1. Introduction

High-repetition-rate, multiparameter measurement tools that are truly noninvasive are particularly relevant for use in high-speed wind tunnels. Quantitative measurements in these test environments are required to validate the computational fluid dynamics (CFD) tools that are used to extrapolate wind-tunnel data toward realistic flight conditions and hardware. The development of fast, time-resolved measurement capabilities that can readily be integrated into extreme conditions present under such test conditions is one of the several major technological challenges associated with the design, building, and operation of these complex test environments. Among the host of physical measurables important for the characterization and modeling of flow around test models in these facilities, the accurate mapping of velocity flow fields remains a significant yet essential challenge. In addition, spatially- and temporally resolved measurements of other flow parameters such as gas density, pressure, and temperature are of paramount importance to characterize the fluid dynamics.

This paper discusses high-speed Interferometric Rayleigh scattering (IRS) that can provide noninvasive velocimetry measurements in any gaseous environment of enough density without the need to seed foreign particles or gases into the flow field. In IRS, Rayleigh scattered light resulting from the interaction of a narrow linewidth laser beam with the flow of interest passes through a Fabry–Pérot interferometer or etalon and is imaged with a high-speed camera or other detector. This image is called an interferogram. The transmission properties of the etalon depend both on the wavelength of light and on the angle of incidence of light on the etalon. Therefore, the interferogram contains both spatial and frequency information, allowing the scattered light Doppler shift and line shape to be measured at multiple locations in the flow. The flow velocity can be found from the measured Doppler shift. Most applications of IRS have been in the measurement of velocity in the study of high-speed jets and wind tunnels [1–5] and in high-speed combustion flows [6]. Both nanosecond pulsed lasers [1,5–8] and narrow linewidth continuous-wave (CW) lasers [2–4] have been employed. In a recent CW laser implementation of IRS [4] light is sampled with a photodetector from one or a few discrete locations in the interferogram at up to 32 kHz rate, effectively time averaging the flow over ∼30 µsec. With a typical pulsed laser system, nearly instantaneous (tens or hundreds of nanoseconds long pulses) interferograms are acquired with a camera and Doppler shift is found by subsequent image processing.

A disadvantage of IRS is that Rayleigh scattering occurs at the same frequency as the incident laser (except for a small Doppler shift and line broadening), and therefore it is susceptible to interference from laser scattering off particles in the flow, wind tunnel walls, etc. The ns-pulsed IRS technique is less susceptible to scattered light interferences than other same-wavelength techniques such as PIV and PDV because localized interferences from particles can be removed through image analysis. Rayleigh scattering is a weak process and the etalon typically rejects 96% to 99% of the signal that would otherwise reach the sensor. Thus, high-energy laser sources are required for precise measurements. High measurement repetition rates and laser pulse lengths below 100 ns are required to temporally-resolve unsteady flow structures such as occurring in turbulent high-speed jets. While the commonly used frequency-doubled pulsed Nd:YAG laser has pulse widths of 5–10 ns, pulse repetition rates at this energy are only of the order of 10 Hz and the maximum usable pulse energy is limited by laser-induced gas breakdown.

The burst-mode laser is a technology developed over the last two decades that will enable transition of ns-pulsed IRS techniques from the laboratory to production wind tunnels [9–11]. The laser wavelength is controlled by a narrow linewidth (of order 1 MHz) CW diode seeding laser that is naturally stable during the experiment. Consequently, it has narrow linewidth of less than 200 MHz at 532 nm, comparable to the etalon linewidth and much less than typical Rayleigh scattering linewidth. The laser could run at repetition rates of up to 1 MHz since the beginning, but the pulse energy was very low at high rate due to insufficient amplification gain. With the current technology the amplification gain is increased, the design is modified for high-rate operation, and for the current laser more amplifiers have been added for high-energy operation at up to100 kHz or more. For IRS, it offers three major advantages relative to conventional ns pulsed lasers. It has high repetition rate over the duration of the burst allowing the temporal resolution of high-speed flow events. It can deliver high pulse energy and good signal-to-noise ratios at pulse repetition rates of 10–100 kHz in atmospheric pressure room temperature air, and at even higher rates at higher density (such as occurs in transonic pressure tunnels and cryogenic tunnels), allowing for the analysis of unsteady and turbulent flows. It has adjustable pulse length (up to 100 ns at 532 nm) so that substantially more energy can be delivered to the measurement volume while avoiding laser-induced breakdown. The laser linewidth is also narrower because the stretched pulse is nearly Fourier transform limit, which is also beneficial to the interferogram quality.

