Dynamics of a 3-D inlet/isolator measured with fast pressure-sensitive paint

Abstract

Fast pressure-sensitive paint (PSP) was applied to an inlet/isolator designed using the Osculating Internal Waverider Inlet with Parallel Streamlines (OIWPS) method. The dorsal isolator surface pressure was measured using anodized-aluminum PSP through transparent cast acrylic that makes up the ventral portion of the isolator. Temperature-sensitive paint was utilized to correct for the PSP’s temperature sensitivity. The model was tested under Mach 5.7 flow at Re $=$ 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel (ANDLM6QT) under conventional noise conditions. Flow phenomena, such as shocks originating in the inlet and flow separation at the throat, were visualized with high spatial resolution. The dynamics measured by the PSP and pressure transducers matched well where the spectral signal-to-noise ratio was above unity. Power spectral densities showed significant frequency content at $\approx$1 kHz in the shock-wave/boundary-layer interaction (SWBLI) regions. Coherence analysis showed a linear relationship between the unsteady pressures at locations underneath different SWBLI in the isolator, with the exception of the Busemann throat shock. Temporal correlation of shock positions indicated that disturbances propagated downstream at 114% of the core-flow velocity; however, improved calculations of the core-flow velocity are needed to refine this assessment. The surface pressure fields at Re = 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m were quantitatively very similar, and the results in the ANDLM6QT were qualitatively similar to previous studies in the Boeing/AFOSR Mach-6 Quiet Tunnel under noisy flow.

1 Introduction

High-performance inlet/isolators are essential for dual-mode scramjet engines that can enable sustained atmospheric flight and cheaper space access (Smart 2007; Heiser and Pratt 1994). The inlet captures incoming air and passes it to the isolator with as much uniformity as possible. The inlet and isolator compress the captured air, raising the static pressure and decreasing the Mach number for efficient combustion (Heiser and Pratt 1994). The isolator prevents the instabilities and back pressure rise inherent to combustion from unstarting the engine (Matsuo et al. 1999; Heiser and Pratt 1994). Unstart is a highly detrimental mode of engine operation where the inlet capture characteristics, such as ingested mass flow, are dependent on the internal flow (Van Wie et al. 1996). It can be caused by several phenomena including off-design operations, excessive flow separation, and thermal choking due to combustion (Wagner et al. 2009, 2010; Im and Do 2018). Unstarted flow is characterized by significant inlet flow spillage, reduced pressure recovery, and significantly increased drag and thermal loads (Chang et al. 2017). The unstart process is dependent on the pressure gradient and inlet-generated shock impingements on the isolator wall. Locating these shock impingements is important for unstart detection and feedback-control of scramjet engines (Li et al. 2019; Tan et al. 2012; Do et al. 2011). Understanding the position of these shocks and their dynamic behavior is also critical for engine design due to the increased mechanical and thermal loads at shock impingement (Li et al. 2018).

Some studies of isolator shock-wave/boundary-layer interactions (SWBLI) investigate shocks caused by the inlet/isolator geometry, whereas others investigate the series of shocks created by combustion-induced choking/separation. Inlet shock dynamics are particularly important for engines operating in a pure scramjet mode, where the inlet shocks are not impacted by combustion-induced separation or flow choking (Leonard and Narayanaswamy 2021). Leonard and Narayanaswamy (2021) investigated an axisymmetric inlet/isolator with no back pressure and found that there existed very low frequency (on the order of $0.03~u_\infty /L$) shock oscillations when the boundary layers were attached. Downstream propagation of pressure fluctuations was primarily due to the convection of boundary-layer structures propagating over several boundary layer thicknesses, and the structures propagated at a velocity of 0.82 $u_{\infty }$. Upstream pressure fluctuation propagation occurred over approximately one boundary-layer thickness and was caused by acoustic waves. Coherence analysis revealed a linear coupling between two SWBLI in the isolator at low frequencies. Wang et al. (2020) investigated inlet shocks induced by a wedge in a cold direct-connect facility. Strong coherence was seen between the two SWBLI investigated, and the downstream shock’s fluctuations lagged the upstream shock’s (Wang et al. 2020).

Streamtraced inlets allow for shape transitions between the inlet capture plate and throat, with enhanced off-design performance compared to conventional inward turning inlets (Billig et al. 1999; Smart 1999; Smart and Ruf 2006; Jacobsen et al. 2006; Stephen et al. 2015; Bisek 2016; Suraweera and Smart 2009; Johnson et al. 2022a; Yeom and Im 2024; Li et al. 2020; Xiong et al. 2020; Musa et al. 2022; Eagan et al. 2023). With complex SWBLI and separation regions, accurately computing the flow of a streamtraced inlet/isolator is very difficult. One such streamtraced inlet is the Indiana Inlet (INlet), a collaboration between the University of Notre Dame, Purdue University, and the U.S. Air Force Research Laboratory (Noftz et al. 2022, 2023a, b, c; Shuck et al. 2023; Bustard et al. 2023a). The INlet is designed for academic inquiry into inlet phenomena with a relatively small contraction ratio of 5.16. It was designed using the Osculating Internal Waverider Inlet with Parallel Streamlines (OIWPS) design method, which defines a three-dimensional inlet geometry through a collection of two-dimensional flowfields (Noftz et al. 2023c). This method is versatile and well-suited for optimization, allowing for a variety of geometric and flow parameters to be specified as inputs. The INlet provides efficient compression, high flow uniformity, and straight leading-edge shocks of equal strength. It has a mixed compression region where the cowl and forebody surfaces contribute to compression, but the flow is not yet fully enclosed by walls at that axial station. The dual spill notches along the sides of the mixed contraction region were a product of the on-design construction of the inlet, but have the added benefit of allowing spillage for off-design operation. This open-source INlet geometry will have similar flow physics to many other streamtraced inlet/isolators.

A preliminary 3D printed INlet model, designed without a viscous correction, was instrumented with pressure transducers and tested with and without back pressure in the Boeing/AFOSR Mach-6 Quiet Tunnel (BAM6QT) at Purdue University (Noftz et al. 2022) with noise levels of less than 0.02% (Mamrol and Jewell 2022). Shock train characteristics and unstart propagation velocities at several jet blowing ratios were measured. A higher fidelity INlet model, designed with a viscous correction, was machined from steel. It was instrumented with pressure transducers and leveraged schlieren imaging to establish the tare flow without back pressure under quiet flow and indicated the approximate location of isolator SWBLI (Noftz et al. 2023b). Under quiet flow, boundary-layer transition occurred in the inlet’s internal contraction, and the boundary layer along the isolator’s dorsal centerline was turbulent.

Computational studies using Improved Delayed Detached Eddy Simulation (IDDES), unsteady laminar, and transitional RANS flow models have shed light on differences between expected and actual compression in the inlet forebody and provided estimates of off-wall quantities for Re $= 10.6 \times 10^6$ /m (Shuck et al. 2023). The Busemann flowfield used as the parent flowfield for the INlet naturally generates a throat shock impinging on the throat shoulder that turns the flow to be parallel to the isolator (Molder and Szpiro 1966), termed the Busemann throat shock in this work. The Busemann throat shock impinged slightly downstream of the throat in the simulations, and was believed to cause a large region of separated flow due to the shock’s adverse pressure gradient (Shuck et al. 2023). The 3D OIWPS design and dual spill notches cause separation to extend farther downstream along the sides than near the centerline.

