectral overlap integral, respectively, which are mainly laser-dependent terms. $t_{\mathrm{det}}$, V, $\varOmega$ and $\eta$ represent detection time of the detector/camera, volume probed by the laser, detection solid angle and the detector efficiency of the camera being used, respectively, and are dependent on the detection system used (Eckbreth 1996). This model is derived from the steady-rate equations based on a two-level energy representation of fluorescence. The two-level assumption has been assessed through comparisons with experimental fluorescence experiments (Drozda et al. 2018). Other experiments utilizing multi-level models to describe NO fluorescence for quantitative measurements often require absolute fluorescence values (Rossmann et al. 2003). The techniques outlined in this paper measure relative changes in fluorescence, which do not require a model greater than two levels. However, multi-level models are needed to account for saturation effectively when more complex effects such as rotational energy transfer are a factor.

The main temperature sensitivity of the LIF signal comes from the Boltzmann fraction ($F_{\mathrm{B}}$), which is defined as:

$$\begin{aligned} F_{\mathrm{b}}= (2J^{\prime \prime } +1) \exp \Bigg [-\frac{1}{k_{\mathrm{b}}}\Big (\frac{g^{\prime \prime }(v^{\prime \prime })}{T}+ \frac{f^{\prime \prime }(J^{\prime \prime },v^{\prime \prime })}{T}\Big )\Bigg ] \end{aligned}$$

(2)

$$\begin{aligned} Z= 2 \sum _{v''=0}^7 \sum _{i=1}^2 \sum _{J^{\prime \prime }=0.5}^{90.5} F_{\mathrm{b}} \end{aligned}$$

(3)

$$\begin{aligned} F_{\mathrm{B}}= \frac{F_{\mathrm{b}}}{Z}, \end{aligned}$$

(4)

where $v^{\prime \prime }$ is the vibrational ground-state level, i is the multiplicity of each level, $g^{\prime \prime }$ is the vibrational ground-state energy, and $f^{\prime \prime }$ is the rotational ground-state energy, $F_{\mathrm{b}}$ is the Boltzmann factor, Z is the partition function, and $k_{\mathrm{b}}$ is the Boltzmann constant. These equations assume thermal equilibrium between the vibrational and rotational temperature, T.

The transition spectral width is affected by the Doppler broadening, which is computed as follows:

$$\begin{aligned} \Delta _D= \frac{2 \nu _{\mathrm{o}}}{c}\sqrt{\frac{ 2 \ln (2) k_{\mathrm{b}} T}{m}}, \end{aligned}$$

(5)

where $\nu _{\mathrm{o}}$ is the transition center wavenumber, c is the speed of light, and m is the molecular mass of NO.

The remaining terms in the equation are either unneeded for the current analysis, or will be addressed specifically in later sections.

3 Experimental set-up

NO-PLIF measurements were obtained in a BL flow over a $10^\circ$ half-angle wedge, oriented with the top surface at a $5^\circ$ angle of attack with respect to the oncoming flow. The wedge was held in position using a sting attached to the rear surface of the wedge (see Fig. 1). An oblique shock forms at the leading edge, followed immediately by the development of laminar hydrodynamic and thermal BLs on the surface. The NO concentration BL begins at the ejection slot, which then extends to the remaining length of the surface. A detailed diagram of the wedge dimensions is shown in Fig. 2, accompanied by the flow conditions in Table 1. The dimensionless plate Reynolds number in Table 1 is calculated from the free-stream velocity and plate length in the x-direction.

Fig. 2

Experimental set-up of a 10-degree half-angle wedge in the NASA Langley 31-in. Mach 10 air tunnel. Images are not drawn to scale

Table 1 Wind tunnel parameters [calculated from properties given by Hollis (1996)]

The laser sheet propagated in the vertical direction and was oriented in the $x{-}y$ plane (refer to coordinate system defined in Fig. 2). A 12-bit, 512 $\times$ 512 pixel (53.2 $\times$ 54.3 mm imaging region), UV-sensitive intensified CCD camera (PI MAX) was oriented perpendicular to the laser sheet outside of the wind tunnel. A LayerTec filter was placed in front of the camera lens (at approximately $15^\circ$ to maximize LIF signal) to block the laser frequency and transmit the fluorescence. The laser used was tuned to a wavelength of 226.222 nm (44,204.37 ${\mathrm {cm}}^{-1}$) and was scanned spectrally over 75 s to 226.232 nm (44,202.31 ${\mathrm {cm}}^{-1}$). Detailed characteristics of the laser sheet are available in Table 2. Three pairs of transitions exist in this spectral range: $Q_1(\hbox {J}^{\prime \prime }=6.5$), $^QP_{21}(\hbox {J}^{\prime \prime }=6.5$); $R_1(\hbox {J}^{\prime \prime }=0.5$), $^RQ_{21}(\hbox {J}^{\prime \prime }=0.5$); $Q_2(\hbox {J}^{\prime \prime }=17.5$), $^QR_{12}(\hbox {J}^{\prime \prime }=17.5$). These transitions are in the (0, 0) band of the $A^2\Sigma ^+$–$X^2 \Pi$ system of the NO molecule. These transitions were strategically chosen to minimize absorption and saturation effects (each transition absorbs and saturates at a similar rate). Therefore, errors associated with these phenomena should cancel in some of the temperature measurement strategies used in this study. Each transition pair is spectrally overlapped (see Fig. 3), the details of which are outlined in Table 3 [obtained from Reisel et al. (1992)]. Note the $R_1$ and $^R Q_{21}$ values are calculated using equations from Reisel et al. (1992).

Table 2 Laser sheet characteristics

Table 3 Constants for the targeted transitions in the experiment.

Fig. 3

Transitions excited in the NO-PLIF experiment ($T = 400$ K, $P = 230$ Pa)

ImageJ, an image processing program created by the National Institute of Health, was used to dewarp, visualize, and sample the experimental PLIF data (Schneider et al. 2012). The average wall temperature ($T_{\mathrm{w}}$ = 314 K) was measured using a thermocouple attached to the hollow interior of the steel wedge model.