Previously, we have demonstrated IRS measurements taken at 10 kHz using a burst-mode laser and an intensified camera [12]. This paper improves upon these previous measurements in that the repetition rate is increased to 100 kHz. This high repetition rate is a critical feature in high-Reynolds-number flow facilities that have significantly shorter run times and require time-resolved velocity measurements to capture unsteady flow phenomena. Additionally, in contrast to the previous measurements, no intensifier is used in the current work. Since intensifiers are non-linear at high signal intensities, working without an intensifier has advantages if the signal levels are substantially greater than dark noise and the quantum efficiency of camera and intensifier are the same. Intensifiers also introduce intensifier-gain noise and reduce spatial resolution. Additionally, the camera will have a higher dynamic range than the intensifier, which is useful for large changes in gas densities such as across a flame front or a shock wave. Finally, intensifiers can be easily damaged by high light intensity produced by an illuminated particle or an object in the field of view. By comparison, CMOS or CCD sensors are not so easily damaged. Therefore, for IRS measurements in large wind tunnel facilities, un-intensified IRS will have wider application.

An advantage of Rayleigh scattering techniques is that Rayleigh scattering occurs for all gases and does not depend on any one absorbing species, temperature range, or wavelength of light. However, Rayleigh scattering line shape and intensity depend on gas composition, temperature, and pressure. If gas composition and pressure are known, as is often the case in a wind tunnel, then temperature can be found from line width. If not, for example in a flame, measurement of velocity by Doppler shift is still possible. A demonstration of IRS to measure temperature and density was in a Hencken burner flame and required a simultaneous measurement in a reference cell of known temperature and density to correct for unknown experimental effects. [8]

In this paper, we demonstrate for the first time simultaneous multi-point, multi-parameter flow measurement using the IRS technique at a 100-kHz repetition rate. Using a burst-mode laser and an un-intensified high-speed camera, flow velocities and flow temperatures are measured (without reference cell) in an under-expanded sonic jet. The results and uncertainties are described.

2. Experiment

A 3D rendering that illustrates the system components and layout of the IRS system is shown in Fig. 1(a) and a schematic that defines some of the variables is in Fig. 1(b). A burst-mode laser provides up to 1000 pulses in a 100 kHz-rate burst with nominal 40 mJ/pulse at 532 nm. Pulses from the laser are focused to form a beam waist of from 60 to 120 µm in diameter in the measurement region via a 300 mm focal length single element focusing lens. Rayleigh-scattered light from the beam-waist is collected using a 100 mm focal length single element lens on a 3-component translation stage. Collected light is collimated by this lens and passes through a free spectral range (FSR) 9.99 GHz fixed-gap air-spaced etalon, which has a measured mirror reflectivity 89.8% at 532 nm for both surfaces and a 30 mm clear aperture. This type of etalon consists of two flat mirrored glass plates rigidly separated by pillars made of a glass that has a very low coefficient of thermal expansion and so, unlike a solid etalon, is not strongly susceptible to the effects of temperature. With an FSR of 10 GHz an IRS system can, in principle, measure without ambiguity velocities of up to 3761 m/s (producing a Doppler shift of 10 GHz) and temperatures of up to 2223 K (producing a Doppler line width of about 5 GHz). (The selection of a 5 GHz etalon would have been more optimum for this experiment.) The IRS measurement region through which the focused laser beam passes is imaged via a 300 mm AFS-Nikkor Nikon lens onto a Phantom Fastcam SA-Z high-speed camera. The camera has a 384×384-pixel field-of-view, which is reduced from the full resolution to allow the camera to operate at 100 kHz. The magnification is thus ×3 and the field of view corresponds 2.56 mm × 2.56 mm in the flow.

 

Fig. 1. Schematic of IRS system for focused-laser-beam interrogation.