Pressure-sensitive paint (PSP) is an instrumentation technique with high spatial and temporal resolution. Its global coverage obviates a priori knowledge of the location of interesting flow features, unlike the installation of point sensors. Several studies have used PSP to investigate internal flows. Idris et al. (2014) applied PSP to the sidewall and bottom of a two-dimensional inlet/isolator and saw three-dimensional pressure distributions. A micro-vortex generator upstream of the inlet shoulder resulted in a reduction of peak pressure at the shock-induced separation attachment point, which was measured by the PSP (Che Idris et al. 2021). Leonard and Narayanaswamy (2021) used porous polymer ceramic pressure-sensitive paint with a frequency response of 4 kHz, fast enough to capture separation bubble and shock oscillations on an axisymmetric isolator. Bustard et al. utilized anodized-aluminum PSP and temperature-sensitive paint (TSP) to measure temperature-corrected time-resolved pressure and temperature fields on the inside of an axisymmetric inlet/isolator model under Mach 6.3 flow (Bustard et al. 2023b). Tare conditions for $9^{\circ }$ and $30^{\circ }$ inlets were quantified. A shock train was visualized at moderate back pressure, with a further increase in back pressure resulting in a high-amplitude oscillatory mode with a dominant frequency of $\approx 225$ Hz.

PSP investigations of streamtraced inlet/isolators are rare and the ones that exist have very limited optical access of the isolator (less than two heights in length) (Johnson et al. 2022a, b). Johnson et al. (2022a, 2022b) applied PSP to a streamtraced Busemann inlet under Mach 4 flow. Mean pressure was measured during the started, steady case and during the unstarted case with back pressure applied (Johnson et al. 2022a). They measured unsteady behavior up to 2 kHz during unstart instigated by jet blowing (Johnson et al. 2022b). This study showed no significant peaks in the spectra when the isolator was started, but elevated frequency content once it had fully unstarted. Cross coherence showed strong linear coupling between the pressure fluctuations in the inlet and isolator, indicating separation across the whole inlet/isolator model during unstart. Eagan et al. (2023) measured the internal pressures in an inlet/isolator by embedding a miniature camera and LED array into the inlet/isolator model itself. They successfully discerned the location of cowl shock impingement adjacent to the inlet throat region and investigated the impingement location’s variation with angle of attack.

Bustard et al. (2023a) investigated the INlet using PSP in the BAMQ6T under noisy and quiet flow. The design of BAMQ6T positions the model upstream of the burst diaphragm, and as a result the model is surrounded by stagnation temperature air before the run. Due to challenges trying to keep the model surface temperature isothermal, and the run-to-run variability in model temperature, the attempted correction of the PSP’s temperature sensitivity was not very successful. However, significant differences in INlet pressure distributions were observed between noisy and quiet flow, exceeding the uncertainty introduced by the lack of a temperature correction of the PSP data. For example, the SWBLI occurred 0.85 isolator heights farther downstream on average for quiet flow than observed for noisy flow.

Providing INlet shock locations and spanwise variation can inform future computational and experimental efforts for a category of inlets not widely studied in the literature. Since improved computational modeling predictions of flow separation behavior is required, global measurement techniques can determine the position of separation and reattachment shocks without a priori knowledge. To the author’s knowledge, no investigations of inlet shock dynamics have been conducted for streamtraced inlet/isolators using time-resolved PSP measurements. PSP’s high spatial resolution complements the point measurements made previously by Noftz et al. (2023a, 2023c). The current experiments were conducted in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel (ANDLM6QT) at Mach 5.7 under conventional noise conditions at freestream unit Reynolds numbers from Re$= 8.5 \times 10^6$ /m to $10.2 \times 10^6$ /m. Temperature-sensitive paint (TSP) measurements of the isolator wall allowed the temperature sensitivity of the PSP to be corrected, resulting in higher accuracy than in the previous BAMQ6T results.

Fig. 1

Schematic of the ANDLM6QT

2 Facility and model

2.1 AFOSR–Notre Dame Large Mach-6 Quiet Tunnel

Experiments were conducted in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel under conventional noise conditions. The facility is a Ludwieg tube with a 60 m long driver tube (Fig. 1). Heating blankets on the driver tube permit stagnation temperatures from 435 to 590 K. A pneumatically actuated fast-acting shutter valve at the contraction exit opens to start a run. The ANDLM6QT is currently in a shakedown phase, with an aluminum surrogate nozzle installed, and not yet running with a quiet freestream flow. As such, the freestream Mach number is 5.70$\pm 0.04$ with noise levels of 2.6% (Hoberg and Juliano 2023), typical for a conventional-noise tunnel (Schneider 2001). The total pressure is measured at the beginning of the nozzle contraction with a XTEH-10L-190SM-1000A Kulite pressure transducer. Data were analyzed over 0.4 s of steady flow during each run, corresponding to two quasi-steady pressure plateaus. Experiments were conducted at two nominal freestream unit Reynolds numbers, $8.5 \times 10^6$ /m and $10.2 \times 10^6$ /m. The unit Reynolds number decreased by 2.5% during the nominally steady portion of the run. The flow conditions for each unit Reynolds number test are summarized in Table 1. The nozzle exit diameter is 0.63 m, the boundary-layer thickness is 0.11 m for noisy flow (Hoberg and Juliano 2023), and the corresponding rhombus of uniform Mach 5.7 flow is 2.30 m in length. The inlet leading edge and back of the model were 0.137 m and 0.594 m downstream of the nozzle exit plane, respectively, therefore the model was fully in the region of uniform Mach 5.7 flow.

Table 1 Experimental test conditions

Fig. 2

Labeled computational rendering of the INlet model

Fig. 3

Bottom-up view of the dorsal isolator. The PSP viewing area is outlined by the blue rectangle. The white circles indicate the positions of the conventional pressure transducers

2.2 INlet model and instrumentation

The INlet geometry is designed with a circular-to-rectangular shape transition and a design operating condition of Mach 6 at $0^{\circ }$ angle of attack. A custom streamtracing tool, OIWPS, was created by the authors to design inward-turning inlets for arbitrary capture and isolator cross-sectional shapes (Noftz et al. 2023c). The tool applies osculating axisymmetric theory by stitching locally irrotational streamline solutions together along a predefined azimuthal sweep. Inward-turning surface streamlines were generated by unique Busemann solutions to the Taylor–Maccoll equations combined with an Internal Conical Flow-A (ICFA) profile at the inlet leading edge. The overall contraction ratio of this inlet is 4.31, and the internal contraction ratio is 1.67. The inlet throat was enlarged to fit a rectangular isolator shape with flat sidewalls and slightly curved corners, dropping the overall contraction ratio to 4.68. A shape transition function blended the viscous-corrected throat shape into the desired isolator shape, the methodology for which is given by Noftz et al. (2023b). The INlet model was sized to ensure the BAM6QT, which is a smaller facility than the ANDLM6QT, would reliably start with the model in the test section (Noftz et al. 2022).

The INlet model is composed of six pieces: inlet forebody, cowl, dorsal isolator, strongback that joins the forebody and dorsal isolator, anodized aluminum PSP plate within the dorsal isolator, and transparent ventral isolator (Fig. 2). A sting adaptor interfaces the model with existing stings used in the ANDLM6QT. The polished, cast acrylic ventral isolator allows for pressure-sensitive paint measurements of the dorsal internal surface of the isolator; the PSP viewing area on the anodized aluminum plate is illustrated in Fig. 3. The isolator height, H, is 13.1 mm at the centerline, and its length is 171 mm. The coordinate x is defined as the distance downstream of the isolator entrance in millimeters; the isolator entrance itself was 5.55 mm downstream of the throat. The INlet model was instrumented with seven XCE-062-70 kPa Kulite pressure transducers along the dorsal centerline of the isolator wall. These pressure transducers are located at $x = 5.74$, 17.0, 28.5, 45.6, 68.5, 91.5, and 114 mm. Three type-K thermocouples (Omega, 0.51 mm diameter) were mounted to the back face of the aluminum plate to measure their pre-run temperature at $x \approx 37$, 80, and 103 mm.

