1 Introduction

The turbulent dynamics of mixing regions in high-speed flows lack a complete description as physical probes interfere with the flow and numerical predictions cannot be verified without reliable experimental comparisons. To provide a better understanding of the turbulent flow in these regions, nonintrusive, optical techniques have been applied to acquire spatially and temporally resolved thermodynamic property measurements. Several techniques such as laser-induced fluorescence (LIF) (DiRosa et al. 1993), Rayleigh scattering (Panda and Seasholtz 1999), filtered Rayleigh scattering (FRS) (Elliott and Samimy 1996; Forkey et al. 1996), and coherent anti-Stokes Raman scattering (CARS) (Woodmansee et al. 2000, 2004) have been successful in obtaining accurate average properties but are typically incapable of capturing the rapid property fluctuations indicative of the turbulent mechanisms governing the flow structure. Although some of these techniques have shown promise for measuring fluctuations about average values, only two studies have acquired fluctuation quantities in high-speed flow. Panda and Seasholtz presented mean and fluctuating density measurements in underexpanded jets using Rayleigh scattering (Panda and Seasholtz 1999). The density fluctuation measurements were possible because of the low system noise associated with the technique. However, Rayleigh scattering is dependent on input laser power and is susceptible to interference scattering, making it difficult to apply in enclosed flows. Gross et al. (1987) employed LIF of nitric oxide to acquire density and temperature fluctuations in a supersonic, turbulent boundary layer, extracting pressure fluctuations through an equation of state. LIF is not impaired by interference scattering; however, it often requires seeding the flow.

Electrostrictive laser-induced gratings (LIG) is capable of making spatially and temporally resolved measurements and is more readily applied than some alternative techniques. The primary objective of this study was to demonstrate the ability of electrostrictive LIG to make mean and fluctuating temperature measurements in a high-speed flow. The technique does not require flow seeding and is not significantly dependent on input laser power. Because of the coherent nature of the signal, electrostrictive LIG is not as severely affected by interference scattering, in which light from the input beams registers during signal detection. In addition, electrostrictive LIG does not employ a molecular filter as used in FRS nor does it require the species-specific dye laser wavelengths common to CARS.

Electrostrictive LIG is a nonresonant, nonintrusive technique that can be employed to probe unseeded air, typically providing the local speed of sound, thermal diffusivity, temperature, or velocity. Several groups demonstrated early success with LIG measurements in gases (Govoni et al. 1993; Hemmerling and Stampanoni-Panariello 1993; Cummings 1994). The technique involves two pump beams of the same frequency that are spatially and temporally overlapped at a probe volume. For the nonresonant or electrostrictive case, the gas is not excited by the pump beams through absorption or dissociation. The interference pattern in intensity that results from the overlap of the pump beams causes a density gradient to form in the gas due to electrostriction, as the air molecules are drawn toward the bright regions of the interference pattern. This fluid motion forms two counterpropagating sound waves and an isobaric density wave at the probe volume (Hubschmid et al. 1995). The results of the motion modulate the index of refraction and induce a material grating (Boyd 1992). A probe beam introduced at the Bragg angle is diffracted from the grating and forms a coherent signal. The intensity of the signal is modulated in time at twice the Brillouin or sound frequency. The signal strength decays at a rate primarily determined by the thermal diffusivity and viscosity, which govern dissipative processes, and the finite size of the probe volume, which affects the rate at which sound waves leave the probe volume (Stampanoni-Panariello et al. 2005). For slow dissipation, the sound frequency is given in good approximation by

$$ \omega=qc_{\rm s} $$
(1)

where q is the magnitude of the grating vector and c s is the local speed of sound (Hubschmid et al. 1996). The grating vector depends solely on the probe-volume geometry and input laser wavelengths. If these experimental parameters and the gas composition are known, the temperature can be extracted from the sound frequency through its relation to the speed of sound. This calculation is simplified if a reference signal is acquired with a fixed experimental setup at a known temperature. The dependence on the grating vector and gas composition can then be eliminated by taking the ratio of the sound frequency of the reference condition to that measured in the experiment, a method used in other LIG flowfield investigations (Barker et al. 1999; Hart et al. 1999). Following the dependence of the speed of sound on temperature, the equation for temperature becomes

$$ T=T_{\rm ref}\left(\frac{\omega} {\omega_{\rm ref}}\right)^2 $$
(2)

where T is the temperature at the measurement location, T ref is the temperature of the reference condition, ω is the sound frequency acquired during the experiment, and ωref is the sound frequency measured for the reference condition.