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A white scattering plate is placed behind the laser beam and a small amount of laser light is directed towards it to provide a uniform, background, diffuse laser scattering source that creates reference rings in the interferograms. (This method of generating the reference rings is simpler than injection of laser light between the collecting lens and the etalon [7], but may not always be feasible.) In some cases, the scattering plate is removed to see clearly the IRS signal without the reference rings. The optical axis is defined as the straight line perpendicular to the etalon mirrors that passes through the optical center of the imaging lens. The imaging lens is set to bring the ring pattern formed by the background light into sharp focus on the sensor, which is equivalent to focusing the lens to image infinity. The center of the ring pattern is at the intersection of the optical axis with the sensor.

A jet flow is generated from a gas bottle via a convergent nozzle. The exit internal diameter is 2.48 mm, and the nozzle exterior is tapered at a shallow angle to a sharp edge at the exit. The nozzle axis is pointed obliquely toward the focus of the laser beam at 45° to the beam axis, see Fig. 1(b). The distance along the nozzle axis from the nozzle exit to the intersection with the laser beam is 14.7 mm. The supply stagnation pressure for the jet flow is 791 kPa. The flow is discharged from the nozzle as an under-expanded jet that transitions to supersonic near the exit, passes through a Mach disc at approximately 4.6 mm downstream, and approaches an approximately self-similar velocity profile as the centerline velocity decays [13,14]. Weaker shocks are reflected in the transonic flow downstream of the Mach disc. Rayleigh scattered light from the focus is collected at 90° to the laser beam by the IRS system. The configuration is such that the Doppler shift of the Rayleigh scattered light collected by the IRS system is caused by the component of velocity parallel to the nozzle axis (u₁). The nozzle and jet flow are scanned in a motion parallel to the laser beam to allow imaging of various positions in the jet. Surveys are only made along the one streamwise line through the jet. It is possible to survey other streamwise lines, but the primary intent is to demonstrate the technique. It is necessary in the analysis of the data to be able to specify the pressure to be 1 atmosphere (due to the dependence of the Rayleigh scattering line shape on pressure), which is not a reasonable assumption near the nozzle exit. The method of testing is to start the laser to provide repeated pulse bursts at approximately 10 s intervals, initiate the jet flow, arm the camera trigger, at which point the camera acquired data at the next laser pulse burst. The flow is then stopped, and data are downloaded from the camera to a laptop. This procedure is repeatable and multiple data sets are acquired at each location although usually only one is analyzed.

3. Interferogram analysis

The approach of the analysis is to represent the interferogram, or some defined sub-region of it, mathematically as a function of parameters of the laser light and the Rayleigh scattered light, the geometry of the focused laser beam, the distribution of background laser scattering, the etalon, and the imaging system. Some of these parameters are well known and some are found by fitting the interferogram. The residual is defined as the sum of the squared difference between the measured intensity at each pixel ${S_{i,j}}$ (i, j are the pixel coordinates in the horizontal and vertical directions on the face of the sensor) and the theoretical intensity ${F_{i,j}}$ (in sensor counts) over all the pixels of the interferogram or subregion. The unknown parameters are found by first providing an initial guess and then following the local slope of the residual in unknown-parameter space to the minimum.

The interferogram pixel value is calculated as the (incoherent) summation of three separate contributions. The first, ${P_{i,j}}$, is a smoothly varying polynomial function that represents the camera background level that would be observed if the laser were switched off. If the camera background is measured and subtracted from the experimental interferogram, ${P_{i,j}}$ is set to zero. The second, ${Q_{i,j}}{f_{i,j}}$, is the contribution to the interferogram of light from the scattering plate, and is the product of two terms, the intensity contribution to the image that would appear in the absence of the etalon, ${Q_{i,j}}$,, and the fraction that is transmitted by the etalon, ${f_{i,j}}$. The third, ${R_{i,j}}_{i,j}{g_{i,j}}$, is the contribution to the interferogram of Rayleigh scattering from the laser beam and is a product of three terms. The product ${R_{i,j}}_{i,j}$ is the intensity contribution to the image in the absence of the etalon, and ${g_{i,j}}$ is the fraction that is transmitted by the etalon. The laser beam shape function $_{i,j}$ has a unit-peak-level, is constant in the direction parallel to the beam, and is Voigt in the direction perpendicular to the beam; it depends on the height that the laser beam appears in the interferogram, ${y_{beam}}$, the angle the beam is inclined to the horizontal, $\alpha $, and the full-width half-maximum (FWHM) of the beam, h. The amplitude function ${R_{i,j}}$ is typically a constant or a function only of the distance parallel to the laser beam. Thus, the theory image on the camera in the absence of the etalon would be ${P_{i,j}} + {Q_{i,j}} + {R_{i,j}}_{i,j}$ while the theory interferogram is given by Eq. (1):

The transmission of light by the etalon is characterized by the FSR, the frequency difference for which the etalon line transmission repeats, and the mirror reflectivity, R₁. The ratio of this transmission to the maximum transmission as it appears on the sensor is given by Eq. (2).