3 Diagnostics and post processing

3.1 Wall pressure transducers

The pressure transducers were sampled at 250 kHz on an HBM Gen7i Genesis High-Speed Data Acquisition System. The DC-coupled transducer millivolt-level signals were amplified by 128 gain with a low-pass filter at 50 kHz. A quadratic least squares fit was used to determine the calibration slope of the transducers. The 95% confidence interval bounds of the least squares fit of each transducer was considered the systematic calibration uncertainty, $\epsilon _{k,\textrm{fit}}$. The systematic uncertainty for the XCE-062-15As due to repeatability, non-linearity, and hysteresis was quoted on the manufacturer data sheet as 0.1% of the full scale of the transducers, $\epsilon _{k,m} =70$ Pa. The total uncertainty for each transducer is:

$$\begin{aligned} \epsilon _k = \sqrt{{\epsilon ^{}_{k,m}}^2+{\epsilon ^{}_{k,\textrm{fit}}}^2} \end{aligned}$$

(1)

3.2 Five-hole probe

A five-hole probe was utilized to align the INlet model with the ANDLM6QT. The five-hole probe is a 30$^\circ$ half-angle cone with a base diameter of 25.4 mm. Four ports located $90^{\circ }$ azimuthally apart were drilled normal to the conical surface. The static-pressure ports are 0.5 mm in diameter and located at a radial distance of 4.2 mm from the cone’s axis. The resulting spatial measurement resolution is 8.4 mm. The probe was instrumented with five XCE-062$-$0.7 bar Kulite pressure transducers. A flush-mounted transducer at the nose tip allowed for stagnation pressure measurements.

3.3 PSP and TSP

Tris(4,7-diphenyl-1,10-phenanthroline)ruthenium(II) was selected as the PSP luminophore as it is well characterized for high-speed testing, has a short luminescent lifetime, and high pressure sensitivity (Sakaue and Sullivan 2001; Sakaue 2005; Liu et al. 2005; Sakaue and Ishii 2010; Kameda 2012; Gregory et al. 2014; Egami et al. 2020). Its peak excitation and emission wavelengths are $\approx 457$ nm and 643 nm, respectively (Liu et al. 2005; Sakaue 2005). Anodized aluminum PSP was utilized for its high oxygen diffusivity, robustness, and thickness/concentration uniformity (Asai et al. 1997; Sakaue and Sullivan 2001; Gregory et al. 2008). The anodization and luminophore application process followed that of Sakaue (2005). This procedure yields anodized aluminum with a pore diameter of 20 nm and pore thickness of $10~\mu$m. The step response of this PSP is 30 $\mu$s (Sakaue et al. 2013). The frequency response of this PSP, defined as the frequency at which the gain falls below $-3$ dB, is $\approx$ 6 kHz (Kasai et al. 2021). A Mitutoyo SJ-310 contact profilometer was utilized to measure the surface roughness of the anodized aluminum plate. Four profiles 4.8 mm in length were taken at different axial stations along the plate. The surface roughness was found from the average standard deviation of the profiles, which was 0.293 $~\upmu$m ± 0.025 $~\upmu$m.

Temperature-sensitive paint was utilized to correct the temperature sensitivity of the PSP. Rhodamine B was selected as the TSP luminophore as it is well characterized for high-speed testing,¹ has a short luminescent lifetime, and has sufficient temperature sensitivity over the expected temperature ranges experienced in the ANDLM6QT facility (Sakaue and Sullivan 2001; Liu et al. 2005; Claucherty and Sakaue 2017). Its peak excitation and emission wavelengths are $\approx 550$ nm and 570 nm, respectively (Liu et al. 2005).

Three 445 nm wavelength blue lasers (Necsel, Blue 445 10 W, with a spectral variability of $\pm 5$ nm) with plano-convex lenses and ground-glass diffusers provided excitation illumination for the PSP. Two 405 nm wavelength LED arrays (SOLIS-405C, 3.9 W) with ground-glass diffusers provided excitation illumination for the TSP. Luminescent intensity was recorded with a 12-bit monochrome Phantom v1840 camera with a 50 mm-focal-length lens. The camera was mounted $\approx 0.72$ m from the model, approximately normal to the painted plate. The spatial resolution of the PSP and TSP measurements was $\approx 0.17$ mm/pixel. A 470 nm wavelength long-pass optical filter was placed in front of the camera lens to prevent the excitation intensity from being measured. For the PSP experiments, the camera frame rate was 10 kHz, and the exposure time was $99.5\,\upmu$s. For the TSP experiments, the camera frame rate was 1 kHz, and the exposure time was $995\,\upmu$s. The minimum value of intensity counts measured during the runs was $\approx 5$% of the camera’s full scale, and the maximum was $\approx 30$%.

Fig. 4

Post-processing steps to convert PSP raw intensity into pressure fields

A flowchart detailing the post-processing to convert raw measured intensity to pressure fields is given in Fig. 4. Surface temperature was calculated from the TSP images. For each run, 300 frames were collected 0.3 s before the run and were averaged together to create a single wind-off reference image. The reference temperature was taken to be the average of the three thermocouples in the isolator during the pre-run period. An average of 5001 dark images, with the same frame rate and exposure time as the wind-on data but with the LEDs off, was subtracted from the wind-off reference image and from each wind-on frame to account for dark-current noise and ambient light (Liu et al. 2005). The wind-on frames were registered to the reference image within 0.01 pixels using a subpixel image registration algorithm (Guizar-Sicairos et al. 2008). The maximum movement between the wind-off and wind-on images during the test time of interest were 0.53 pixels and 1.2 pixels in the streamwise and spanwise directions respectively. A ratio was taken between each wind-on image and the wind-off reference image to get the intensity ratio, $I/I_{{\textrm{ref}}}$. An a priori calibration collected in a static pressure-temperature chamber was utilized to convert the intensity ratio to a temperature at each pixel location for each frame, T(i, j, t). An image mapping code utilized a perspective transform matrix generated from the known axial and azimuthal position of fiducial marks and their corresponding pixel coordinates to map the images to x–y space, where y is the spanwise distance relative to the model centerline, following the method outlined by Gordeyev et al. (2014). The temperature data were then linearly interpolated onto a 106 by 832 point structured grid for each frame, corresponding to the physical area of 17.8 mm by 142.2 mm. To minimize noise introduced through the temperature correction, the temperature was averaged over the span within 5 mm of the centerline, yielding the streamwise temperature distribution, T(x, t). Finally, a cubic polynomial was fit to the temperature data T(x, t) over the steady wind-on portion of the run at each streamwise position to further reduce noise.

Surface pressure was calculated from the PSP images. For each run, 2001 frames were collected 0.2 s before the run and were averaged together to create a single wind-off reference image. The reference pressure was taken to be the average of the seven wall pressure transducers in the isolator during the pre-run period. An average of 5001 dark images was subtracted from the wind-off reference image and from each wind-on frame. Each wind-on frame was registered to the wind-off reference image. The maximum movement between the wind-off and wind-on images during the test time of interest were 0.8 pixels and 1.4 pixels in the streamwise and spanwise positions, respectively. A ratio was taken between each wind-on image and the wind-off reference image to get the intensity ratio, $I_{{\textrm{ref}}}/I$. The intensity ratios were mapped to x–y space and then linearly interpolated onto the same grid as the TSP. The intensity ratios were then temperature-corrected using the TSP data, T(x,t), to account for the temperature-sensitivity of the PSP (Kurita et al. 2006). An a priori calibration conducted in a pressure-temperature chamber was utilized to characterize the temperature-sensitivity of the PSP. The pressure-sensitivity of the PSP at each pixel was characterized by an a priori calibration conducted in the wind tunnel. Wind-off intensities were recorded at each pixel under varying test-section pressures, and a quadratic fit of intensity ratio at each pixel at each pressure formed the calibration. A representative calibration at a single pixel is given in Fig. 5 for Re$= 10.2 \times 10^6$ /m. The calibration was applied at each pixel to convert intensity ratio into pressure, p(x, y, t), at each pixel for each frame.