Measurements have been obtained by applying electrostrictive LIG to heated (Hemmerling et al. 2000; Neracher and Hubschmid 2004; Kozlov 2005) and unheated subsonic gas flows (Kozlov et al. 2000; Hart et al. 2001, 2002); however, only two investigations, Hemmerling et al. (2002) and Schlamp et al. (2005), have applied electrostrictive LIG to supersonic flow. Hemmerling et al. (2002) measured velocity in a recirculation region of a supersonic flow where the local Mach number was subsonic but were unable to acquire measurements in a supersonic region. Schlamp et al. (2005) studied the effects of a pre-heated supersonic jet flow on electrostrictive LIG signals. Their measurements confirm the difficulty of applying electrostrictive LIG to such flows, as turbulent and convective effects hindered signal acquisition. The only other known applications of LIG to supersonic flow are two studies that employed resonant or thermal gratings, which markedly increases signal strength by employing flow seeding (Schlamp and Allen-Bradley 2000) or deep UV laser radiation (Barker et al. 1999). Therefore, the current study presents the first known findings in a supersonic flowfield investigated with nonresonant, electrostrictive LIG, most notably within the compressible shear layer of an underexpanded jet.

The major disadvantage in applying electrostrictive LIG to high-speed flow is the temporal duration of the signal. While the grating generation occurs during the pump-beam pulse (on the order of 10 ns), the diffracted signal must be detected over a relatively longer period (on the order of 500 ns). This is significantly longer than techniques such as CARS, for which the signal duration is on the order of the pump beam-pulse width. Therefore, the detected signal can be altered by the bulk flow velocity, which convects the density gradient downstream, and turbulent mixing, which disrupts the pattern of the grating. Two features of the current experimental setup allowed for a signal to be acquired in the underexpanded jet despite these challenges. First, the close spacing of the input laser beams and the associated narrow-crossing angle between the pump beams reduce the effects of beam steering as all input beams are steered by similar portions of the flowfield (Schlamp et al. 2005). The decrease in crossing angle also increases the signal duration. Second, for the underexpanded jet experiments, the probe beam was slowly expanding before being focused at the probe volume, rather than entering collimated, so that the focal spot diameter at the probe volume was roughly twice the focal spot diameter of the pump beams. This allows the grating formed by the pump beams to convect downstream at the bulk velocity yet still be spatially coincident with the probe beam. This is similar to the technique used by Schlamp and Allen-Bradley, in which the probe beam was displaced downstream a prescribed distance from the probe volume so that the grating formed upstream passes through the focal point of the probe beam (Schlamp and Allen-Bradley 2000).

As the probe volume traverses the compressible shear layer, the detection of a signal is further complicated by the components of fluid velocity associated with turbulent mixing. The mixing disrupts the regular pattern and results in an increased decay rate observed within the shear-layer data. However, a discernible signal was acquired at all locations probed within the flowfield. Even with the bulk velocity and turbulent mixing affecting the grating, the signal obtained and temperature extracted are indicative of the properties at the probe volume. The grating is generated during the short laser pulse of the pump beams and then convected downstream with the initial grating spacing unchanged (Schlamp and Allen-Bradley 2000). If the signal can be acquired after it leaves the probe-volume location, the temperature at the initial measurement volume will be extracted. Thus, the lengthy signal duration should not imply that the technique does not measure the instantaneous temperature at the probe volume, only that it increases the complexity of signal detection.