The spectra of the laser beam and of the Rayleigh scattered light are now described. The laser beam is centered at the nominal laser wavelength ${\lambda _{laser}}$ with an assumed Gaussian shape and FWHM of $\Delta {\nu _{laser}}$. The light from the scattering plate has the same spectrum as the laser beam and contains no Doppler shift. The Rayleigh scattering for air at density less than atmospheric and a 90° viewing angle is thermally dominated, has a near-Gaussian line shape, and a frequency linewidth FWHM given by Eq. (4).

To derive the etalon transmission functions for the light from the scattering plate and the Rayleigh scattered light, ${f_{i,j}}$ and ${g_{i,j}}$, the etalon transmission given by Eq. (2) is multiplied by the laser light spectrum or Rayleigh scattering spectrum intensity and integrated with respect to wavelength, as given in Eqs. (6), (7). These evaluations are performed at each pixel of the image, i, j.

The procedure for fitting the experimental interferograms to theory using the methods described previously follows. The known camera, etalon, and laser parameters are specified: pixel size, R₁, FSR, $\lambda $. The laser line width $\Delta {\nu _{laser}}$ is specified if it is <200 MHz but can be fitted if greater. While the imaging lens focal length is known in principle, it is generally more accurate to fit for it. The focal length fits close to the expected value, consistent with uncertainties in fabrication and lens glass refractive index. A sequence of interferograms measured in atmospheric air, where Doppler shift is zero, and temperature and pressure are known, is fitted for C, ${f_2}$, h, w_blur, ${x_{center}}$, ${y_{center}}$, ${y_{beam}}$, $\alpha $. An initial guess of these parameters is made by trial and error. It is enough for convergence that the rings and Rayleigh scattering patches of the theoretical interferogram computed from these initial parameters overlap the corresponding features of the experimental interferogram. The values of ${f_2}$, w_blur, ${x_{center}}$, ${y_{center}}$, $\alpha $ thus obtained are averaged over the sequence and fixed in subsequent fitting. The fitted parameters for interferograms that include light from the scattering plate (with rings), are then C, T, $\Delta {\nu _{Doppler}}$, h, and ${y_{beam}}$. The average C obtained in a prior calibration using interferograms with rings is also fixed for fitting of interferograms without rings. Note that h and ${y_{beam}}$ are fitted for every interferogram in the sequence since they can change though a pulse burst due to thermal lensing within the laser. The fitting algorithm converges monotonically and rapidly for any well-posed set of fit variables. Examples of an ill-posed problem would include simultaneous fit within a small subregion of w_blur and $\Delta {\nu _{laser}}$, since these parameters both broaden the rings or, when there is no scattering plate (no rings), the simultaneous fit of C and $\Delta {\nu _{Doppler}}$, since these parameters both move the Rayleigh scattering patches radially. Convergence is terminated when the root-sum-square of the change in the normalized parameters in an iteration is less than 10⁻⁴. Typically, less than 1% of fits fail to converge to this degree.

4. Results and discussions

4.1 Interferogram fitting

Interferogram sequences from laser pulse bursts obtained in room air and in high-speed flows at a 100-kHz rate are shown in Fig. 2. Without the etalon, the image would show a uniform background from the scattering plate with, superimposed on top, a focused laser beam line that extends horizontally across the vertical center of the image. As explained previously, spatial and frequency-dependent transmission of the background by the etalon produces the reference rings, and transmission of the Rayleigh scattering from the laser beam generates a horizontal line of patches that are broadened and shifted relative to the rings. In Fig. 2(a), since the flow speed is zero, no Doppler shift is observed, and the transmitted Rayleigh-scattering patches are centered on the reference rings. The diameters of the laser rings do not change during the burst duration, indicating the stability of both the laser wavelength and the etalon. With high-speed flows, the Doppler shift effect is clearly observed in Fig. 2(b).