The pressure measured by the PSP was normalized by the instantaneous freestream static pressure. The freestream static pressure was calculated using the isentropic relations:

$$\begin{aligned} p_\infty = p_0\left[ 1+\frac{\gamma -1}{2}M^2\right] ^{\frac{-\gamma }{\gamma -1}} \end{aligned}$$

(2)

The pressure was decomposed from its instantaneous pressure, p, into its mean and unsteady components, $\bar{p}$ and $p'$. The mean component $\bar{p}$ was calculated by taking the average value of instantaneous pressure over the run time of interest. As the freestream static pressure slightly decreased during the quasi-steady portion of the run, simply subtracting the mean component of pressure was insufficient to calculate $p'$. A quadratic fit of p(t) over the steady portion of the run was subtracted from p(t) over the same time period to yield $p'$. The unsteady component was then spatially averaged with a 1.19 mm diameter circular filter; this filter diameter is similar to the the sensing area of the Kulite pressure transducers.

Fig. 5

A priori PSP calibration for Re $= 10.2 \times 10^6$ /m. Dashed lines are 95% confidence interval of the fit

Fig. 6

PSP mean uncertainty for Re $= 10.2 \times 10^6$ /m

The TSP uncertainty $\epsilon _{\textrm{TSP}}$ was composed of the calibration uncertainty of the TSP $\epsilon _{\textrm{TSP,cal}}$ and the uncertainty in reference temperature measured by the thermocouples $\epsilon _{T{\textrm{ref}}}$. The TSP calibration uncertainty $\epsilon _{\textrm{TSP,cal}}$ was the 95% confidence level of the calibration’s quadratic fit of TSP intensity ratio and temperature. The uncertainty in reference temperature is:

$$\begin{aligned} \epsilon _{T{\textrm{ref}}} = \frac{\sigma _{\mathrm{T/C}} t_{2,95\%}}{\sqrt{N}} \end{aligned}$$

(3)

where $\sigma _{\mathrm{T/C}}$ is the standard deviation of the three thermocouple measurements, $t_{2,95\%}$ is the Student’s T-distribution with 95% confidence, and N is the number of thermocouples. The total TSP uncertainty is:

$$\begin{aligned} \epsilon _{\textrm{TSP}} = \sqrt{{\epsilon ^{}_{\textrm{TSP,cal}}}^2+{\epsilon ^{}_{T{\textrm{ref}}}}^2} \end{aligned}$$

(4)

The temperature uncertainty of the TSP was 0.53 K.

The mean PSP uncertainty analysis considered four uncertainty sources identified by Liu et al. (2005): uncertainty in measurement of the reference pressure, photodegradation of the PSP, uncertainty in PSP calibration, and uncertainty in correction of the PSP’s temperature sensitivity. The uncertainty in measurement of the reference pressure $\epsilon _{p,{\textrm{ref}}}$ is the average pressure transducer uncertainty (calculation described in Sect. 3.1) for the measurement of reference pressure in the pre-run. The photodegradation uncertainty $\epsilon _{p,d}$ was characterized by comparing two reference images from successive runs. The decrease in the intensity counts between reference images was subtracted from the average wind-on intensity image, propagated through the a priori calibration, and compared to the originally calculated pressure to estimate the uncertainty due to photodegradation. The PSP’s pressure calibration uncertainty $\epsilon _{\textrm{PSP,pcal}}$ was the 95% confidence level of the calibration’s quadratic fit of PSP intensity ratio and corresponding wind-off pressure. The total uncertainty in PSP intensity ratio due to the uncertainty in temperature correction is:

$$\begin{aligned} \epsilon _{IR,TC} = \sqrt{{\epsilon ^{}_{\textrm{IR,Tcal}}}^2+\left( \epsilon _{\textrm{TSP}}*b_{Tcal}\right) ^2} \end{aligned}$$

(5)

where $b_\textrm{Tcal}$ is the coefficient describing the linear relationship between the PSP’s intensity ratio and temperature. The PSP’s temperature calibration uncertainty $\epsilon _{\textrm{IR,Tcal}}$ was the 95% confidence level of the calibration’s quadratic fit of PSP intensity ratio and temperature values used for the calibration. The uncertainty in intensity ratio due to temperature, $\epsilon _{IR,TC}$, was subtracted from the average wind-on intensity image, propagated through the a priori calibration, and compared to the originally calculated pressure to estimate the uncertainty in temperature correction of the PSP, $\epsilon ^{}_{{\textrm{PSP}},T}$. The total uncertainty for PSP mean pressure is:

$$\begin{aligned} \epsilon _{{\textrm{PSP}}}=\sqrt{{\epsilon ^{}_{p,{\textrm{ref}}}}^2+{\epsilon ^{}_{p,d}}^2+{\epsilon _{\textrm{PSP,pcal}}}^2+{\epsilon ^{}_{{\textrm{PSP}},T}}^2} \end{aligned}$$

(6)

The PSP’s uncertainty in mean pressure is given in Fig. 6. The component of mean uncertainty and total mean uncertainty for each diagnostic, averaged across each transducer/pixel, is given in Table 2. Mean uncertainties in pressure are expressed as a percentage of mean pressure.

Table 2 Mean uncertainty for each diagnostic

The unsteady PSP error analysis closely followed the process outlined by Funderburk and Narayanaswamy (2019). The camera noise was assumed to be dominated by photon shot noise (Schairer 2002). The camera noise can then be calculated from a measured intensity using the following relationship in the small perturbation limit:

$$\begin{aligned} I_{\textrm{noise}} \approx a \sqrt{I} \end{aligned}$$

(7)

where $I_{\textrm{noise}}$ is the camera noise and a is a scaling constant with units of ${\textrm{counts}}^\frac{1}{2}$. To determine the value of a, the intensity trace at a single pixel is extracted from a 2001 frame wind-off set of reference images with the same exposure time, frame rate, and spatial filtering as the wind-on unsteady data. The standard deviation of the mean-subtracted intensity trace, $\sigma _{{\textrm{SN}}}$, is compared to the mean intensity of the same trace, which indicates a value of a of 0.12 ${\textrm{counts}}^\frac{1}{2}$ for the present study. The camera noise was calculated at each pixel using the mean wind-on intensity using Eq. 7. Propagating both I and $I-I_{\textrm{noise}}$ through the PSP calibration and taking the difference between those calculated pressures at each pixel yields the minimum resolvable standard deviation $\sigma _{p,\textrm{min}}$. This value varied across the model, but was typically in the range of 0.02–$0.05\,\,\sigma _{p,\textrm{min}}/\bar{p}$ when normalized by the mean pressure at each respective pixel.

4 Angularity measurements

The five-hole probe (5HP) was utilized to quantify the yaw and angle of attack of the model. The first step was to calibrate the probe. The 5HP was mounted on the sting in the tunnel and a digital inclinometer provided the pitch angle relative to gravity. The 5HP was rolled such that the ports were in the pitch and yaw planes. Four runs were done to calibrate the 5HP at different pitch angles, as measured by the inclinometer. A linear fit between pitch angle and the pressure difference between the two transducers on the pitch plane was calculated. The angle-of-attack of the probe relative to the flow was found by dividing the pressure differential by the slope of the linear fit. It is believed one of the four ports was blocked with debris, greatly reducing its response time; as a result, only one set of ports were calibrated. Yaw was quantified by rotating the sting $90^{\circ }$ in order to use the same set of ports used to quantify AoA, using the same calibration found in the pitch plane. The uncertainty in the angle of attack/yaw is:

$$\begin{aligned} \epsilon _{\textrm{angle}} = \sqrt{{\epsilon ^{}_{\textrm{5HP},\textrm{cal}}}^2+\left( {\epsilon _k*m_{AoA}}\right) ^2} \end{aligned}$$

(8)

where $\epsilon ^{}_{\textrm{5HP},\textrm{cal}}$ is the 95% confidence level of the calibration’s linear fit between pressure differential and angle and $m_{AoA}$ is the slope of the linear fit. The measured angle of attack was $0.14\pm 0.38^{\circ }$, and the measured yaw angle was $1.47\pm 0.38^{\circ }$. Facility constraints resulted in a value of yaw that was an order of magnitude larger than angle of attack.