To investigate the performance of electrostrictive LIG at conditions typically encountered within high-speed flows, data are obtained under static conditions in an optically accessible pressure vessel over the subatmospheric pressure range of 0.1–0.9 atm. The uncertainty of the static measurements provides a baseline system noise level for comparison with the fluctuation levels obtained in the flowfield. This experiment was followed by temperature measurements along the centerline and over two radial traverses in an unheated, underexpanded air jet at a fully expanded jet Mach number of 1.19, which demonstrate the successful application of the technique to high-speed flow. The temperature fluctuation measurements presented herein represent the first of their kind in the compressible shear layer of an underexpanded jet.

2 Experimental setup

Figure 1a presents a schematic of the electrostrictive LIG experimental setup. A similar arrangement was employed for the pressure-vessel experiment. A frequency-doubled Nd:YAG laser (Spectra-Physics PIV 200-10) supplies the 532-nm pump beams. The pulse energy in the pump beams is controlled by a half-wave plate and a polarizing beamsplitter cube. A beamsplitter is used to separate the two pump beams with approximately equal pulse energies. To account for the beam path difference and to ensure temporal overlap at the probe volume, one of the pump beams is directed through an adjustable delay line. Once a signal is acquired, the difference in beam paths is minimized to optimize the signal strength (Stampanoni-Panariello et al. 2005). The continuous-wave probe beam is supplied by an argon ion laser (Spectra-Physics Stabilite 2017). The 488-nm line is selected to provide spectral separation between the pump and the signal beam to reduce interference scattering from the pump beams. The diameter of the probe beam is expanded through a combination of lenses to match the diameter of the pump beams, increasing the overlap at the probe volume. The pump and probe beams are focused at the probe volume by a 500-mm focal-length lens. Due to the small frequency difference between the pump and probe beams, the pump-beam-crossing angle and the Bragg angle are very similar. Therefore, the probe beam is brought into the probe volume above one of the pump beams. This choice not only facilitates the alignment of the probe beam, but also allows for spatial separation of the signal beam from the pump beams. The pump, probe, and signal beams are recollimated by another 500-mm focal-length lens. The narrow probe-volume spacing as viewed on the collimating lens is displayed in Fig. 1b. The signal beam is steered toward the detector by a series of prisms while the pump and probe beams are directed into beam dumps.

Fig. 1
figure 1

Schematic of a the electrostrictive LIG experimental setup and b the probe-volume beam spacing as viewed on the collimating lens with predicted signal beam location indicated

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The signal beam is passed through two spatial filters separated by a 100-mm focal-length lens that focuses the signal onto the detector aperture. To further reduce the interference from scattered pump beam light, a 488-nm high-rejection laser line filter is placed just before the detector aperture. A photomultiplier tube (Hamamatsu R1516) is used to detect the signal, and a digital storage oscilloscope (LeCroy WavePro 7200) with a 2-GHz bandwidth and 20-GS/s sample rate recorded the time-series data.

During the subatmospheric pressure-vessel study, pump-beam pulse energies varied between 7 and 26 mJ, and the probe power was held constant at 620 mW. For the underexpanded jet measurements, several pulse energies were employed over a similar span for the pump beams, and the probe beam power ranged between 120 and 700 mW, with a typical power of 350 mW. The reduction in probe beam power compensated for the increase in Rayleigh scattered light from the jet due to the higher density and, at times, interference scattering from the nozzle lip. The total crossing angle between the pump beams was 2.177°, resulting in a Bragg angle of 0.998°. The probe-volume width and length are experimentally estimated as 90 μm and 2.2 mm, respectively. For the underexpanded jet experiments, the focal spot diameter of the probe beam is slightly larger than the pump beams, approximately 200 μm, which is adjusted by the lens system that initially expands the probe beam.