 

Fig. 2. Interferogram sequence at 100 kHz rate for (a) zero flow velocity, and (b) high-speed flow.

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Figure 3 shows four distinct interferograms of both flow and no-flow conditions from different sequences with and without the reference rings. Note that the Rayleigh scattering patches move out relative to the rings due to the Doppler shift. Since the laser wavelength is very stable over time, the interferograms can be analyzed without reference rings (once calibrated). This approach potentially provides slightly easier interferogram fitting due to one less fit parameter and more accurate velocity measurement (so long as the laser wavelength is constant) since the rings no longer interfere with the Rayleigh scattering patches. With the reference rings removed, it is easier to see that the effect of the jet flow is also to narrow and brighten the patches due to reduced line broadening and greater gas density at the lower gas temperature.

 

Fig. 3. Interferograms with reference rings (a) for zero flow velocity and (b) high-speed jet flow. Interferograms without reference rings (c) for zero flow velocity (d) and high-speed jet flow. The color scale is the same in all these images.

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Figure 4 shows a comparison of interferograms obtained with and without an intensifier on the camera. The intensified image is obtained with an intensifier gain of 55%. Although the un-intensified interferogram has lower signal intensities, signal counts that are ∼1/10 the amplitude of the intensified signal, the signal-to-noise-ratio (SNR) for the un-intensified case and the intensified case are similar (peak signal divided by background noise standard deviation ∼30:1) due to the read noise in the un-intensified case offsetting the intensifier-gain noise in the intensified image. Furthermore, as the laser energy is increased the SNR for the un-intensified case will improve more than it will for the intensified case due to reduced impact of the fixed read noise in the un-intensified case. Note that the un-intensified image is also in better focus, though both setups are optimized for focus, due to intensifier blurring above. Our interferogram fitting indicates 60-80 micron FWHM broadening associated with the intensifier as compared to the 20-micron pixel size.

 

Fig. 4. Comparison of 100-kHz interferograms with/without image intensifiers: (a) intensified, (b) un-intensified.

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The procedure for fitting the experimental interferograms to theory using the methods is described in section 3. Figure 5 shows the results of fitting 100 instantaneous interferograms obtained during a pulse-burst near the center of the jet and averaging the results. Parts (a) and (b) are averaged fitted theory interferograms while parts (c) and (d) are the averaged residuals after subtracting the theory interferogram from the experimental. Parts (a) and (c) are with rings while (b) and (d) are without. Because the conditions in the jet vary within the field of view, a series of automatically selected rectangular subregions of each interferogram centered around each beam-ring intersection are separately fitted and the individual subregion fits are combined to form a single fit interferogram. Note that the region of the experimental interferogram not obviously fitted in these images is not used in the fit and is only retained in these figures to provide context. The average fits shown are typical for the study of the jet flow in which the jet nozzle is translated to build up a picture of the flow field. In performing fitting with rings the fit phase parameter C changes by less than 2×10⁻⁴ during any given pulse burst. If this change is attributed entirely to laser frequency change, it amounts to less than 2 MHz, a very small amount. For a series of laser pulse-bursts occurring over a period of less than an hour, fitted C changed up to 0.012. If this change is attributed entirely to the laser frequency, it amounts to up to 120 MHz. But this change could also be explained by thermal or vibrational effects that slightly change the etalon mirror spacing or the location of the center of the rings on the sensor. The typical residual, defined as the average over all fitted pixels of the root-mean-squared difference between the fit interferogram and the experimental interferogram, is 23 counts/pixel, and in a typical fit sequence of 100 consecutive interferograms the standard deviation of this typical residual is 0.7 counts/pixel. Peak image signal level is 1100 to 1500 counts. There is no significant difference between the residual for fit subregions towards the edge of the interferogram and in the center.

 

Fig. 5. Examples of interferogram fitting in the high-speed jet: Fitting interferograms (a) with rings and (b) without rings. The residuals after subtracting the fit interferogram from the experimental (c) with rings and (d) without rings.