5 Representative temperature measurement

The temperature history of one representative streamwise location ($x =$ 70 mm) measured by the TSP at Re$= 10.2 \times 10^6$ /m is shown in Fig. 7. The valve begins its actuation at $t =$ 0 s, and the temperature of the plate increases as small amounts of air pass through the valve. At $t \approx$ 0.55 s, the inlet starts, leading to a short spike in temperature followed by a slight decrease. The valve has fully opened at $t \approx$ 0.75 s, confirmed by a PLC signal and transducer data in the isolator, and the temperature gradually increases. At $t \approx$ 1.4 s, it is believed that the oblique shocks from the over-expanded nozzle flow impinge on the inlet as the end of the run nears, resulting in a drop in surface temperature. Finally, the tunnel unstarts at $t \approx$ 1.7 s resulting in large fluctuations. The ‘on-condition’ portion of the run is from t = 0.75  to 1.4 s. A cubic fit for T(t) in this interval is shown as a black line. Figure 8 shows the T(x, t) contour during the on-condition portion of this run. The temperature of the plate increases by $\approx$ 0.5 to 1 K with the farthest upstream and downstream portions of the plate having a higher temperature than the middle.

Fig. 7

Curve-fit of temperature data at $x = 70$ mm for Re $= 10.2 \times 10^6$ /m

Fig. 8

Spanwise averaged streamwise temperature time history for Re $= 10.2 \times 10^6$ /m

Fig. 9

Normalized mean pressure for Re $= 10.2 \times 10^6$ /m

Fig. 10

Normalized mean pressure streamwise profiles for Re $= 10.2 \times 10^6$ /m

6 Pressure measurements

The normalized mean pressure contour for Re $= 10.2 \times 10^6$ /m is given in Fig. 9, and streamwise pressure profiles at $y = -2.5$ mm and $-7.5$ mm are given in Fig. 10. The slight asymmetry is attributed to the non-zero yaw of the model. Using the IDDES analysis by Shuck et al. (2023), we can postulate what is occurring in the current experiment. Namely, the increase in mean pressure from $x \approx$ 3 mm to 12 mm corresponds to the Busemann throat shock impinging on the separation downstream of the throat. The pressure rise occurs farther downstream away from the centerline, indicating separation extends farther downstream near the isolator corners. A gradual increase in pressure is seen at $y = -7.5$ mm after the Busemann throat shock up to $x \approx 23$ mm and the pressure rise is greater than the $y = -2.5$ mm profile. This corresponds to the throat reattachment shock. At $x = 30$ mm, both profiles exhibit a pressure rise induced by the throat-bottom shock. This shock is caused by a compression ramp at the transition between the inlet and isolator surfaces due to the shape transition from the inlet’s viscous-corrected throat to the desired isolator geometry (Noftz et al. 2023b). The pressure along both profiles gradually decreases starting at $x \approx$ 40–42 mm, likely corresponding to the impingement of expansion waves from the throat SWBLI. The centerline pressure increases due to a isolator compression shock at $x \approx 75$ mm, and the pressure along both profiles increases beginning at $x \approx 82$ mm due to the third throat-bottom shock impingement. The pressure rises gradually until $x \approx$ 105–110 mm, then decreases gradually until the end of the viewing area, likely due to another set of expansion waves. The compression ratio is 10.6, calculated using the spanwise average of normalized mean pressure at the end of the viewing area. The average difference between the transducer and PSP mean pressures at $y = -2.5$ mm is 9.6%, and the largest difference is 13.9%. The results are typically within the uncertainty, shown as low opacity bands on the PSP profiles and uncertainty bars on the transducers. That said, the PSP generally indicates higher pressure than the transducers.

In the supplemental video showing the time history of normalized pressure, occasional events cause a perturbation in the SWBLI regions that transiently push the shocks downstream. This is most apparent for the throat-bottom, isolator compression, and third throat-bottom shocks. These occasional events are termed “propagating disturbances" and cause downstream movement for all shocks in the isolator.

Fig. 11

Maximum frequency where spectral SNR is greater than one for Re $= 10.2 \times 10^6$ /m

Fig. 12

Normalized band-integrated RMS pressure distribution for Re $= 10.2 \times 10^6$ /m. Markers correspond to PSD locations in Figs. 16 and 17

6.1 Spectral and skewness results

Power spectral densities (PSDs) of the PSP and transducer data were calculated to investigate the frequency content. The spatially averaged unsteady pressure was normalized by the mean pressure at that pixel before its PSD, $S_{xx}$, was calculated. A Hanning window with 50% overlap of signal content was used to maximize signal quality. Window averaging with a frequency resolution of 100 Hz is presented. The PSD window size is 100 points for the PSP and 2500 for the transducers, corresponding to 79 windows for the spectral calculation. Each window was detrended by subtracting a quadratic fit of the signal for that window. Spectral signal-to-noise ratio (SNR) limits the maximum resolvable frequency for PSP spectral analysis; the calculation of spectral SNR closely followed the process outlined by Funderburk and Narayanaswamy (2019). The first step to determining the spectral SNR is to calculate wind-off spectra to determine the shape of the noise spectra. A reference intensity trace at an arbitrary wind-off pixel was converted to pressure using the a priori calibration. The spatially averaged unsteady component was calculated. This unsteady reference pressure was used to calculate a power spectral density over 0.2 s using the same calculation procedure as the wind-on spectra, except for using only 39 windows to account for the difference in sample time. This spectra was then scaled by a factor of $\left( \sigma _{SN}/\sigma _{{\textrm{SN}},{\textrm{ref}}}\right) ^2$, to account for the non-uniform intensity and shot noise distribution over the model during a run. The spectral SNR is then calculated as:

$$\begin{aligned} SNR = \frac{S_{xx,{\textrm{PSP}}}-S_{xx,\textrm{noise}}}{S_{xx,\textrm{noise}}} \end{aligned}$$

(9)

The maximum frequency at each pixel where the spectral SNR is above unity, $f_\textrm{max}$, is given in Fig. 11. Pixels whose maximum frequency is less than 100 Hz are colored white, and the maximum value of $f_{max}$ is the Nyquist frequency of 5000 Hz. In general, there is sufficient SNR to resolve the spectra underneath and adjacent to SWBLI regions, and in the upstream region immediately following the throat.

Fig. 13

Streamwise band-integrated RMS pressure distribution for Re $= 10.2 \times 10^6$ /m

To provide a single value for spectral content at each pixel, band-integrated root-mean-square (RMS) pressures were calculated (Funderburk and Narayanaswamy 2019; Running and Juliano 2021a). The spectra at each pixel were integrated using the trapezoidal rule from 0 Hz to the Nyquist frequency of 5000 Hz. The band-integrated RMS was calculated using the following equation:

$$\begin{aligned} \sigma _{p,0-5\textrm{kHz}} = \sqrt{\int _{0~\textrm{Hz}}^{5000~\textrm{Hz}}S_{xx} \,\textrm{d}f} \end{aligned}$$

(10)

The band-integrated RMS (BIRMS) pressure distribution for Re $= 10.2 \times 10^6$ /m under noisy flow is given in Fig. 12. The effects of shocks can be seen at $x \approx$ 3 to 15, 30, 75, and 90 mm. Comparing the band-integrated RMS contour to Fig. 11, there are regions of elevated wind-on spectra resulting in high spectral SNRs across the entire spectra frequency range. Figure 13 gives the band-integrated RMS streamwise profiles at $y = -2.5$ mm and $-7.5$ mm. Well-defined peaks in the band-integrated RMS are interpreted as the results of shocks impinging on the dorsal isolator surface, and their locations are given in Table 3.

Table 3 Shock positions and intermittent region length

The average difference between the transducer and PSP band-integrated RMS pressures at the $y = -2.5$ mm profile is 54.9% and the largest difference is 144.3%, which occurs at the $x =$ 68.5 mm transducer. The large difference between the transducer and PSP RMS measurements is attributable to the low spectral SNR of the PSP in regions without separation or shock impingement and strong gradients in RMS adjacent to some of the transducers. The PSP overcalculates the RMS compared to the transducers at $x =$ 28.5, 45.6, 68.5, and 114 mm; locations with low spectral SNR (Fig. 11). The transducers at $x =$ 5.74 mm and 17 mm are located in regions of high RMS gradients, which could explain the lack of agreement with the PSP. While differences exist between the two measurement techniques, these results show the benefit of PSP compared to traditional point sensors as give information on important flow features on the order of millimeters away from the sensor would otherwise be missed.