The reference condition for the underexpanded jet measurements is determined before and after the jet is run. The LIG signal is acquired in room air, and the reference temperature determined by the same thermocouple that is used to monitor the stagnation temperature of the jet. Once a reference signal is acquired, the experimental setup is not adjusted so as not to affect the probe-volume geometry and the reference sound frequency. After the jet is shut off, another reference signal is acquired to verify that the reference sound frequency did not change during the data acquisition in the jet. The nozzle pressure ratio for the jet was 2.39, resulting in a fully expanded jet Mach number of 1.19. The air for the jet is provided by compressed air bottles that feed into a manifold with the stagnation pressure controlled by a series of regulators. The nozzle linearly contracts the flow from an entrance diameter of 19.3 mm to an exit diameter of 7.5 mm.

3 Results

Figure 2 displays the ensemble-averaged and single-shot time traces of the electrostrictive LIG signal along with sound frequency histograms obtained in the pressure vessel over a pressure range of 0.1–0.9 atm. Each ensemble contains approximately 1,000 single-shot data traces. The data are compared to the mathematical model developed by Cummings et al. (1995). The analytical results are similar to that presented in the review of Stampanoni-Panariello et al. (2005), with both analyses resulting in an electrostrictive LIG signal that is modulated at twice the sound frequency. The predictions based on the work of Cummings et al. (1995) are used for comparison here as they include a consideration of the finite size of the probe volume. This consideration modifies the decay rate of the LIG signal in a manner analogous to that observed in the pressure-vessel data. The full version of the Cummings et al. (1995) model, contained in an appendix, includes a convolution of the temporal evolution of the LIG signal with the pump-beam pulse duration, approximated here as a Gaussian with a full width at half maximum of 10 ns. As seen in Fig. 2a–c, there is good agreement between the ensemble-averaged data and theory, including sound frequency, decay rate, and intensity distribution. Even at a pressure of 0.1 atm, three peaks are detected and used to determine the sound frequency. Increased noise is evident in the single-shot data in Fig. 2d–f; however, the modulation of the electrostrictive LIG signal intensity is still discernible even at 0.1 atm, and there is general agreement between single-shot data and theory. The peaks occurring after the signal has decayed, most notably in Fig. 2d after 100 ns, result from the detection of single photons that are Rayleigh scattered from the probe volume. As described in the following paragraphs, these peaks are removed during the frequency analysis to eliminate confusion with signal peaks.

Fig. 2
figure 2

a–c Comparison of ensemble-averaged experimental and theoretical time traces, d–f single-shot experimental and theoretical time traces, and g–i sound frequency histograms for pressure levels of 0.1 atm (a, d, g), 0.5 atm (b, e, h), and 0.9 atm (c, f, i) in the pressure vessel

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As seen in Fig. 2, there is a decline in signal intensity with decreasing pressure, which is expected from the quadratic dependence of signal intensity on density (Govoni et al. 1993; Cummings et al. 1995). This effect is partially mitigated during the experiment by increasing the pump-beam pulse energy with decreasing pressure. The pressure dependence of the integrated intensity of the single-shot data is shown in Fig. 3. Each point represents the ensemble average of temporally integrated intensity of the single-shot time traces normalized by the pulse energy of the pump beams. A quadratic fit demonstrates the expected dependence of the signal intensity on pressure. The uncertainty bars correspond to one standard deviation of integrated intensity for the ensemble and are approximately 10% of the average over the range investigated. The decline in intensity is not a concern for the underexpanded jet experiment, as the density will not drop to similar levels.