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Figure 6 shows a comparison of fits of measurement to theory for interferograms with the laser beam alone (no rings) at atmospheric (no flow) conditions to the Tenti S6 model for Rayleigh scattering and to the Gaussian model that is correct in the limit of low density. Also shown are the corresponding residual lines, defined as the fitted theory minus the measured interferogram. Note that the peaks in this plot are spaced by 10 GHz in optical frequency, which provides a reference by which spectral line width of the collected scattered light may be gauged. Both sets of fits are for the same 10-shot-sequence and the results are averaged. The plots are for an average of a 5-pixel high region of the interferogram centered on the centerline of the beam. The temperature for each subregion fit is averaged across the ten shots and then averaged across the subregions. For the Tenti S6 fit the average temperature is 259 K and for the Gaussian it is 301 K. However, the fit using the Gaussian line shape is poor compared to the Tenti S6 model, and this discrepancy is likely to be worse at higher density. This error in temperature will be discussed further in relation to the measurements in the jet flow.

 

Fig. 6. Fitting of the interferograms in room air with the Tenti S6 model.

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4.2 Flow analysis and uncertainties

Figure 7 shows the mean and the standard deviation of measured velocity and temperature inferred from a single 100-interferogram sequence with rings obtained in the atmospheric air, and plotted as a function of the location on the sensor around which each fit subregion is located projected onto the space of the flow field. The mean systematic error in temperature is the mean temperature minus the nominal expected value of 293 K while the error in velocity is the mean velocity itself since the nominal velocity is zero. The standard deviation provides a measure of the random error. Each point corresponds to a fitted subregion and its position is the center of the subregion. The errors in velocity are as little as ± 5 m/s at the center of the interferogram and as much as −30 m/s at the edge, where the spectral resolution is poorer. These errors compare favorably to the dynamic range of the instrument, defined as the velocity that produces a Doppler shift of one FSR, which is 3760 m/s. The temperature is biased low by between zero and −30 K. The standard deviation is roughly 10 m/s in velocity near the center rising to 20 m/s at the edge and 15 K in temperature near the center rising to 40 K at the edge. The velocity change that produces, via Doppler shift, a one-pixel shift of a transmission patch on the sensor is found from small r analysis of Eqs. (1) and (2) to be 62.5 (m/s) × r (mm). Thus, a velocity error of 20 m/s near the edge of the sensor is equivalent to an error in determining the Doppler-shift-induced-motion of a patch of 0.11 pixels. It is suspected that these small errors are from aliasing and that aliasing errors similarly impact the measurement of temperature. If aliasing is the source of error in velocity and temperature, then this error is not easily calibrated for and corrected as it depends on where the transmitted Rayleigh-scattering patches fall relative to the pixels, and thus depends on temperature, Doppler shift, and small motions of the center of the rings. This type of error is unlikely to increase in any systematic way as temperature or velocity increase and can be reduced by using an IRS system with higher spectral sensitivity (rate of change of wavelength with position on the sensor), which can be accomplished with a longer imaging lens focal length or with smaller pixels. Random error would be reduced by greater signal-to-noise level at each pixel.

 

Fig. 7. Profiles of mean and standard deviation (a) velocity and (b) temperature in atmospheric air.

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Figure 8 shows the results of fitting each instantaneous interferogram in a series of 100-shot pulse-burst-sequences with reference rings in which the jet position is translated in increments of 1 mm in the direction of the laser beam. Each sequence contributes results from 9 positions in space. Shown are profiles of the mean and standard deviation of (a) velocity and (b) temperature. Also shown is the average over all the positions on the sensor of the standard deviation measurements obtained for zero flow. The horizontal axis represents the position of the velocity or temperature measurement along the laser beam axis, and depends on the translation position of the jet and the position of the measurement point in the interferogram. Since the laser beam axis (X) intersects the nozzle axis (x₁) at 45°, the flow is mapped along a line at 45° to the jet and increasing X coincides with increasing downstream distance from the nozzle – see Fig. 1(b). The data points from adjacent sequences overlap in position: where they overlap there may be some differences in the measurement, but these are usually within the range of the previously-discussed errors. The measurements of mean velocity in the jet show an approximately Gaussian profile shape that is slightly asymmetrical with velocity, lower at the right, i.e., at greater distance from the nozzle exit. The Mach number at X = −1 mm is ∼1.3, while at X = 1 mm it is ∼1.1. It is speculated that this abrupt fall in Mach number and velocity (relative to axial symmetry) is due to an axially-symmetric shock wave that crosses the jet slightly downstream of the laser beam on the left-hand side of Fig. 8, and slightly upstream of it on the right-hand side. Conversely, the temperature drops toward the center of the jet due to roughly adiabatic expansion from nominally atmospheric temperature in the nozzle plenum. The standard deviations in velocity and temperature peak near the middle of the shear layer on each side of the jet, as expected, and are not significantly different than the zero flow values (noise) near the center of the jet, indicating that the actual flow fluctuation levels there are not significantly greater than zero. The mean velocity and temperature measurements scatter about their respective trend lines (not shown) by an amount consistent with the uncertainties based on the measurements at zero velocity, i.e., approximately ± 20 m/s in velocity and ± 30 K in temperature. The temperature plot also shows temperature calculated from the experimental mean velocity by assuming adiabatic flow from stagnation temperature (assumed equal to atmospheric temperature ${T_0}$=293 K) using Eq. 8 [19].