Skewness, which describes the asymmetry of a distribution about its mean (Mears et al. 2019), was calculated for the unsteady pressure distribution. Shocks are particularly well resolved by skewness due to their unsteady oscillations about their mean and the sharp pressure rise across them. High skewness indicates a small percentage of samples have a disproportionate impact increasing the mean. In relation to SWBLI intermittency, this indicates the region is upstream of the mean shock location and rarely sees the pressure rise behind the shock. Conversely, low skewness indicates a small percentage of samples have a disproportionate impact decreasing the mean. This would be associated with the region downstream of the mean shock location. Zero skewness indicates that the signal distribution is symmetric, and the shock spends equal time upstream and downstream of this position. This is defined as the mean shock location obtained from skewness.

Fig. 14

Skewness distribution for Re $= 10.2 \times 10^6$ /m

Fig. 15

Streamwise skewness distribution for Re $= 10.2 \times 10^6$ /m

The skewness distribution for Re $= 10.2 \times 10^6$ /m is given in Fig. 14. Shocks identified from skewness are at $x \approx$ 5 to 16, 35 to 36, 76, and 89 mm. Curiously, the majority of the skewness contour outside of the SWBLI regions is slightly positive. This may be due to a non-equilibrium turbulent boundary layer in the SWBLI interaction region and recovery region which has not had sufficient streamwise distance to recover from multiple SWBLI. The increased importance of pressure diffusion and pressure strain in theses regions (Vyas et al. 2019), in addition to this being a confined flow rather than external, may also contribute to non-zero skewness away from the SWBLI.

Skewness profiles are given in Fig. 15. The mean shock location is the spatial location with zero skewness. The region over which the shock moves is bounded by the maximum value of skewness directly upstream and the minimum value of skewness downstream (Running and Juliano 2021b). The shock locations and the intermittent region lengths are given in Table 3.

Figure 12 shows the locations PSDs are plotted for Figs. 16 (circles), 17a (squares), and Fig. 17b (diamonds). A comparison of spectra between the transducer at $x =$ 91.5 mm and four PSP pixels 2 mm away from the transducer is shown in Fig. 16a. Overall, the PSP measurements shows good agreement with the pressure transducer measurements, with the PSP positions at the same axial position as the transducer showing the best agreement. This transducer is underneath the third throat-bottom SWBLI, with a peak frequency of 900 Hz followed by a decrease in power as frequency increases. The spectral SNR drops below one for the downstream PSP position at approximately 3.5 kHz (Fig. 16b). This matches what is observed in its spectrum, with the PSP PSD beginning to increase in amplitude compared to the transducer starting at 2 kHz, signifying that its spectrum are dominated by noise at these frequencies.

Figure 17a shows the spectra of the PSP along a $y = -2.5$ mm profile; highlighting the spectral content of SWBLIs whose locations were identified by peak band-integrated RMS. The spectrum underneath the Busemann throat shock has high power broadband frequency content, with significantly more power than the other SWBLI in the frequencies below 900 Hz and above 2.4 kHz. The spectra underneath the throat reattachment and throat-bottom shocks, at $x =$ 18.1 mm and 35.9 mm, respectively, have broad peaks from 0.8 kHz to 2 kHz, with a drop off in power at higher frequencies. The spectra underneath the isolator compression and third throat-bottom shocks, at $x =$ 76.5 mm and 89.2 mm, have a similar shape but their peaks are more pronounced at frequencies around 1 kHz.

Figure 17b shows the spectra of the PSP along a $y = -7.5$ mm profile: note that the Busemann throat and throat reattachment shocks could not be distinguished from the band-integrated spectra at this spanwise position, and four spectra are shown. The third throat-bottom shock’s spectrum $x =$ 11 mm is significantly less broadband than its $y =-2.5$ mm equivalent, with much lower power at low and high frequencies and a peak at 900 Hz. The throat-bottom shock’s spectrum at $x =$ 34.4 mm is similar to the other profile with a more pronounced peak at 900 Hz. The isolator compression shock’s spectrum at $x =$ 78.7 mm has lower power across all frequencies than its equivalent at the other azimuth. The third throat-bottom spectrum at $x =$ 89.2 mm has a very similar shape and amplitude as its $y = -2.5$ mm equivalent.

Fig. 16

Comparison between transducer and PSP spectra at $x =$ 91.5 mm. Re $= 10.2 \times 10^6$ /m

Fig. 17

PSP spectra at various streamwise positions for Re $= 10.2 \times 10^6$ /m

Fig. 18

Coherence contours averaged from 900 to 1200 Hz. Re $= 10.2 \times 10^6$ /m

6.2 Coherence results

Coherence is the measure of the linear relationship between two time series in the frequency domain (Stoica et al. 2005). It is useful for investigating SWBLI because it provides a frequency-dependent assessment of the correlation between two signals, pointing to potential communication pathways between separate SWBLI (Funderburk 2019; Running and Juliano 2021b; Johnson et al. 2022b; Jenquin et al. 2023). The magnitude-squared coherence spectrum $\gamma _c^2$ can be defined:

$$\begin{aligned} \gamma _c^2 = \frac{|S_{12}(f)|^2}{S_{11}(f)S_{22}(f)} \end{aligned}$$

(11)

where $S_{12}$ is the cross-spectral density between the signals, and $S_{11}$ and $S_{22}$ are the power spectral densities of each signal. A magnitude-squared coherence spectrum value of 1 indicates a perfect linear relationship, while a value of 0 indicates no relationship between signals. The phase spectrum is:

$$\begin{aligned} \phi (f) = \arctan \left( \frac{Q_{12}(f)}{C_{12}(f)}\right) \end{aligned}$$

(12)

where $Q_{12}$ is the quadrature spectral density function and $C_{12}$ is the coincident spectral density function. A phase spectrum value of 0 indicates the signals are perfectly in phase while a value of $\pi$ indicates the signals are perfectly out of phase. The phase spectrum is useful for determining the order SWBLI fluctuate due to disturbances and determining the time delay between this movement.

The same unsteady pressure signal, number of windows, and time interval used in the PSD calculation was used to calculate coherence. Numerous locations were investigated for the coherence spectrum with the following presented for brevity. The results of the coherence analysis taking a pixel underneath the isolator compression shock ($x = 76.5$ mm, $y = -0.25$ mm) are presented in Fig. 18. Calculating coherence relative to the isolator compression shock shows the linear relationship between the unsteady pressures underneath most of the SWBLI. Fig. 18a shows the average value of the magnitude-squared coherence spectrum over the 900–1200 Hz frequency band. These frequencies were selected because they showed the highest correlation for the various SWBLI in the isolator. The unsteady pressure underneath the throat reattachment, throat-bottom, and third throat-bottom shocks are coherent with the unsteady pressure underneath the isolator compression shock — $\gamma _c^2$ $\ge$ 0.5. There is additionally a region of elevated coherence between the isolator compression and third throat-bottom shocks. Note there is unexpectedly low coherence between the Busemann throat shock and the rest of the other shocks. The reason is currently unknown, but may have to do with the Busemann throat shock impinging on the throat’s separated flow. As many streamtraced inlets utilize truncated Busemann parent flowfields, it is possible this nonlinear relationship between the Busemann throat shock and others would be seen in other streamtraced inlet/isolator configurations.