4 Computational fluid dynamics

CFD predictions of temperature results are used to assess the thermometry techniques used in the present work. Some of the CFD data used in this study were obtained from simulations performed in previous works (Arisman et al. 2013, 2015). A brief summary of the simulation setup is provided here. Full simulation details can be found in the original works (Arisman et al. 2013, 2015).

Simulations were performed using the open-source software, OpenFOAM. The density-based compressible flow solver, rhoCentralFOAM, was used, which uses the flux scheme of Kurganov and Tadmor (2000), providing high-order, non-oscillatory results near flow discontinuities such as shock waves. rhoCentralFOAM has been extensively verified and validated in the literature. Specifically, it has shown good agreement with high-speed flow experiments (Hinman and Johansen 2016a, c; Hinman et al. 2017), popular commercial solvers (Arisman et al. 2013, 2015), and reduced-order solvers (Hinman and Johansen 2016b). As rhoCentralFOAM does not natively support multi-species transport or chemical reactions, these modifications were implemented by the current authors.

The simulations were performed in two dimensions on a quad-dominant grid generated using the built-in meshing utility, blockMesh. Laminar flow was assumed due to the low non-dimensional Reynolds number compared to the literature ($3.4 \times 10^5$, where transition occurs beyond $10^6$) and the smooth flow pattern observed in the PLIF flow visualization images from a previous study (Danehy et al. 2015; Deem and Murphy 1965).

Because of the low temperature and high speeds in the Mach 10 tunnel, thermophysical properties were modeled in CFD using modified NASA polynomials for enthalpy and heat capacity, and a custom low-temperature viscosity law (Hollis 1996). The NO seed gas was injected from a 2D slot in the simulation. It was determined that NO chemistry effects were negligible at these conditions (Arisman et al. 2013). As well, the effect of a high Knudsen number on the leading edge was examined and found to be negligible. A reduced-order laminar flow solver is also implemented, HyPE2D, which agreed with rhoCentralFoam results to within 1% (Hinman and Johansen 2016b). A combination of rhoCentralFoam and HyPE2D is used in the present study. Three corrections were implemented in the current work to improve the accuracy of the simulations, which are explained below.

4.1 Temperature boundary condition uncertainty

Although initial simulations were completed with a surface boundary condition of $T_{\mathrm{w}}=314$ K, this temperature represented the time-averaged thermocouple temperature. The temperature measured from the thermocouple increased linearly from $T_{\mathrm{t,i}}=306$ K to $T_{\mathrm{t,end}}=320$ K throughout the run. Note that this temperature was measured inside the wedge model, approximately 9.5 mm from the surface.

An unsteady 1D heat transfer analysis through the 9.5-mm-thick steel plate on the upper surface of the wedge model indicated a 1-K temperature difference. The true wall temperatures at the beginning and end of the run are corrected to $T_{\mathrm{t,i}}=307$ K to $T_{\mathrm{t,end}}=321$ K.

Rarefied effects at the tunnel conditions result in temperature slip, which can be computed as follows (White and Corfield 2006):

$$\begin{aligned} T_{\mathrm{gas}} = 0.87 M C_{\mathrm{f}} (T_{\mathrm{r}} - T_{\mathrm{w}}) +T_{\mathrm{w}}, \end{aligned}$$

(6)

where $T_{\mathrm{gas}}$ is the gas temperature at an infinitesimally small distance from the surface, M is the free-stream Mach number, $C_{\mathrm{f}}$ is the skin friction coefficient, $T_{\mathrm{r}}$ is the adiabatic wall temperature, and $T_{\mathrm{w}}$ is the measured (thermocouple) wall temperature. The value of $T_{\mathrm{gas}}$ was calculated for the lower (307 K) and upper (321 K) bounds on the surface temperature, resulting in corresponding near-surface gas temperatures of $T_{\mathrm{t,i}} = 315$ K and $T_{\mathrm{t,end}} = 329$ K.

Since the surface temperature boundary condition was time dependent, the flow was simulated using the upper and lower limits of surface temperature as boundary conditions in HyPE2D. The simulated temperature fields from the upper and lower temperature limits form a confidence band for the CFD temperature at any location in the flow. This band is used to assess NO-PLIF thermometry results.

5 Image pre-processing

5.1 Spatial filtering

Shot noise is the main source of noise observed in the NO-PLIF images when the SNR is large, which is common for similar-type experiments (Inman et al. 2009; Thurber et al. 1997). At low SNR levels, other noise sources become non-negligible, such as intensifier noise. To increase the SNR, a 2D moving average filter was applied to each PLIF image. A 3 $\times$ 3 pixel window size for the filter was found to be optimal for improving the quality and reliability of the spectra fits for each thermometry technique, while minimizing blurring.

5.2 Reflection correction technique

Reflection of fluorescence radiation off of the brushed-steel side wall on the opposite side of the tunnel from the camera was observed. Reflection was estimated to be between 5 and 10% of the maximum signal in the BL. The radiation reflection for the $J^{\prime \prime } = 6.5$ transition pair is shown in Fig. 4. More details on the effect of fluorescence reflection on the accuracy of the PLIF thermometry techniques can be found in McDougall et al. (2018a). The main source of fluorescence (near-surface) reflects off of the side wall, artificially increasing the PLIF signal at the edge of BL and freestream.

The reflectance observed in this experiment is a combination of specular and diffuse reflections. This reflected radiation is corrected by fitting a linear function to the reflectance region outside the concentration BL. This region is defined as 1% of the wall-value mole fraction calculated by rhoCentralFoam for the BL profile at any x-location. This linear fit is extended into the fluorescence region, and subtracted from the profile. This method is repeated for all x-locations and all NO-PLIF images in the sequence, and is then masked for 1$\%$ of the CFD mole fraction profile. The result of this correction is illustrated in Fig. 4.

Fig. 4

$J^{\prime \prime } = 6.5$ reflection correction at $x = 85$ mm

6 Analysis methods

This section describes several temperature measurement and correction techniques. None of the techniques require calibration to a known region in the flow, and are therefore absolute measurements. The saturation models below account for laser-saturation effects occurring in the NO laser absorption interaction. An absorption correction is outlined using diffusion modeling results obtained from the CFD simulations.