 

Fig. 8. Scanned profiles of mean and standard deviation (a) velocity and (b) temperature in the jet.

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A potential source of error in temperature arises from the sensitivity of the Tenti model to pressure. The pressure is assumed to be 1 atm. and the average pressure may be close to 1 atm, but shock waves and associated expansion waves are expected to cause substantial variations. If the spectra near the center of the jet are instead fitted assuming pressure of 0.67 atm., 1. atm., and 1.5 atm., the temperature fits to 222 K, 215 K., and 207 K respectively, plotted as X symbols in Fig. 8. Pressure variations can explain much of the apparently random variation of temperature relative to trend. Another important source of potential error arises from the sensitivity of the velocity and temperature to the viewing angle to the laser beam. The Rayleigh scattering linewidth scales as $\Delta {\lambda _{Rayleigh}}\sim \sqrt T sin({\theta /2} )$ from Eq. (4), while the Doppler shift depends upon the viewing angle by $u \propto {\Delta }\lambda \sin ({\theta /2} )$ from Eq. (5). Thus the relative error in temperature associated with error in measuring viewing angle, for a viewing angle of 90°, is $dT/T ={-} d\theta $, and the relative error in velocity is $du/u = d\theta /2$. A −7% error in temperature, consistent with the error found in the experiment, could be caused by a 4° error in angle, which is within the range of uncertainty in the angle measurement for this experiment. (This could easily be improved.) The relative error in velocity from this source would then be + 3.5%. If it were known for certain that the −7% error in temperature at atmospheric air conditions is due to error in the viewing angle then the measurement at atmospheric air could be be treated as a calibration; all measurements of temperature in the jet could be corrected by adding 0.07T and all measurements of velocity in the jet could be corrected by subtracting 0.035u. The line shape model may also provide a source of error. Doll [20] developed an empirical model for use with filtered Rayleigh scattering that provided better measurement of temperature and velocity in air than the Tenti S6. In the future, we will investigate the suitability of alternative line shape models, including curve fits that are computationally fast to evaluate [21].

Measurement of temperature by Rayleigh scattering is inherently more uncertain than velocity. This greater error is compounded by the effect of micron or larger sized particles in the flow. Particles originate in the laboratory air, are stirred up by the jet action, and their number density is the greatest in the mixing layer and at the edge of the jet. Particles produce relatively very bright dots in the interferogram that can spill over to multiple adjacent pixels, dominating Rayleigh scattering in the vicinity and, when present in a fit region, contaminate the fit. Temperature depends on Rayleigh scattering line shape, which is more sensitive to particles than velocity. Velocity depends on the location of the line center, which is the same for both Rayleigh and particle light scattering provided the particle follows the flow. Fits with relatively high fit error are rejected (<1% of fits) which tends to eliminate these particle events.