The corresponding phase spectrum is presented in Fig. 18b and is also averaged from 900 to 1200 Hz. Defects (noise) in the phase spectrum are due to difficulties processing the signal’s phase in regions of low coherence. As seen, the isolator compression and third throat-bottom shocks have negligible phase difference. The region in between these shocks is almost perfectly out of phase with the shocks flanking it. There is non-zero positive phase for the throat-bottom shock and increased positive phase for the throat reattachment shock relative to the isolator compression shock. This implies that the propagating disturbance is traveling downstream, and there is a time delay between the perturbations for each successive shock. This observation is similar to previous studies, which attributed this phenomenon to the convection of boundary-layer structures in isolator flows with and without back pressure (Sugiyama et al. 1988; Xiong et al. 2018; Leonard and Narayanaswamy 2021).

6.3 Proper orthogonal decomposition results

A snapshot Proper Orthogonal Decomposition (POD) method was applied to the unsteady pressure in the style of Gordeyev et al. (2014). A space-averaged correlation matrix $A(t,t')$ was calculated:

$$\begin{aligned} A(t, t') = \overline{p'(s,t)p'(s,t')} \end{aligned}$$

(13)

where the over bar denotes spatial averaging and $t'$ is the shifted time. Next, a set of temporal modes $b_k(t')$ and their energies $\lambda _k$ were calculated by solving the eigenproblem:

$$\begin{aligned} \frac{1}{T}\int \limits ^T_0A(t,t')b_k(t')ds = \lambda _k b_k(t) \end{aligned}$$

(14)

The spatial modes $\varPhi ^{(k)}(s)$ were found from:

$$\begin{aligned} \varPhi ^{(k)}(s) = \frac{1}{T}\int ^T_0u(s,t)b_k(t)\textrm{d}t \end{aligned}$$

(15)

Modes were generated in order of their relative energy content $\lambda _{kn}=\lambda _k/\sum ^N_{i=1}\lambda _i$. In this case, N = 4000 modes were computed. To reduce low energy noise from the system, the unsteady pressure signal can be reconstructed using a reduced number of modes:

$$\begin{aligned} U_{rec}(s,t) = \sum _{i=1}^M a_k(t)\varPhi ^{(k)}(s) \end{aligned}$$

(16)

where the temporal coefficients are $a_k(t) = \int u(s,t)\varPhi ^{(k)}(s)\textrm{d}s$ and M is the number of modes used for the reconstruction.

Figure 19 shows the fraction of the total energy each mode has and the cumulative sum of that mode and all previous modes’ energies. The first mode has 7.7% of the total energy, the first 206 modes have 50% of total energy, and the first 1499 modes have 85% of total energy. The distribution of energy across so many modes is likely due to the highly unsteady turbulent boundary layer and SWBLI with varying length scales when scaled at the global level. The lack of concentration of energy into a few modes makes it impractical to reduce shot noise by reconstructing the unsteady pressure signal with a limited number of modes, as significant information about the true flow dynamics will be lost. However, reconstructing a signal that captures the propagating disturbance, while reducing noise and smaller scale dynamic content, is valuable as it conditions the data for shock-tracking methods. Therefore, the unsteady PSP pressure field was reconstructed with the first 15 modes, corresponding to 29.3% of the total energy. The mean pressure contour was spatially averaged with the same moving circle filter, 1.19 mm in diameter, as the unsteady data. It was then added to the reconstructed unsteady pressure field to yield a reconstructed instantaneous pressure field at each point in time, $\widetilde{p}'$. This reconstructed pressure is compared to the raw pressure across the centerline in an included supplemental video. The POD-reconstructed pressure field has significantly reduced noise compared to the raw signal while still capturing the shock motion induced by the propagating disturbances.

Fig. 19

Energy of the POD modes for Re $= 10.2\times 10^6$ /m

6.4 Shock-tracking results

A shock-tracking algorithm utilizing the POD-reconstructed instantaneous pressure $\widetilde{p}'$ was used to identify movement of the isolator compression shock. Streamwise pressure profiles were taken from the reconstructed instantaneous pressure. These profiles were averaged over a span of $y = -2.5$ mm to $-4.1$ mm and had a streamwise moving average of five pixels applied to the profile. Finally, the streamwise pressure gradient was calculated by taking the difference in pressure for each successive streamwise pixel. The maximum of the streamwise pressure gradient over the range of $x \approx$ 69 mm to 84 mm was taken to be the position of the isolator compression shock for that frame $x_s$, and the shock position was calculated over 0.4 s for the steady portion of the run. A representative profile for a single frame is given in Fig. 20, with the isolator compression shock at that frame identified by the circle marker.

Fig. 20

Streamwise $\varDelta p$ profiles from reconstruction of POD modes for Re $= 10.2\times 10^6$ /m

The average isolator compression shock position, $\bar{x_s}$, calculated by shock-tracking algorithm was 76.9 mm. The intermittent region was taken as the region within which the shock was present 99% of the time: $2\cdot 2.58\cdot \sigma _{x_s}$. The length of the intermittent region was 2.47 mm according to the shock-tracking algorithm, 15% lower than the calculation from skewness. The spectra of $x_s$ is compared to the spectra of the unsteady pressure underneath the isolator compression shock ($x =$76.5 mm, $y = -2.5$ mm) in Fig. 21. The $x_s$ spectra calculation parameters were exactly the same as when calculating the pressure spectra, and has units of $\textrm{mm}^2/\textrm{Hz}$. Note that the $x_s$ spectra has been scaled by a constant factor across all frequencies to better compare the shape of the spectra; the amplitude cannot be compared directly, as the spectra are calculated from different quantities. The spectral shape of the unsteady pressure is very similar to that of $x_s$, with both spectra having a peak frequency at 900 Hz.

Fig. 21

Comparison of $x_s$ and pressure spectra under the shock

Conditional averaging was conducted to demonstrate that the shock-tracking algorithm accurately detected frames containing the propagating disturbances. The raw pressure data with no temporal or spatial filtering was averaged based on three conditions identified from the shock tracking algorithm. The three conditions were: shock forward ($x <$76.5 mm), shock back ($x >$77 mm), or shock in its approximate median position (76.5 mm$< x <$77 mm). Figure 22 shows the pressure profile focused on the isolator compression and third throat-bottom shock region, conditionally averaged under the three different conditions. The shock-forward condition results in the rise in pressure at each shock occurring furthest upstream. The shock-back condition results in the the rise in pressure at each shock occurring furthest downstream. The pressure profiles obtained from conditional averaging exhibited the expected outcome, and thus the shock-tracking algorithm is accurately able to identify moments when the propagating disturbances are present.

Fig. 22

Conditionally average streamwise pressure profiles for Re $= 10.2\times 10^6$ /m

Fig. 23

Streamwise mean pressure profiles at $y = -$2.5 mm

Cross-correlations between the transducers at $x =$ 28.5 mm and 91.5 mm, adjacent or underneath the throat-bottom and isolator compression shocks, were used to calculate the velocity of the propagating disturbance. A cross-correlation using 0.01 s of $p'$ data was calculated for each propagating disturbance occurrence in time. A propagating disturbance occurrence was defined to be a moment where the isolator compression shock was downstream of x = 77 mm, as detected by the shock-detection algorithm. The maximum of the normalized cross-correlation was identified and its time lag was taken to be the time delay between the two sensors. Intervals whose maximum normalized correlation coefficient was at least 0.3 were included. The average time lag, $\varDelta t$, was used to calculate the disturbance propagation speed:

$$\begin{aligned} U_{d} = \frac{\varDelta x}{\varDelta t} \end{aligned}$$

(17)

where $\varDelta x$ is the distance between the two transducers, 63 mm. The average disturbance speed was 855 m/s. Due to the temporal resolution of the transducers which were sampled at 250 kHz, the uncertainty in time delay was 2 $\mu$s, and the corresponding uncertainty in disturbance speed was 33 m/s. The average core-flow velocity between these two sensors according to qualitatively similar IDDES computations is 749 m/s (Shuck et al. 2023). This disturbance propagates at 114% of the computed core flow velocity, larger than in previous investigations (under different conditions), in which the disturbances convected slightly slower than the core-flow velocity (Leonard and Narayanaswamy 2021). However, there is significant uncertainty in the core-flow velocity found via the IDDES simulations, which lack a transition model and may over/underpredict separation bubble size (Shuck et al. 2023). Therefore, the authors cannot rule out that this propagating disturbance convects at a similar normalized speed to previous studies, and emphasize the need for improved computational models for streamtraced inlet/isolators.