Fig. 3
figure 3

Pressure dependence of the integrated signal intensity

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To determine the capabilities of the technique in acquiring instantaneous temperature measurements, a frequency analysis was applied to the single-shot data to estimate the sound frequency. A series of data-grooming techniques was applied to the time traces before calculating a power spectral density (PSD) function. To remove the high-frequency noise, a 20-pole bandpass filter was applied between 30 and 160 MHz (corresponding to a sound frequency range of 94–503 MHz). The time trace was then trimmed based on the ensemble-averaged time trace to eliminate Rayleigh scattered noise occurring after the signal decay. For example, the time trace acquired at 0.1 atm (Fig. 2d) was trimmed after 100 ns. The original time trace was acquired at 20 GS/s for 10,000 samples, which would provide a frequency resolution in the PSD before trimming of 2 MHz (6.3 MHz in sound frequency). To enhance this resolution, the time trace was zero padded before and after the retained portion, extending the series to 65,536 data points. Padding artificially extends the beginning and end of the time trace; however, because the signal begins from and decays to zero, the same effect could be accomplished during the data acquisition procedure, significantly lengthening acquisition and processing. The padding technique used herein has been successfully applied in the past in processing electrostrictive LIG signals (Neracher and Hubschmid 2004) and reduces the PSD resolution to 305 kHz (960 kHz in sound frequency). Finally, the mean of the data trace is set to zero, a Hanning window is applied, and a PSD is calculated. Any individual PSD without a peak between 20 and 60 MHz (sound frequency range of 63–188 MHz) is removed from the ensemble. The dominant peak within this range is taken as the sound frequency for each single-shot time trace.

The histograms displayed in Fig. 2g–i represent the results of the frequency analysis for the single-shot pressure-vessel data at 0.1, 0.5 and 0.9 atm. The narrow distribution in Fig. 2h and i is indicative of low system noise. As the pressure decreases, the distribution widens and uncertainty increases (Fig. 2g); however, similar to the integrated intensity results, the conditions experienced in the underexpanded jet are not expected to reach a density level this low. The uncertainty of the frequency measurements is estimated by one standard deviation in frequency for each ensemble. The trend in frequency uncertainty is displayed in Fig. 4 over the entire pressure range examined. For pressure levels above 0.2 atm, the uncertainty is below 5 MHz, corresponding to an average uncertainty of approximately 2.2%. The uncertainty in temperature is proportional to twice the uncertainty in frequency; hence, the baseline detectable normalized temperature fluctuation is estimated to be 4.4% over the pressure range of 0.2–0.9 atm. This uncertainty compares well with previous CARS (Beyrau et al. 2004; Vestin et al. 2006); LIF (Gross et al. 1987), and LIG (Hart et al. 1999) studies at room temperature.

Fig. 4
figure 4

Uncertainty in sound frequency

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Figure 5 presents the ensemble-averaged and single-shot time traces from three positions along a radial traverse in the underexpanded jet at a downstream location of z/d = 2.5 along with the temperature histograms calculated at each location. Each ensemble represents approximately 500 single-shot time traces. These three locations are chosen so that one is inside the jet core at r/d = 0.2 (Fig. 5a, d, g), one is near the middle of the shear layer at r/d = 0.6 (Fig. 5b, e, h), and one is outside of the shear layer in the ambient air at r/d = 1.0 (Fig. 5c, f, i). As with the pressure-vessel data, the single-shot flowfield data include increased levels of high-frequency noise and Rayleigh-scattered photon signals toward the end of the time traces. The trimming process was effective in eliminating the Rayleigh scattered light; for example, the trace in Fig. 5e was trimmed after 160 ns, leaving six intensity peaks for the frequency analysis.

Fig. 5
figure 5

a–c Ensemble-averaged experimental time traces, d–f single-shot experimental time traces, and g–i temperature histograms acquired at a downstream location of z/d = 2.5 and radial locations of r/d = 0.2 (a, d, g), 0.6 (b, e, h), and 1.0 (c, f, i)

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The contrast between thermodynamic and kinematic conditions for the three locations is evident in Fig. 5 for both the ensemble-averaged and the single-shot time traces. The rapid signal decay seen in Fig. 5b and e in the shear layer is faster than that would be predicted, as the density and decay rate at this location should be between the conditions measured in the jet core (Fig. 5a, d) and the ambient air (Fig. 5c, f). The increased decay rate seen in the shear layer is attributed to the increased shear experienced by the density grating formed at the probe volume, which works to disrupt the pattern. The grating vector is aligned with the downstream flow direction, so the cross-stream velocity that develops in the shear layer distorts the density grating. Although the disruption reduces the signal duration, the time traces in the shear layer still maintain enough peaks for frequency and temperature to be extracted. The temperature histograms in Fig. 5g–i display results typical of those found in the jet: narrow distributions centered around the average temperature in the jet core and ambient air (Fig. 5g, i), and wide distributions in the shear layer (Fig. 5h). Of note, the temperature histogram for the shear-layer location, Fig. 5h, displays a small but significant number of realizations at the core and ambient temperatures, with the majority of the measurements occurring near the mean temperature. This result implies that the shear layer is not always a uniform mixture of core and ambient fluid, but at times can be composed of small pockets of unmixed core or ambient fluid.