6.1 Thermometry techniques

The PLIF thermometry techniques are largely based on methods described in prior experimental studies (Sánchez-González et al. 2012; Inman et al. 2013; Palma et al. 2003; Bessler and Schulz 2004). The techniques were implemented in MATLAB and applied to PLIF data sampled using ImageJ. Each technique targets different temperature-dependent features of the LIF spectra. Specifically, two temperature-dependent features are used: The Boltzmann factor $F_{\mathrm{b}}$ and the Doppler broadening width $\Delta _D$. In each technique, the triple-Gaussian approximation is used to replace the spectral overlap integral term, which represents the overlap of the laser spectral line and the transition absorption line. The triple-Gaussian approximation has been assessed in prior works (Ivey et al. 2011). Due to the low-pressure environment of the facility ($60{-}230$ Pa), collision (pressure) broadening ($\Delta _C$) is generally a small component ($<2\%$) of the overall broadening of the transition line.

Each thermometry method uses a Non-Linear Least-Squares (NLLS) regression to fit the LIF spectra. NLLS is an iterative process that minimizes the sum of squares of the residuals between the model and the experimental data points, reaching a minimum when the model best fits the experimental spectra. Each model includes temperature as a free parameter (within the equations for $F_{\mathrm{b}}$ and $\Delta _D$), which is varied during iterations, eventually reaching an optimal value when the sum of squares is minimized.

6.1.1 Spectral peak thermometry (SPT)

SPT targets the rotational temperature dependency of the Boltzmann factor term in the fluorescence equation to determine temperature from the NO-LIF spectra. The Boltzmann factor affects the relative amplitudes of the transitions. SPT fits the relative amplitudes for temperature, while fitting free parameters to width, overall amplitude, and translation factor terms. The resulting model that is fit to the LIF spectra is denoted $\mathrm{SPT}(T,k)$, and is given by:

$$\begin{aligned} \mathrm{SPT}(T,k)&= \delta _u + W_A \sum _{k=1}^6 \Bigg ( B_{12}(k) F_{\mathrm{b}}(T,k) \\&\quad {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _{\mathrm{W}}^2 }}\Bigg ), \end{aligned}$$

(7)

where $\nu$ is the laser wavenumber, $\nu _{\mathrm{o}}$ is the transition center wavenumber, $\delta _u$ is the amplitude translation factor, $\delta _{\mathrm{s}}$ is the spectral translation factor, and $\Delta _{\mathrm{W}}$ is the spectral width parameter. $W_A$ is a free parameter given by:

$$\begin{aligned} W_A=\frac{\chi _{_{\mathrm{NO}}} N_{\mathrm{T}} I_{\mathrm{o}} \varPhi }{Z \sqrt{\Delta _D(T)^2 + \Delta _L^2 }} t_{\mathrm{det}} V \frac{\varOmega }{4 \pi } \eta , \end{aligned}$$

(8)

where $\Delta _D$ is the Doppler broadening width, and $\Delta _L$ is the laser linewidth. $W_A$ is derived from the remaining terms in Eq. (1) that are not defined in Eq. (7). This derivation is completed in a similar way for each thermometry method scaling parameter, W.

6.1.2 Doppler broadening thermometry (DBT)

DBT targets the translational temperature dependency of the Doppler broadening term in the fluorescence equation to determine temperature from the NO-LIF spectra. Note that the translational and rotational temperatures are expected to be in thermal equilibrium for this boundary layer flow. DBT fits the widths of the transitions for temperature, while fitting free parameters to relative amplitude, and translation factor terms. The resulting model that is fit to the LIF spectra is denoted ${\mathrm{DBT}}(T,k)$, and is given by:

$$\begin{aligned} {\mathrm{DBT}}(T,k) = \delta _u + \sum _{k=1}^{6} \Bigg ( W_{\mathrm{B}}(k) {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _C^2 + \Delta _D(T)^2 + \Delta _L^2 }}\Bigg ), \end{aligned}$$

(9)

where $W_{\mathrm{B}}$ is a free parameter given by:

$$\begin{aligned} W_{\mathrm{B}}=\frac{\chi _{_{\mathrm{NO}}} N_{\mathrm{T}} I_{\mathrm{o}} \varPhi }{Z \sqrt{\Delta _C^2 +\Delta _D(T)^2 + \Delta _L^2 }}B_{12} \frac{F_{\mathrm{b}}}{Z} t_{\mathrm{det}} V \frac{\varOmega }{4 \pi } \eta \end{aligned}$$

(10)

6.1.3 Full spectra thermometry (FST)

FST targets the temperature dependency of both the Boltzmann factor and Doppler broadening terms in the fluorescence equation (assuming thermal equilibrium), combining components of SPT and DBT. FST fits the relative amplitudes, and the transition widths, for temperature. The overall amplitude and translation factor terms are left as free parameters. The resulting model that is fit to the LIF spectra is denoted FST(T, k), and is given by:

$$\begin{aligned} {\mathrm{FST}}(T,k)&= \delta _u + W_A \sum _{k=1}^6 \Bigg ( B_{12}(k) F_{\mathrm{b}} (T,k)\nonumber \\&\quad {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _C^2 +\Delta _D(T)^2 + \Delta _L^2}}\Bigg ), \end{aligned}$$

(11)

where $W_A$ is the same scaling constant used in the SPT model.