6.5 Freestream unit Reynolds number comparison

Figure 23 compares the mean pressure profiles at $y = -$2.5 mm for Re$=$ 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m. Overall these profiles are extremely similar, showing that a 20% increase in unit Reynolds number doesn’t appreciably affect the mean flow structure. Also included is a pressure profile of the INlet at Re$=$ 8.8 $\times 10^6$ /m collected at Purdue in BAMQ6T (Bustard et al. 2023a). The BAMQ6T profile is very similar to the ANDLM6QT profiles despite the different facilities, a different anodized plate, and the lack of temperature correction for the BAMQ6T data. The transducer measurements between the two facilities show qualitative agreement, but show worse agreement compared to the PSP profiles. Five of the seven transducers’ normalized mean pressures in BAMQ6T are higher than those measured in ANDLM6QT. The exact reason for this is unknown; slight differences in angularity, freestream noise, Mach number, or unresolved transducer temperature sensitivity effects in BAMQ6T could help explain the difference in results between the two facilities.

Fig. 24

Streamwise band-integrated RMS profiles at $y = -$2.5 mm

Figure 24 compares the band-integrated RMS profiles at $y = -$2.5 mm for Re$=$ 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m. These profiles are very similar. The magnitude of the RMS fluctuations tends to decrease as the freestream unit Reynolds number is increased. Additionally, it appears the shocks are moved slightly downstream as Reynolds number increases. This is likely due to slightly thinner boundary layers at higher Reynolds numbers, which increases the core flow area and reduces the confinement of the flow. This increased core flow area results in a larger vertical distance for the shocks to travel before interacting with the boundary layer, resulting in shock impingements farther downstream. The shocks, as identified by the peak RMS values, occur an average of 0.34 mm farther downstream for Re $= 10.2 \times 10^6$ /m compared to Re $= 8.5 \times 10^6$ /m. Qualitatively, the BIRMS profile at Re $=$ 8.8 $\times 10^6$ /m in BAMQ6T is very similar to those of the PSP data collected in ANDLM6QT. Note the minimum resolvable BIRMS is lower for BAMQ6T. Higher intensities were measured for the BAMQ6T data, increasing the SNR and decreasing the effect shot noise had on the integrated spectra.

7 Discussion

This study employed a variety of complementary PSP data analysis methods for a Mach 6 inlet/isolator study. High spatial resolution time-averaged results were used to identify the positions of flow features. Instantaneous pressure movies were useful for picking out trends in the unsteady shock motions, namely the existing of the propagating disturbance. The contour plot of maximum resolvable spectral SNR is a novel graphical representation based on the procedure introduced by Funderburk and Narayanaswamy (2019), and is well-suited to determining the validity of spectra/integrated spectra results that are displayed in a contour form. Using skewness and band-integrated RMS results, the mean shock location was determined with over an order of magnitude greater precision compared to conventional pressure transducers. SWBLI intermittent region length was determined from a profile of the skewness of the fluctuating wall pressure. The use of power spectral densities for investigating low-frequency content underneath SWBLI was improved using this more precise estimate for shock location, as the spectrum was interrogated within ± 0.085 mm of the mean shock.

Coherence analysis has been used in PSP studies of inlet/isolators before (Johnson et al. 2022b). However, the contour plots of average magnitude-squared coherence spectrum and phase magnitude are novel for inlet/isolators investigations, despite similar techniques being used for external SWBLI investigations (Funderburk 2019; Running and Juliano 2021b; Jenquin et al. 2023). A comparison of magnitude-squared coherence spectrum at every pixel revealed that the fluctuating pressure under the Busemann Throat shock was not correlated with the rest of the shocks in the isolator, which was an unexpected result found using the global contour representation afforded by the PSP. Snapshot POD analysis of this turbulent inlet/isolator showed that the POD mode energy was distributed across many modes, rather than being concentrated in a select few. While this makes fully reconstructing the turbulent inlet/isolator with severely reduced shot noise impractical, POD-reconstruction neglecting 70.7% of the total mode energies was beneficial for tracking the propagating disturbance. Shock tracking provided an estimate for the SWBLI intermittent region length and was used to identify instances when the propagating disturbance was present in the isolator. Using conditional wall pressure time correlations informed by the PSP shock-tracking, the speed of the propagating disturbance was determined. After the Re $= 10.2 \times 10^6$ /m test condition had been analyzed, mean and band-integrated RMS profiles were used to demonstrate the impact of Re and compare results collected in ANDLM6QT to those collected in an earlier BAMQ6T entry. This study has leveraged an unprecedented number of analysis techniques compared to existing inlet/isolator PSP studies, demonstrating an extended utility of PSP for internal flow investigations.

8 Conclusions and future work

Fast pressure-sensitive paint was applied to the Indiana Inlet to investigate the shock location, shock spanwise variation, and inlet shock dynamics for a streamtraced inlet/isolator. The model was tested under Mach-5.7 flow at Re = 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m. A five-hole probe was utilized to determine the model’s angle of attack and yaw. The measured angle of attack was $0.14^{\circ }$ ± 0.$38^{\circ }$, and the measured yaw angle was $1.47^{\circ }$ ± $0.38^{\circ }$. Temperature-sensitive paint was utilized to temperature-correct the PSP’s intensity ratios. The dorsal isolator surface pressure was measured using time-resolved anodized-aluminum PSP sampled at 10 kHz and calibrated a priori at every pixel.

The keys results from these experiments are:

• (i) The INlet’s top wall temperature increase during the experiments was measured to be a modest 0.5–1 K under these conditions.

• (ii) INlet flow features, such as SWBLI and expansion waves, impinging on the top isolator wall were visualized with PSP. Identification of these flow features was achieved using 3D IDDES computations from Shuck et al. (2023).

• (iii) Mean pressures between the centerline transducers and PSP showed good agreement, with an average difference of 9.6%. Significant spatial gradients in pressure, occurring on order millimeter length scale, made direct comparisons with the accompanying wall pressure transducers more challenging, especially near the SWBLI locations. This highlights the importance of using global surface measurement techniques and/or large numbers of precisely placed point sensors to study phenomena in these inlet/isolator flows.

• (iv) Coherence analysis showed a strong linear relationship between the unsteady pressure underneath all SWBLI(s) in the isolator, with the exception of the Busemann throat shock.

• (v) Downstream propagating disturbances caused successive motion of the isolator shocks and occurred at frequencies of 900–1200 Hz. The velocity of these disturbances was found to be 114% of the core-flow velocity, a value higher than that found in other studies (Leonard and Narayanaswamy 2021). However, improved calculations of the core-flow velocity are needed to refine this assessment.

• (vi) Negligible differences were seen in the pressure distributions when comparing the Re $=$ 8.5 $\times 10^6$ /m and 10.2 $\times 10^6$ /m runs. PSP measurements of the INlet in BAMQ6T (under noisy flow) and ANDLM6QT were also very similar, with strong agreement between shock impingement location and pressure levels.

Future work is being planned to further characterize the convection of the propagating disturbance. The reason for the lack of the linear relationship between the unsteady pressure underneath the Busemann throat shock and the rest of the shocks should also be identified. Additional studies investigating the difference between INlet flows under quiet and noisy flow are planned. Using temperature-corrected PSP data for these investigations would verify the conclusions found by Bustard et al. (2023a). Studies will be conducted with jet blowing to investigate unstart of this inlet using PSP. Emphasis will be on measuring shock-train leading edge velocities during unstart with higher spatial resolution and more focus on spanwise variation than previous studies. Jet blowing studies will establish the magnitude of temperature increase unstarting flow induces on the top wall, which will influence the temperature-correction of the PSP. Several unstart-detection methods will be compared, which may inform the design of active control systems to avoid unstart.