Figure 6a–c displays the ensemble-averaged temperature distribution along the centerline and over two radial traverses through the shear layer of the underexpanded jet. The centerline measurements characterize the oscillations in mean temperature that occur because of the successive crossings of the intercepting shock waves and expansion fans that form the shock cells. The spacing of the shock cells is in good agreement with the density results of Panda and Seasholtz (1999) for similar operating conditions. The relative magnitude of the temperature oscillations and the gradual rise in minimum shock-cell temperature with axial distance are also verified (Panda and Seasholtz 1999).

Fig. 6
figure 6

a–c Ensemble-averaged temperature and d–f rms temperature measured along one axial and two radial traverses in the underexpanded jet

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Characterization of the shear layer is provided by the two radial traverses at downstream distances of z/d = 0.4 and 2.5. The upstream location represents the nearest position to the nozzle lip that could be obtained without significant scatter from the nozzle itself. The downstream traverse was chosen to coincide with the location of peak density fluctuations measured by Panda and Seasholtz (1999). As can be seen in both traverses, the temperature is at a minimum in the jet core before increasing through the shear layer and then reaching equilibrium in the ambient air. The decrease in temperature across the jet core seen in Fig. 6c between a radial location of r/d = 0.0 and 0.3 is consistent with the average density trend obtained in a radial trace at a similar location reported by Panda and Seasholtz (1999). The width of the shear layer grows noticeably between the two downstream locations from a normalized width of approximately w/d = 0.1 to 0.3.

The spatial resolution of the experimental setup is demonstrated by the large temperature gradient captured in the radial traverses, especially across the shear layer in the upstream location. Similar to the procedure employed for planar imaging techniques, the relative resolution can be calculated to provide a quantitative perspective on the spatial resolution of the LIG setup (Clemens and Mungal 1995). The relative resolution is defined as the ratio of the spatial resolution of the experimental technique to the Batchelor scale. The Batchelor scale for the downstream radial trace is approximately 4.7 × 10−7 m. This provides a relative resolution of approximately 200 based on the width of the probe volume and 4,700 based on the length of the probe volume. The value for the width of the probe volume is comparable with other imaging studies in similar flows (Clemens and Mungal 1995). The narrow width of the probe volume is the reason for aligning the length of the probe volume primarily in the circumferential direction for which the property gradients are minimized compared with those in the axial and radial directions. The alignment is similar to that utilized in previous studies of underexpanded jets using Rayleigh scattering (Panda and Seasholtz 1999) and CARS (Woodmansee et al. 2004).

The rms temperature fluctuation along the centerline and through the two radial traverses is shown in Fig. 6d–f. As expected, the fluctuation magnitude along the centerline is low and nearly constant at 6.5 K, resulting in a normalized temperature fluctuation magnitude of approximately 3.5%. This value compares well with the baseline detectable limit predicted from the pressure-vessel experiment. The small oscillation in fluctuation magnitude at the first few centerline positions is indicative of the movement of the first intercepting shock in this region. The temperature fluctuations along the radial traverses (Fig. 6e, f) are similar in magnitude to the centerline traverse for the portions of the traces in the jet core and ambient air. The fluctuation magnitude peaks sharply in the shear layer for both radial traverses, demonstrating the ability of the technique to detect the rapid and significant changes in temperature that occur in this region. The large peak in fluctuation magnitude at the upstream radial traverse (Fig. 6e) is possibly amplified by the low signal-to-noise ratio that occurred for the outer positions of this traverse. Evidence of this effect is seen in the rise in fluctuation magnitude in the ambient air outside the shear layer, which is roughly a factor of 1.5 larger than that of other ambient-air locations in the downstream radial traverse. Because the probe volume is very close to the nozzle lip, probe beam light is scattered into the detector. Even with the spatial filters employed, it was difficult to eliminate this effect without reducing the input laser power to the probe volume and, correspondingly, the signal-to-noise ratio. However, the qualitative description of fluctuation magnitude across the upstream traverse is considered representative.