6.2 Saturation correction methods

Saturation occurs when the laser energy is relatively high, depleting the ground-state population of the tracer species (Eckbreth 1996). When a transition is saturated, the relationship between the laser energy and the fluorescence signal becomes non-linear, exhibiting a plateau shape. Therefore, a saturated transition produces less fluorescence than would otherwise occur in an unsaturated transition with the same laser energy (Lucht et al. 1983; Daily 1977, 1978). The saturation irradiance, defined as the term $I_{\mathrm{SAT}}$, is dependent on transition properties and thermodynamic properties of the surrounding fluid. The saturation effect on the transition amplitude can be computed from the following equation:

$$\begin{aligned} S_{\mathrm{SAT}} = \frac{S_{\mathrm{f}}}{1+\frac{I_{\mathrm{o}}}{I_{\mathrm{SAT}}}}, \end{aligned}$$

(12)

where $S_{\mathrm{SAT}}$ is the saturated fluorescence radiation, $S_{\mathrm{f}}$ is the linear fluorescence radiation (if saturation did not occur), $I_{\mathrm{o}}$ is the laser irradiance, and $I_{\mathrm{SAT}}$ is the transition-specific saturation irradiance. The saturation effect on spectral width of transitions is determined by the following equation:

$$\begin{aligned} \gamma _{\mathrm{SAT}} = \gamma _{\mathrm{f}} \sqrt{1+\frac{I_{\mathrm{o}}}{I_{\mathrm{SAT}}}}, \end{aligned}$$

(13)

where $\gamma _{\mathrm{SAT}}$ is the saturated width, and $\gamma _{\mathrm{f}}$ is the unsaturated width. The laser-normalized saturation parameter, s, is defined as:

$$\begin{aligned} s = \frac{I_{\mathrm{o}}}{I_{\mathrm{SAT}}} \end{aligned}$$

(14)

The s parameter is used to correct for the effects of saturation, as discussed later.

The spectra shape is affected by saturation, resulting in changes in the relative amplitudes and spectral widths of the transition. The relative amplitudes change due to the unique saturation irradiances for each transition. The transition lines artificially broaden as well, due to saturation broadening, where the wings of the transition are less saturated. The saturation effect on two transitions, at three different laser energies, is shown in Fig. 5. The figure is not scaled by the original laser energy to better visualize the saturation effect on a single axis. Three saturation correction methods have been assessed in the current work. Each method builds on the models developed in previous fundamental literature (Lucht et al. 1983; Daily 1977, 1978).

Fig. 5

Effects of laser saturation on normalized spectra amplitudes and widths

6.2.1 Analytical correction (AC)

The AC method relies on a well-known algebraic model for saturation (Eckbreth 1996):

$$\begin{aligned} I_{\mathrm{SAT}} = \frac{A_{21} + Q_{21}}{B_{12} + B_{21}}O(\nu _{\mathrm{o}}), \end{aligned}$$

(15)

where $B_{21}$ is the Einstein stimulated emission coefficient, $A_{21}$ is the Einstein spontaneous emission coefficient, $O(\nu _{\mathrm{o}})$ is the overlap fraction between the laser lineshape and absorption lineshape at the center wavelength of the transition, and $Q_{21}$ is the collision quenching term. Rotational energy transfer (RET) effects are assumed to be negligible due to the low pressure (approximately 230 Pa) and, therefore, low collision rate in the boundary layer (Ebata et al. 1984). The pressure dependence of RET rates has been discussed in the literature (Daily et al. 2005; Naik and Laurendeau 2004). The overlap fraction at the center wavelength $O(\nu _{\mathrm{o}})$ determines the overlap of the narrow-band laser at the center frequency of the transition, which calculates the fraction of the population being saturated by the laser irradiance $I_{\mathrm{o}}$. The equation for $Q_{21}$ is given below:

$$\begin{aligned} Q_{21} = N_{\mathrm{T}} \sqrt{\frac{8 k_{\mathrm{b}} T}{\pi m_{\mathrm{a}}}} \sum _{i} \chi _i \sigma _i \left( 1+\frac{m_{\mathrm{a}}}{m_i}\right) , \end{aligned}$$

(16)

where $m_{\mathrm{a}}$ is the mass of the absorbing species, $\sigma _i$ is the collision cross section for species i, and $m_i$ is the mass of species i. The collisional quenching term has been shown to be independent of the excited transition (McDermid and Laudenslager 1982).

Using Eq. (15), the saturation irradiance ($I_{\mathrm{SAT}}$) can be calculated. The laser irradiance ($I_{\mathrm{o}}$) is determined using measurements of the laser sheet dimensions and energy per pulse.

Equations 12 and 13 are used to calculate the saturated amplitude and saturated width of the transitions and are used as correction factors in each thermometry model (SPT, DBT and FST). An example is shown with the FST technique:

$$\begin{aligned} {\mathrm{FST}}_{\mathrm{AC}}(T,k)= & {} \delta _u + W_A \sum _{k=1}^6 \Bigg ( \frac{1}{1+s(k)} B_{12}(k) F_{\mathrm{b}} (T,k) \nonumber \\&\quad {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ (1+s)(\Delta _C^2 +\Delta _D(T)^2 + \Delta _L^2)}}\Bigg ). \end{aligned}$$

(17)

The thermometry results from these corrected techniques will be denoted ${\mathrm {SPT}}_{\mathrm {AC}}$, ${\mathrm {DBT}}_{\mathrm {AC}}$, and ${\mathrm {FST}}_{\mathrm {AC}}$.

6.2.2 Maximum correction (MC)

The MC method assumes that the transitions are fully saturated, and corrects the relative magnitudes accordingly. RET is assumed to be negligible in this saturation correction as well. This method does not correct for saturation broadening. This form of the saturation correction is taken from Eckbreth (1996), where an analytical correction to the fluorescence equation is derived based on the assumption that the laser intensity is very large. This correction is shown below for the SPT and FST thermometry methods:

$$\begin{aligned} \mathrm{SPT}_{\mathrm{MC}}(T,k)&= \delta _u + W_A \sum _{k=1}^6 \Bigg (\frac{B_{12}(k)}{B_{12}(k)+B_{21}(k)}\nonumber \\&F_{\mathrm{b}}(T,k) {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _{\mathrm{W}}^2 }}\Bigg ) \end{aligned}$$

(18)

$$\begin{aligned} {\mathrm{FST}}_\mathrm{MC}(T,k)&= \delta _u + W_A \sum _{k=1}^6 \Bigg ( \frac{B_{12}(k)}{B_{12}(k)+B_{21}(k)} \nonumber \\&F_{\mathrm{b}} (T,k) {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _C^2 +\Delta _D(T)^2 + \Delta _L^2}} \Bigg ). \end{aligned}$$

(19)

The MC technique does not apply to DBT, as this technique only corrects the relative transition amplitudes. Using the above equations, the temperature can be determined while assuming maximum saturation. The thermometry results from these corrected methods are denoted ${\mathrm {SPT}}_{\mathrm {MC}}$ and ${\mathrm {FST}}_{\mathrm {MC}}.$