Figure 9 shows a sequence of 100-kHz measurements separated by 10 µs. (This is the longest sequence that could be plotted while allowing the points and lines to be easily distinguished and is not related to the maximum possible data set that could be acquired or analysed.) Fig. 9(a) shows velocity at X = 0.95 mm and 1.1 mm, while (b) shows the corresponding temperature sequences. These locations are at the right-hand edge of the jet core where it is speculated that the velocity is affected by shock interaction. The measurements at the adjacent points in the flow appear to show some correlation in both velocity and temperature, but correlation is not perfect. Figure 10 show examples of Fast Fourier Transform (FFT) analysis of the data at the X= 1.1 mm point. These analyses are of flow velocity and temperature in a single pulse burst sequence. An obvious frequency peak at ∼7 kHz is observed for both velocity and temperature. This frequency is not found at other locations in the flow and may be associated with unsteadiness in shock interactions. Other peaks maybe spurious, in which case they could be removed by averaging multiple data sets, or they may reflect real gas dynamical phenomena that would require much more detailed study to explain. This example demonstrates the ability of the high-speed IRS to capture important flow dynamics.

 

Fig. 9. An example of 100-kHz IRS data series of (a) flow velocity, and (b) flow temperature at X = 0.95 mm and 1.1 mm.

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Fig. 10. Examples of FFT analysis of (a) velocity and (b) temperature measured at 1.1 mm jet exit. Both FFT spectra show a frequency peak at 7 kHz.

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The normalized two-point correlation coefficient, calculated using Eq. 9, can provide information on the turbulent length scale of the flow:

 

Fig. 11. (a) The correlation coefficient calculated from multi-point velocities in a single interferogram sequence and (b) the calculated integral length scales.

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The integration is implemented by fitting a cubic spline to the measurements and numerically integrating between the limits of the data. Figure 11(b) shows the length scale computed in the manner described above, two center point positions, X, obtained for each scan position of the jet flow, plotted against X. The length scale is found to be in the range 0.4 mm to 0.55 mm across most of the jet, except around X = 1 and X = 3 mm. The lower values of length scale around X = 1 mm coincide with the previously observed deficit in velocity and the 7 kHz frequency that are attributed to unsteady shock waves. The implication is that the shock waves disrupt the turbulence, reducing the length scale.

Error in the correlation coefficient arises due to instrument noise. Assuming the instrument noise is not correlated with the flow fluctuation or the instrument noise at different points, which is reasonable since these points correspond to separately fitted subregions, the normalized correlation coefficient is given by Eq. 11.

The integral length scale is a measure of the size of the largest eddies of the jet that carry much of the energy of the turbulent fluctuations. The frequency of fluctuation in the flow associated with convection of an eddy of such a size may be estimated as a typical shear layer velocity (300 m/s) divided by the length scale (∼0.5 mm), which is ∼ 600 kHz. Higher frequencies would also exist in the jet flow associated with the smallest scale eddies, but such scales could not be spatially resolved by the current instrument. These results demonstrate the ability of the 100-kHz IRS technique to measure and spatially resolve integral length scales of this small high-speed jet and temporally resolve some important frequencies. Turbulence frequencies of interest could potentially be resolved in flows of an order magnitude larger length scale that are more relevant to most practical applications.

5. Conclusions

In conclusion, high-repetition-rate, burst-mode-laser-based IRS diagnostics were demonstrated at 100 kHz in a high-speed turbulent jet with a mean peak velocity of ∼396 m/s. Compared to 10-kHz measurement rate which is suitable for subsonic flows, 100-kHz IRS can resolve the turbulence evolution for supersonic flows. The pulse-train within the 10-ms-long burst enabled the measurement of simultaneous multi-point time-resolved velocity and temperature in the high-speed jet flow. Although density results are not shown here, the measured high-speed IRS signal intensities, i.e., the brightness of the Rayleigh scattering patches in the interferogram, are directly proportional to flow density and measurement of density is possible. An un-intensified camera was employed for better signal-to-noise ratio and better tolerance to sensor damage by excessive light. Compared to earlier 10-kHz rate demonstrations, high-frequency flow dynamics were better resolved by the 100-kHz measurement. FFT, two-point correlation, and integral length scale analysis show the ability of 100-kHz velocimetry to resolve turbulent length scales and flow dynamics. These advancements significantly improve the ability to make time-resolved measurements in high-speed, unsteady, turbulent flow and combustion environments, where un-seeded and non-intrusive flow field measurements are required. This work is the first demonstration of IRS temperature measurement without external reference cell and may be the first demonstration of simultaneous non-intrusive measurement of velocity and temperature at 100 kHz using any diagnostic technique.