5.1. Total pressure measurements

Total pressure measurements were conducted using a constructed Pitot pressure probe. A Kulite XCEL-100-5A pressure sensor was flush mounted in precision stainless steel tubing. Three sections of increasing diameter tubing were connected downstream of the sensor head with epoxy-smoothed transitions to bring the outer diameter from the size of the transducer (0.26 cm) up to 0.6 cm (further described in Mai (Reference Mai2014)). The Pitot probe could then be mounted in a sting assembly to position it in the test section. The Kulite sensor was sampled at 300 kHz and the signal was low pass filtered at 50 kHz before being recorded on the computer's data acquisition system.

Total pressure measurements were taken along the centreline of the nozzle exit plane in vertical and horizontal directions. Fifty runs were recorded at each of 9 locations across the nozzle exit plane while the probe was moved manually using a transition stage. Target locations of the probe were measured with a ruler across the nozzle exit and marked on the transition stage, accuracy of the location of the probe is of the order of 0.2 cm. For each location a 2 ms section of data was taken from each run in the region of stable pressure. The same 2 ms section of data from the settling chamber pressure was normalized against itself and used to correct the Pitot signal to account for shot-to-shot influences over time. This was done for both vertical and horizontal rakes.

5.2. Laser system description and experiment configuration

Two identical pulsed laser systems were used in this study for excitation of nitric oxide in various LIF techniques. Each system consisted of an injection seeded Spectra Physics PRO-290-10 Nd:YAG laser and a Sirah Cobra Stretch dye laser with a sum frequency mixing unit. The YAG laser operated at 10 Hz with a linewidth of 0.003 cm$^{-1}$, and was tuned for output in both second and third harmonics (532 nm and 355 nm respectively). A fundamental beam was generated in the dye laser by pumping a dye solution consisting of Rhodamine 610 and Rhodamine 640 in methanol with the first harmonic output of the YAG. The resulting fundamental beam was tuneable in the range of 600 to 630 nm. The dye fundamental beam was then mixed with the third harmonic output from the YAG in a Sirah SFM-355 mixing unit, resulting in an ultraviolet (UV) 223 to 227 nm output beam. Typical linewidth and power of the UV beam was 0.08 cm$^{-1}$ and 5–12 mJ pulse$^{-1}$ respectively.

The experimental configuration used for both laser based methods is depicted in figure 4. The model wedges were installed flush with the exit plane of the nozzle and spaced 5.1 cm apart vertically. The optimal vertical spacing for the wedges was determined via schlieren imaging flow visualization, where the Mach stem's height was maximized while remaining downstream of the wedge geometry. A Pitot pressure probe was mounted outside of the laser diagnostic area, in the lower corner of the core flow, to monitor tunnel conditions. A Princeton Instruments PI-MAX4 ICCD camera was mounted for viewing from the right-hand side of the tunnel.

Figure 4.

Model wedge configuration for MTV and PLIF thermometry experiments. (a) Image of actual configuration in tunnel; (b) diagram showing model wedge and Pitot probe position from behind, looking straight into nozzle exit.

The tunnel N$_{2}$ supply was pre-seeded with 0.5 %–1.0 % NO and pre-heated to achieve 375 K in the tunnel settling chamber. The initial laser was timed at 30–32 ms following initial tunnel valve opening and the second laser followed at a 1.7 $\mathrm {\mu }$s time delay. This time delay was selected to minimize the uncertainty in the velocimetry technique, and was used in both laser based methods for consistency. The camera was liquid cooled to $-$35 C with the intensifier enabled. The dual image feature setting was used and camera gates were set to 3/100 ns for the velocimetry ‘write’/‘read’ exposures and 10–15 ns for both PLIF exposures.