6.2.3 Width correction (WC)

The WC method measures the saturation broadening width to estimate the level of saturation for each transition, which is used to correct the relative amplitudes. Each transition broadens at different rates, due to the distinct saturation irradiances of different transitions (Palma et al. 1999). In contrast, the unsaturated broadening (due to Doppler and collision effects) should be transition independent. Therefore, Eq. (13) is rearranged for $1+s$, and substituted into Eq. (12). This method cannot correct for the exact saturated width, only the relative difference between transition pairs, and therefore can only be used with SPT. It should be noted that this is an approximate technique, but offers the advantage of not requiring prior knowledge of the laser energy. In the SPT equation (Eq. 7), this allows the $\gamma _{\mathrm{us}}$ term to be factored out, and included as an additional scaling constant. This correction applied to SPT is shown in Eq. (20).

$$\begin{aligned} \mathrm{SPT}_{WC}(T,k)&= \delta _u + W_A \gamma ^2 _{\mathrm{f}} \sum _{k=1}^6 \Bigg ( \frac{B_{12}(k) F_{\mathrm{b}}(T,k)}{\gamma _{\mathrm{sat}}^2(k)}\nonumber \\&\quad {\mathrm{e}}^{\frac{-4 \ln (2)(\nu -\nu _{\mathrm{o}}(k) - \delta _{\mathrm{s}})}{ \Delta _{\mathrm{W}}^2 }} \Bigg ). \end{aligned}$$

(20)

6.3 Absorption correction (AB)

The AB method is used to correct for absorption effects as the laser propagates through the NO concentration BL. This method requires an NO mole fraction distribution, which could be obtained experimentally (McDougall et al. (2018b)), approximated from empirical relations (Arisman et al. (2015)), or from CFD. Since the goal of the AB analysis is to assess the effectiveness of the correction for different thermometry techniques, CFD is used in the current work to obtain the NO mole fraction distributions. Beer’s absorption law is used when correcting for absorption in LIF experiments:

$$\begin{aligned} I(k,y) = I_{\mathrm{o}} {\mathrm{e}}^{-k_a y}, \end{aligned}$$

(21)

where $k_a$ is the transition absorption coefficient, and y is distance traveled into the BL. The absorption coefficients are transition dependent, and are given by the following equation:

$$\begin{aligned} k_a(k) = \frac{1}{1+s} \chi _{_{\mathrm{NO}}} N_{\mathrm{T}} F_{\mathrm{B}} B_{12} h \nu _{\mathrm{o}} c \end{aligned}$$

(22)

This term scales the laser lineshape magnitude as the laser propagates the BL. Since more absorption occurs in the center of the transition than in the wings, the transition is broadened. In experiments with large absorption effects, the laser lineshape can be altered significantly, to the point where the triple-Gaussian approximation is not valid (Danehy and O’Byrne 1999). Since the spectral width of the laser is smaller than the transition width, absorption is expected to have a minor effect on the laser lineshape. To account for minor broadening, the laser lineshape width in the triple-Gaussian assumption is scaled by the following term.

$$\begin{aligned} \Delta _L(k,y,\nu ) = \Delta _{L_{0}} {\mathrm{e}}^{\frac{k_a y}{2}}. \end{aligned}$$

(23)

This correction assumes that the peak magnitude at which the laser is absorbed artificially broadens the laser lineshape at the same rate. This correction is derived in an analogous method to the saturation broadening correction by Demtröder (2008). More intricate absorption models are available that are better suited to highly absorptive medium (Danehy and O’Byrne 1999). The temperature term in the $N_{\mathrm{T}}$ and $F_{\mathrm{B}}$ terms is determined using the SPT technique. The $\chi _{_{\mathrm{NO}}}$ term is determined using rhoCentralFoam results. Since the term is y-dependent, a step-wise program was developed to recalculate the irradiance I at each location, for each transition. The saturation term s is assumed to be zero, unless the AC method is applied.

By including the irradiance I(k, y), replacing $I_{\mathrm{o}}$ in each basic thermometry technique, the irradiance becomes a transition and spatially dependent term. The term is included inside the sums as an additional correction term for each thermometry technique. The techniques that correct for absorption include a superscript AB (i.e., ${\mathrm {SPT}}^{\mathrm {AB}}$).

Fig. 6

NO-PLIF thermometry technique matrix. Combines X basic thermometry techniques with Y saturation correction techniques and Z the absorption correction techniques. Matrix elements (18), subscripts, and superscripts represent the thermometry, saturation, and absorption techniques that were assessed, respectively

7 Evaluation of analysis methods

Various aspects of the NO-PLIF thermometry techniques detailed above are evaluated: (1) accuracy and robustness of extracting temperature from the PLIF data, (2) the minimum spectral resolution and maximum laser linewidth required to perform these measurements, and (3) the most temperature-sensitive transitions within the spectral domain used. Overall, the best thermometry techniques should fit the spectra to a high degree of precision, while providing temperature measurements that agree well with the predicted temperature field from CFD. The sections below describe each evaluation criterion in greater detail.

7.1 Comparison to CFD temperature

All temperature fields output by the thermometry techniques described in Fig. 6 are compared quantitatively to the CFD temperature confidence band to assess their accuracy. To determine the most accurate thermometry technique, the least-squares difference between the PLIF temperature field and the CFD temperature confidence band is calculated. Based on this analysis, the three best performing thermometry methods are chosen for extended discussion in Sect. 8.

7.2 Undersampling of NO-PLIF images

The three most accurate thermometry techniques are analyzed in greater detail by undersampling the PLIF data in the frequency domain. The data are undersampled at consistent intervals between the full resolution of 0.0031–0.25 ${\mathrm {cm}}^{-1}$. Note that the laser linewidth is 0.07 ${\mathrm {cm}}^{-1}$. This was done to extend the applicability of this experiment to future studies looking to employ these thermometry techniques at lower spectral resolutions.