The magnitude of the temperature fluctuations at the downstream radial traverse is considered more reliable, as the interference scattering does not pose a problem at these locations. Therefore, the results of the downstream radial traverse will be the focus of the comparison with previous investigations, although the upstream traverse provides similar meaningful quantitative comparison as well. The normalized fluctuation magnitude of the downstream radial traverse is confirmed by comparisons with two other studies (Panda and Seasholtz 1999; Freund et al. 2000). Panda and Seasholtz (1999) reported density fluctuation measurements in the shear layer using a phase-averaged technique. Because of this, they report the density fluctuations that exist in addition to the recurring density oscillations caused by a standing pressure wave in the shear layer. To provide a comparison, the phase-averaged density oscillation and rms density fluctuation need to be combined, as no phase averaging was performed in the current study. In addition, Panda and Seasholtz (1999) report their results normalized by the jet core density, which must be taken into account. The peak-to-trough phase-averaged density oscillation normalized by the average density at the position under consideration is approximately 0.21, and the similarly normalized density fluctuation on top of this oscillation is 0.08. These two quantities are combined to arrive at a rough estimate for a total possible normalized density fluctuation of 0.29 if phase averaging had not been employed. Based on the data in Fig. 6, the maximum temperature fluctuation normalized by the mean temperature is 0.24 at a location of r/d = 0.5 and z/d = 2.5. Therefore, the relative magnitude of the maximum temperature fluctuation agrees with the approximated value for density from Panda and Seasholtz (1999).

Another comparison can be made with the direct numerical simulation results of Freund et al. (2000) in which they simulate annular mixing layers. Because of the differences between the two types of flows investigated, the convective Mach number is used to evaluate which operating conditions from the simulations are similar. For the underexpanded jet, the convective Mach number is 0.55, which is approximated best by the results for a convective Mach number of 0.59 studied by Freund et al. (2000). For this case, they predict a radially integrated normalized temperature fluctuation of 0.05, which compares well with a value of 0.068 acquired for the downstream radial traverse. In a separate publication, Freund reports maximum normalized pressure and density fluctuations of 0.2 under similar conditions (Freund 1997). This value is similar to the estimate of 0.29 from Panda and Seasholtz for density (1999) and that of 0.24 for temperature in this study. The comparisons with these two studies help to confirm the relative magnitude of the temperature fluctuations measured in the shear layer using electrostrictive LIG and lend credence to the capability of the technique for measuring temperature fluctuations in similar flows.

4 Conclusions

Electrostrictive LIG has been employed to acquire instantaneous temperature measurements in an underexpanded jet. The nonresonant technique is readily applied to supersonic flow, avoids some of the interference issues observed with Rayleigh scattering, and eliminates the seeding requirement typically found with fluorescent techniques. The ability of electrostrictive LIG to obtain temperature on an instantaneous basis was verified under static conditions at subatmospheric pressure, providing a baseline detectable temperature fluctuation of 4.4% for a pressure between 0.2 and 0.9 atm.

Mean and fluctuating temperatures were measured along the centerline and through two radial traverses in the underexpanded jet, representing an average temperature range of 160–300 K. Mean temperature measurements provided a description of the shock-cell structure and the shear-layer width. Temperature fluctuation levels in the jet core and ambient air agreed well with the predicted baseline detectable level from the pressure-vessel study. The temperature fluctuation measurements reached a maximum in the shear layer with a maximum normalized fluctuation magnitude of 0.24 that is supported by values from two other studies. The fluctuation measurements represent the first of their kind in a supersonic, compressible shear layer. Future experiments employing the technique will allow for the turbulent nature of the mixing regions to be investigated in more detail.