5.3. Molecular tagging velocimetry

Velocity measurements were made across a normal shock using a single component MTV approach. Two lasers were used in the MTV technique, one ‘write’ laser to initially excite the ground state NO, and one ‘read’ laser that would later target the tagged $v''=1$ NO, as previously demonstrated in Sánchez-González, Bowersox & North (Reference Sánchez-González, Bowersox and North2014). The final turning mirrors and sheeting optics for both lasers were mounted to an optical breadboard, which in turn was mounted to a linear transition stage on a rail system. A diagram of this set-up is shown in figure 5. The linear transition stage was used to optimize the location of the ‘write’ laser lines in relation to the normal shock wave during operation of the tunnel. The camera gate was set to 100 ns exposure so that signal from the scattered laser between the focused lines could be detected in order to reveal the shock location. The transition stage was then adjusted until the ‘write’ line was just downstream of the shock wave, usually with a separation of roughly 5 pixels. Once the lines were in optimal location, the transition stage was locked to prevent accidental motion.

Figure 5.

Diagram of experiment set-up for MTV measurements.

For each Reynolds number condition a series of images was taken. First a dotcard image was taken to align and focus the camera on the laser excitation region, as well as provide a distance conversion factor to be used in post-processing. The background intensity was corrected by taking a ‘dark-charge’ image of the covered camera lens and subtracting this from each frame. In order to confirm the location of the shock wave in post-processing, a long term PLIF exposure of NO(0,0) was recorded and averaged over 100 frames. Finally, the MTV images were recorded at several locations. Five hundred image pairs were taken in the ‘optimal’ position described previously. The transition stage would then be adjusted to shift the laser lines 5–10 pixels farther downstream, and 500 image pairs were taken at the new location. This was repeated until 4 total locations had been recorded.

All velocimetry images in this study were processed using a custom program written in Matlab. Cross-correlation was performed using the built in ‘$x$-corr’ Matlab function. First, each image pair was divided into ‘sub-images’ representing a region of single pixel height surrounding the location of the initial ‘write’ laser line. A threshold value was subtracted from each sub-image to remove the background signal and avoid corruption of the cross-correlation results. The ‘$x$-corr’ function was then applied and the resulting correlation matrix was cropped around the peak value and fitted with a third-order polynomial with centring and scaling applied.

A goodness of fit ($r^2$) value was recorded for the correlation function so that the user could remove poor correlation results. An example cross-correlation result is shown in figure 6. It was found that the correlation fit would underestimate the correlation peak value in most cases with the polynomial fit. While the peak value itself is not meaningful, in order to make sure the location of the peak was as accurate as possible a secondary fitting procedure was applied. If the initial polynomial fit returned a goodness of fit value above a user specified threshold (typically 0.95), the application would crop even closer around the peak and perform another polynomial fit. Including more points initially allows for poor correlations, without strong peaks, to be excluded from further processing; and it was found to be within 0.5 pixels of the true peak. The secondary fitting further reduced this uncertainty to 0.1 pixels.

Figure 6.

Cross-correlation example of a selected sub-image pair. The far (a) shows the normalized and thresholded sub-images, where the delayed image has been shifted by the correlation peak. In the (b) is the primary cross-correlation fit, and to the (c) is the secondary cross-correlation fit. Each is cropped around the peak value. A third order polynomial fit is applied with 95 % confidence bounds, and the resulting peak and goodness-of-fit $r^2$ value are displayed at the top.

The recorded sub-pixel correlation peak was then converted to a velocity value via the pixel-to-metre conversion factor and time delay. This was repeated for all ‘sub-image’ regions across each image pair recorded in the series. The velocity and goodness of fit values were then sent to post-processing where mean and fluctuating values could be determined across the region of interest.

5.4. PLIF thermometry

Temperature measurements were made across a normal shock wave by a two-line PLIF thermometry technique, similar to that of Sánchez-González et al. (Reference Sánchez-González, Bowersox and North2014). The ‘read’ lasers in this study were tuned to probe transitions in the $A^2\varSigma ^+(v'=0)- X^2\varPi (v''=0)$ band for ground state NO. For the nominal flow Mach number of 4.4, the expected mean temperature conditions were 77 K in the free stream and 361 K downstream of the normal shock wave. Target rotational states were chosen based on the Boltzmann population distributions of ground state NO at these temperatures. The rotational states $J''=1$ and $J''=8$ were chosen to maximize temperature sensitivity, particularly in the region downstream of the normal shock, while retaining reasonable signal potential from both regions of flow. Laser wavelengths were set for these rotational states and the power was balanced to $\sim$1.5 mJ pulse$^{-1}$ for both lasers.