7.3 Two-line thermometry

The importance of each transition to the multi-line thermometry techniques is also investigated. The NO-PLIF data were undersampled in the frequency domain by cropping out one transition. This process was repeated for all three transition pairs, resulting in three combinations of transition pairs available for two-line thermometry. The pairs are assigned a subscript, as defined by Table 4. Each pair is designated as a subscript preceding the acronym corresponding the the thermometry technique (i.e., $_{\mathrm {A SPT}}$).

Table 4 Two-line thermometry transition pairs

8 Results

The results from each thermometry technique applied to the Mach 10 NO-PLIF experiment are shown below. The temperature fields are compared to a CFD temperature confidence band, output by HyPE2D.

8.1 Full spectral resolution thermometry results

Results have been cropped to the concentration BL, as defined by rhoCentralFoam simulations (1% of the surface value). The maximum and minimum SNR observed for each transition pair within this region are as follows: SNR$_{17.5} = [0.7, 9.3]$, SNR$_{6.5} = [3.3, 9.8]$, SNR$_{0.5} = [7.5, 8.8]$. The following sections will review the three most accurate techniques, as well as the remaining techniques that did not perform to the same level of accuracy.

8.1.1 Best performing techniques

The most accurate techniques, as determined by least-squares minimization, were SPT, SPT$^{\mathrm {AB}}$, and ${\mathrm {SPT}}_{\mathrm {WC}}^{\mathrm {AB}}$. Three temperature profiles, corresponding to the start, middle and end of the imaging region, are provided in Fig. 7. Error bars are provided on the SPT data points, which represent 95% confidence intervals calculated from the NLLS fit added in quadrature with spatial error due to minor wedge model movement (approximately 0.14 mm in the y-direction) over the duration of the experiment. This spatial error was calculated by quantifying the range of temperatures reported by SPT within the spatial movement of the wedge model. The SPT, ${\mathrm {SPT}}^{\mathrm {AB}}$, and ${\mathrm {SPT}}_{\mathrm {WC}}^{\mathrm {AB}}$ techniques deviated on average from the CFD-HyPE2D confidence band by 31.6 K (12.6%), 30.7 K (11.2%), and 35.8 K (16.3%), respectively. Relative error was calculated against CFD temperature for each point in the flow, and then averaged for the temperature field.

Fig. 7

Temperature profiles for the best performing thermometry methods

Fig. 8

Spectra at location (i) in Fig. 7a

Fig. 9

Spectra at location (ii) in Fig. 7b

Fig. 10

Spectra at location (iii) in Fig. 7c

Table 5 Uncertainty in each thermometry technique (based on 95% confidence of fit)

Three profiles are shown in Fig. 7 that correspond to important regions of the flow, which were used to evaluate the spectra-fitting quality of each thermometry technique, that is shown in Figs. 8, 9 and 10. The 95% temperature confidence interval for each thermometry technique at each profile location is given in Table 5.

At each location, the SPT technique fits the spectra to the highest degree of accuracy. This indicates that the transitions chosen for the present work are not largely affected by absorption or saturation for the majority of the BL region. The overshoot in fluorescence signal for the middle transitions from the ${\mathrm {SPT}}^{\mathrm {AB}}_{\mathrm {WC}}$ model in Figs. 8 and 9 indicates shortcomings of the the width correction method in estimating the saturation broadening.

Near the BL edge, ${\mathrm {SPT}}^{\mathrm {AB}}_{\mathrm {WC}}$ fits the spectra well (see Fig. 10), indicating saturation may be occurring in this region. However, the poor SNR combined with a sensitive saturation correction in ${\mathrm {SPT}}^{\mathrm {AB}}_{\mathrm {WC}}$ results in large error in the temperature measurement for a couple of points near the edge of the BL (see Fig. 7a). The ${\mathrm {SPT}}^{\mathrm {AB}}_{\mathrm {WC}}$ technique has not fit the spectra to a high degree of accuracy elsewhere in the BL, only near the outer edge. The most likely reason to explain this observation is that the saturation irradiances ($I_{\mathrm{SAT}}$ values) are temperature dependent. The saturation behavior of the transitions is different at the edge of the BL (where temperatures are approximately 150 K) than the near-surface region (where temperatures can reach 400 K). This is illustrated by Fig. 11a and b, where the fluorescence signal for each transition is plotted against the laser irradiance at two different temperatures. A valuable observation from these results is that saturation broadening can be identified and used to correct saturated transitions in some instances. The WC method needs to be further validated (both analytically and experimentally) to understand its value as a viable saturation correction technique.

Absorption effects did not appear to be a factor in this analysis, regardless of the BL region. This observation indicates that the transitions used in the present work are largely unaffected by laser absorption. This result is expected, as the NO concentration is low and the concentration boundary layer thickness is very small (short path length through NO region).

Overall, the most accurate thermometry technique was determined to be SPT, due to good agreement with the CFD temperature field and consistently high-quality spectra fits. The SPT temperature profiles fell within error of the CFD temperature profiles for the large majority of the imaging region. The full temperature map of the imaging region is provided in Fig. 12, with the CFD temperature field in Fig. 13 for comparison. The other two high-accuracy techniques, ${\mathrm {SPT}}^{\mathrm {AB}}$ and ${\mathrm {SPT}}^{\mathrm {AB}}_{\mathrm {WC}}$, perform comparably to SPT.

Fig. 11

Simulated saturation curves for each transition. The vertical dotted line indicates the laser energy used

Fig. 12

SPT measured temperature field

Fig. 13

CFD simulated temperature field

8.1.2 Under-performing techniques

Analysis of the under-performing techniques can provide useful information about the LIF modeling, laser setup, and the flow.

Figure 14 shows the disagreement between these methods and CFD. AC and MC normally induce large changes to the fluorescence model (as opposed to the more minor changes induced by the WC method), which can cause the fitting algorithm to perform poorly. However, since transitions were chosen that saturate similarly, large changes are not seen in the thermometry results unless the transitions were partially saturated. As seen in the previous section, SPT fits the spectra to a high degree of accuracy in the majority of the BL due to absorption and saturation effects canceling to first order for the chosen transitions. The AC method corrected the temperature profiles of the SPT, FST and DBT methods to varying degrees of success, and in some cases over-corrected. These errors are likely due to inaccuracies in the laser irradiance measurement, or inadequate physical modeling within the models themselves. It has been reported that applying analytical saturation corrections can be challenging when the laser used is not characterized to a high degree of accuracy (Palma et al. 1999).

The DBT technique over-predicted temperature in all regions of the BL, as shown in Fig. 14. The spectra were broader than expected at these temperatures, indicating saturation broadening could be occurring, inflating the DBT temperature. Also, since the laser linewidth was not known to a high degree of accuracy (see Table 2), the DBT technique inherently has a large associated error, indicated by the error bars on Fig. 14. The DBT temperature profile in Fig. 14 indicates less saturation (better agreement with CFD) at the near-surface BL, and more saturation (disagreement with CFD) at the BL edge. Most likely, an underestimate of the laser linewidth, and saturation broadening at the BL edge, combine to produce the inaccurate DBT profile. Since these factors cancel out to first order in SPT, better results are obtained for that technique.

Fig. 14

Temperature profiles of other thermometry techniques at $x =$ 85 mm

The FST technique is also affected negatively by saturation broadening also shown by the DBT results at the BL edge. The FST technique combined with the AC method corrects the temperature profile in some regions of the BL, especially near the edge. In other regions, the profile is overcorrected, as seen in AC applied to other thermometry techniques.

The techniques not presented in the figures had similar performance to their base thermometry technique.

8.2 Under-sampled thermometry results

Three methods were used to evaluate the effect of spectral resolution and laser linewidth on the temperature measurement using the best performing thermometry technique, SPT. Method 1 (M1) involved undersampling the PLIF data at consistent intervals from full resolution (0.0013 ${\mathrm {cm}}^{-1}$) to 0.2 ${\mathrm {cm}}^{-1}$. At each undersampled interval, the surrounding data points were averaged, simulating a laser with larger spectral linewidth. Note the original laser spectral linewidth is 0.07 ${\mathrm {cm}}^{-1}$. Method 2 (M2) undersampled the data at the same intervals as M1, but did not average the surrounding data points. Therefore, M2 simulated a lower spectral resolution with a constant laser linewidth of 0.07 ${\mathrm {cm}}^{-1}$. This corresponds to a faster spectral scan. Method 3 (M3) averaged surrounding data points while increasing the averaging bin width, simulating a larger laser linewidth, while retaining the same step size. Relative error between the thermometry techniques and CFD was determined at the peak temperatures at each x-position in the BL. This point was chosen because SPT was most accurate at this location at full resolution.

Each data point in Fig. 15 represents an average standard error (SE) obtained from NLLS fits for methods 1 and 2. At each resolution, the spectra were shifted from 0 to 0.2 ${\mathrm {cm}}^{-1}$ in 20 equal intervals. This figure clearly indicates the regions where the temperature fitting techniques are no longer viable, which is denoted as the “minimum spectral resolution method 1” (MSR1) and “minimum spectral resolution method 2” (MSR2). The MSR1 (approximately 0.102 ${\mathrm {cm}}^{-1}$) and MSR2 (approximately 0.074 ${\mathrm {m}}^{-1}$) in Fig. 15 are qualitative approximations for the minimum spectral resolution needed to extract meaningful temperature information from the PLIF signal using SPT-based techniques. Figure 16 shows the spectra undersampled at various rates, above and below the MSR1, to highlight the effect of spectral resolution on the spectral line measurements. Figure 17 shows the effect of increasing the laser linewidth, and indicates a maximum linewidth (MLW) that could be used to perform thermometry using the SPT technique with the selected transitions. The MLW was determined to be 0.49 ${\mathrm {cm}}^{-1}$.

Fig. 15

Standard error (SE) at different sampling methods 1 (M1) and 2 (M2) for the SPT technique

Fig. 16

Spectra under-sampling at different spectral resolutions using sampling method 1

Fig. 17

Absolute temperature error at larger laser line-widths (Method 3)

To extend to future studies, the MSR1, MSR2 and MLW can be non-dimensionalized (ND). MSR1 and MSR2 are ND by the spectral width of the transitions, while MLW is ND by the separation distance between the closest transition pairs ($J^{\prime \prime }$ = 0.5 and $J^{\prime \prime }$ = 17.5). Shown below is the ND equations, along with the ND MSR1, MSR2 and MLW values for this study:

$$\begin{aligned}&\frac{\hbox {MSR1}}{\sqrt{\Delta _\nu ^2 + \Delta _{D}^2 +\Delta _{C}^2}} = 0.97 (+\,0.23 , -\,0.21) \end{aligned}$$

(24)

$$\begin{aligned}&\frac{\hbox {MSR2}}{\sqrt{\Delta _\nu ^2 + \Delta _{D}^2 +\Delta _{C}^2}} = 0.70 (+\,0.17, -\,0.15) \end{aligned}$$

(25)

$$\begin{aligned}&\frac{\hbox {MLW}}{\nu _{\mathrm{o}}(0.5) - \nu _{\mathrm{o}}(17.5)} = 1.55 (+\,0.10, -\,0.06). \end{aligned}$$

(26)

8.3 Two-line thermometry results

Additional analysis was performed to determine the most temperature-sensitive transitions in this experiment. Instead of using all three transition pairs, an analysis was done using three combinations of two spectral lines (noted in previous studies as two-line thermometry, (Palma et al. 2003; Sánchez-González et al. 2012)). Each combination is given a subscript, shown in Table 4. The two-line temperature profiles are shown in Fig. 18. The B and C combinations used with SPT agree well with CFD temperature. This is due to the $J^{\prime \prime }$ = 6.5, 17.5 and $J^{\prime \prime }$ = 0.5, 17.5 thermometry pairs having the most temperature sensitivity in their associated Boltzmann fractions, shown in Fig. 19. The relative magnitude of the Boltzmann fraction for the B and C transitions changes more dramatically with temperature than the A transitions, resulting in more accurate thermometry. However, near the BL edge, the A combination agrees better with CFD due to higher sensitivity at lower temperatures.

Fig. 18

Two-line thermometry comparison between A, B and C line pairs

Fig. 19

Boltzmann fraction over a range of temperatures observed in the hypersonic boundary layer

9 Discussion

Eighteen NO-PLIF thermometry techniques were applied to a hypersonic BL experiment on a wedge in a Mach 10 flow.