A schematic of the experimental set-up for PLIF thermometry is shown in figure 7. Both lasers were introduced through the top window of the tunnel. In order to optimize the overlap of the laser sheets, the angle between the beams was minimized by mounting the final turning mirrors as close to each other as possible without clipping the beams. Both lasers were sent through a cylindrical lens to alter the circular profile into a planar sheet oriented in the direction of flow. A Princeton Instruments PI-MAX4 ICCD camera was mounted on the right-hand side of the tunnel for image collection.

Figure 7.

Diagram of experimental set-up for PLIF thermometry.

A dotcard image was taken first, to focus the camera on the plane of the laser sheets. The background intensity was corrected in situ by subtracting a ‘dark-charge’ image of the covered camera lens from each frame. Five hundred image pairs were taken of the normal shock region for each Reynolds number condition and processed as described below. This was repeated in the same region of flow with the model wedges removed to further characterize the free-stream conditions. Additional image pairs were taken in regions of known temperature for calibration and image correction purposes during processing.

Thermometry images were processed using a custom application, written in Matlab. Free-stream images were initially processed following a conventional approach for linear LIF thermometry to establish baseline conditions. It was believed that the high laser intensity needed to obtain signal in the low density free stream led to a LIF response in the higher density post-shock region that was at the edge of the linear limit, possibly in the transitional to saturated regime for LIF thermometry. A separate processing routine was written to examine the Mach-stem PLIF images, where each region of the flow was individually analysed for an assumed mean temperature based on the previously established conditions. Because the quantities of interest in this study were fluctuations, and the mean flow conditions are well known conventionally, it was determined that this ‘in situ’ processing would be sufficient for the scope of this analysis.

In this two-line PLIF technique, temperature is determined via (5.1), where $R_{12}$ is the ratio of LIF signals from rotational states ‘1’ and ‘2’, $C_{12}$ is a calibration constant based on experimental conditions, ${\rm \Delta} E$ is the energy separation between the states, $k$ is Boltzmann's constant and $T$ is the rotational temperature.

(5.1)\begin{equation} R_{12}=C_{12}\exp({-{\rm \Delta} E/kT}).\end{equation}

The processing routine analysed the upstream and downstream locations as separate images. Calibration images were not used in this routine. Instead, a mean temperature was assumed based on the previous free-stream measurements and conventional shock-jump relations. The calibration constant ($C_{12}$) was then specified to match this assumed temperature for the image intensity ratio ($R_{12}$) values at each pixel. This temperature specification method essentially acts as a laser profile correction step, combined with a temperature calibration, as is typically seen in the conventional approach. The raw images were cropped to view only the normal shock region with fluorescence signal, and then a region in front of and behind the normal shock was selected for further processing. Next, the images were binned in two dimensions for both pre- and post-shock regions, in this case a bin size of $4\times 4$ was selected to optimize both signal and resolution. The mean value of the pixels within each bin, excluding negative or ‘not a number’ values, was recorded and returned as the value for the entire bin.

Each pixel was then analysed individually to determine the appropriate $C_{12}$ value and resulting mean and fluctuating temperatures. For each pixel coordinate, $R_{12}$ was first determined for all image pairs. Spurious $R$ values beyond three standard deviations from the mean $R$ value were removed for the determination of ${C_{12}}^{\prime }$. A ${C_{12}}^{\prime }$ value was then selected such that the resulting mean temperature, calculated for the range of $R$ values, would match the specified mean temperature of the region. This was done through manipulation of the standard PLIF thermometry equation into (5.2) to find ‘$\left \langle {{C_{12}}^{\prime }}\right \rangle$’ based on the given $R_{12}$ range and specified mean temperature.

(5.2)\begin{equation} \langle {{C_{12}}^{\prime}}\rangle=\langle {\exp({(\ln(R_{12})+{\rm \Delta} E/k\bar{T})})}\rangle . \end{equation}

Once the appropriate ${C_{12}}^{\prime }$ value was determined for a given pixel coordinate, temperature values for each image pair at that coordinate were calculated and recorded. This was repeated across both regions of interest, and the resulting temperature data were sent to post-processing. Mean and fluctuating temperatures were analysed for each region, the results of which are discussed in the next section.