The SPT-based techniques were in closest agreement with the CFD predictions. This may be due to absorption and saturation effects canceling at first order in the SPT technique. Other techniques, such as DBT, do not cancel to first order (both absorption and saturation act to broaden the transition leading to significant error). Since these effects cancel to first order in the SPT technique, corrections do not necessarily improve performance at certain absorption and saturation levels.

Deviations in the measured temperature field occurred at the surface. This disagreement has been seen before in near-surface measurements of temperature using PLIF (Yoo et al. 2010; O’Byrne 2002). It is possible that local laser heating as seen in O’Byrne (2002) artificially increases the surface temperature. Also, minor model motion (0.25 mm upward drift) during the experiment has large effects on the spectra at the surface, which could also account for the deviation. Further work is required to reduce laser heating effects to allow for accurate NO-PLIF temperature and heat flux measurements. Recently, initial results in the literature have been presented that use the integrated temperature profile to measure heat flux, which is then used to correct the near-surface temperature gradient (McDougall et al. 2018b). These methods, once refined, could correct many of the common near-surface temperature discrepancies observed in PLIF thermometry experiments.

The absorption correction methods used in this study did not have a large effect on thermometry techniques. This may indicate that absorption does not affect thermometry to a high degree using the chosen transitions in this flow in which only a thin layer of NO is probed at low pressure. This result is useful for future design of seeded BL PLIF experiments, as the result indicates that absorption effects are negligible if transitions with similar Einstein absorption $B_{12}$ coefficients are chosen. Alternatively, the CFD mole fraction profile used for the absorption correction may not be accurate. Analysis of the accuracy of CFD diffusion modeling has been completed by McDougall et al. (2018b), but further work is required to determine if the mole fraction field needs correction. Additionally, a more intricate absorption model may have performed better if the laser lineshape was significantly deformed along the laser path (Danehy and O’Byrne 1999).

Previous studies have used a multi-line-based thermometry technique similar to FST to determine temperature (Bessler and Schulz 2004; Denisov et al. 2014). In those studies, the laser was tuned to a lower energy to avoid saturation effects. However, spatial laser non-uniformities, such as a not-perfectly collimated laser sheet or local hot spots can cause an unpredictable laser energy distribution. The AC method had moderate success correcting this effect, while overcorrecting near the wedge surface.

The under-sampled thermometry results identified distinct minimum spectral resolutions and spectral laser widths to extract meaningful temperature information. As an alternative to a simple linear scan completed in the current study, future LIF thermometry experiments could be completed by collecting a small cluster of samples around the peak of each transition. This method would reduce the number of unusable laser pulses (between spectral lines), increase the SNR at the transition peak, and minimize the effect of pulse-to-pulse laser fluctuations. The minimum spectral resolutions identified above would still hold for this strategy, and could be used as a guideline to ensure resolving the peak of the transition. It should be noted that in flows with large velocity gradients, clustered sampling around spectral peaks would be difficult due to the Doppler shifting of spectral lines. Therefore, some knowledge of the velocity field is needed for this type of analysis. If the Doppler shift is small enough to resolve the peaks with clustered sampling, the temperature and velocity could be measured in a single experimental run, while reducing the number of required laser pulses. This technique would benefit experiments completed at facilities with restricted run times, as the length of the spectral scan could be greatly reduced with the use of clustered sampling.

The two-line thermometry analysis points to the importance of transition selection in NO-PLIF thermometry, thoroughly documented by previous authors (Palma et al. 2003; Bessler and Schulz 2004; Sánchez-González et al. 2012).

The SPT technique achieved 12.6% average agreement with CFD for the majority of the boundary layer. This level of agreement demonstrates how CFD can be used to evaluate new experimental diagnostic techniques if tested and simulated in a simple flow.

10 Conclusion

High spectral resolution PLIF thermometry was applied for the first time in the NASA Langley Research Center’s 31-in Mach 10 facility. The best performing thermometry technique (Spectral Peak Thermometry—SPT) achieved agreement of 31.6 K (relative error of 12.6%) with computational fluids dynamics simulations and was largely within the identified experimental error bounds. The SPT technique also proved to be the most robust, fitting the spectra the closest based on the sum of the residuals. While the analysis shows that saturation and absorption are expected in this experiment, overall closer agreement with CFD was not achieved by implementing appropriate corrections. This result shows that experimental design considerations, such as selecting transitions that saturate and absorb similarly, are an effective method to prevent these non-linear effects from propagating large errors into the temperature measurement. It was found that in regions of the flow where saturation was expected to be significant (e.g., edge of BL), appropriate corrections improved measurement agreement to CFD. However, these corrections resulted in worse agreement with CFD in other regions of the flow (e.g., near the wall) where minor saturation was expected to occur. Unless the laser is accurately characterized and the mole fraction distribution is known, correcting minor absorption and saturation effects is highly difficult. Therefore, it is recommended that quantification of minor absorption and saturation effects be used to estimate measurement uncertainty for PLIF thermometry instead of forming the basis for a correction. The undersampled thermometry results showed a distinct minimum spectral resolution for methods 1 and 2 (0.102 ${\mathrm {cm}}^{-1}$ and 0.074 ${\mathrm {cm}}^{-1}$, respectively) that is required to apply the thermometry techniques outlined in this paper. The maximum laser linewidth to perform thermometry using the chosen transitions was determined to be 0.49 ${\mathrm {cm}}^{-1}$. Laser linewidths greater than this threshold would provide erroneous measurements. Comparable accuracy for SPT could be achieved with only two transition pairs (as opposed to the default 3 transition pairs), as long as there was a large relative difference between the transition’s energy states. For this experiment, the $J''=17.5$ transition pair needed to be included for an accurate measurement. The level of agreement between the best performing technique, SPT, and CFD temperature results indicates that CFD can be used to evaluate new experimental diagnostic techniques, given that the flow is simple. Overall, this work utilizes a simplified LIF model to obtain quantitative temperature using a straight-forward NLLS fitting method, and outlines the required parameters to apply the best thermometry techniques effectively.