3.3.1. Power spectral density

The PSD of the surface pressure fluctuation was computed over the entire measurement domain to characterize both the streamwise and spanwise variations of the surface pressure dynamics caused by the three-dimensional flow motions. Figure 5 shows the PSD of the incoming boundary layer over different azimuthal locations. The coordinates are normalized to compare the PSD with the literature. Figure 5 shows that the PSDs across all the azimuthal locations are nearly identical, evidencing the two-dimensional nature of the inflow boundary layer. The PSD exhibits a decay at very low frequencies followed by a plateau for non-dimensional frequencies greater than $10^{-3}$.

Figure 5. Comparison of wall pressure fluctuation PSD obtained beneath the incoming boundary layer at different azimuthal locations. Comparisons with previous measurements from Beresh et al. (Reference Beresh, Henfling, Spillers and Pruett2011) and the empirical predictions of Lowson (Reference Lowson1968) are also presented.

The empirical curve fit of Lowson (Reference Lowson1968) also predicts a PSD plateau at very low non-dimensional frequencies and the predicted PSD magnitude by the curve fit is presented in figure 5 along with the more recent measurements of a Mach 2.5 boundary layer PSD by Beresh et al. (Reference Beresh, Henfling, Spillers and Pruett2011). Comparing the plateau with the literature shows that the measured plateau is over an order of magnitude higher than the curve fit of Lowson (Reference Lowson1968) and Beresh et al. (Reference Beresh, Henfling, Spillers and Pruett2011). One possibility is that the value of the displacement thickness $\delta ^*$ in the normalization was estimated as 1/8 of the boundary layer thickness ($\delta$) as suggested by Lowson based on incompressible boundary layer datasets; typically, $\delta ^*$ of the compressible boundary layers are a higher fraction of $\delta$.

Another possibility for the discrepancy is that the noise floor of the PSD becomes dominant and the plateau that is obtained in figure 5 is simply the noise floor. To evaluate this possibility, wall pressure measurements were obtained using Kulite transducers within the incoming boundary layer. Figure 6 shows the comparison between the frequency-premultiplied PSD of the pressure fluctuations obtained from the PSP and Kulite transducers; physical units are retained in figure 6 to make a direct comparison. It can be observed the measured PSDs from both PSP and transducers agree very well with one another over the entire frequency range that could be resolved. A noticeable difference is the spike in the PSD from the PSP method at 200 Hz. This spike pervades the PSD across the entire measurement domain, and it does not correspond to a physical value, as seen from its absence in the Kulite measurements. In the remaining sections, the spike is ignored in the discussions that accompany the PSD. It should also be noted that the incoming boundary layer poses the greatest limitation on the dynamic pressure measurements using PSP within a SBLI unit because of the very small pressure fluctuations in the incoming boundary layer, as shown by Funderburk & Narayanaswamy (Reference Funderburk and Narayanaswamy2019b). The fact that there is a very good agreement between the PSP- and transducer-based PSD measurement provides enough confidence about the accuracy of the PSD in other locations of the SBLI units that is presented subsequently.

Figure 6. Direct comparison of the frequency-premultiplied wall pressure fluctuation PSD obtained using PSP and pressure transducer beneath the incoming boundary layer, measured at the same location.

The corresponding frequency-premultiplied pressure fluctuation PSDs within the separation bubble obtained at $x = -11.6\ {\rm mm}\ (x \approx -3 \delta )$ are presented across different azimuthal locations in figure 7(a). The PSD exhibits a monotonically increasing trend with frequency, which suggests that the peak of the PSD occurs above the frequency range that could be measured by the PSP. It should be noted that there is indeed a significant low-frequency content in the separation bubble PSD as was observed in earlier works (e.g. Pasquariello et al. Reference Pasquariello, Hickel and Adams2017). The reason why the low-frequency content is not emphasized in figure 7(a) is because the plots are frequency-premultiplied PSD. We note that it is possible that the low-frequency content is too weak in the frequency-premultiplied PSD such that the current measurements cannot capture the peak. Between the different azimuthal locations, the PSDs are nearly identical to one another. The overall spread in the PSD between $\phi = 0^\circ$ and $\phi = -45^\circ$ never exceeded 20 % within the measured frequency range, which demonstrates a consistent evolution of the shear layer over the separation bubble along the azimuthal direction.

Figure 7. Frequency-premultiplied wall pressure fluctuation PSD obtained at characteristic locations within the SBLI unit over different azimuthal locations: (a) separation bubble ($x \approx -3\delta$), (b) intermittent region ($x \approx -9.5\delta$) and (c) reattachment location ($x \approx +2.1\delta$).

The intermittent-region PSDs, obtained at $x = -8.5 \delta$, are presented in figure 7(b) to determine how the separation shock pulsations vary in the azimuthal direction. Both physical frequency unit and a Strouhal number scale based on the spanwise-average separation length are presented. The PSDs in the vicinity of the azimuthal centre exhibit a broadband peak at 350 Hz ($St_L = 0.0193$) and the broadband PSD reaches a minimum at 4 kHz ($St_L \approx 0.24$). These peak Strouhal numbers are within the range of $St_L = 0.02 \pm 0.01$ reported in Dolling (Reference Dolling1993) and Dupont et al. (Reference Dupont, Haddad, Ardissone and Debieve2005) who collected multiple experimental works on unswept and swept compression ramp interactions and impinging shock interactions. The minimum in the PSD is followed by another elevation in the PSD that extends until (and beyond) the maximum frequency that could be measured. This elevation is likely caused by the passage of the boundary layer structures that causes jitter motions in separation shock, as suggested by Erengil & Dolling (Reference Erengil and Dolling1991b). Overall, the PSDs were very consistent between $\phi = 0^\circ$ and $\phi = -45^\circ$ and the maximum shift in the peak $St_L$ did not exceed 50 Hz ($\Delta St_L \approx 0.0025$).

The pressure PSDs of the reattachment region that capture the dynamics of reattachment shock, obtained at $x = +2.1 \delta$, are presented in figure 7(c). The PSDs in this region exhibit a monotonically increasing PSD with frequency with a broad hump at 7 kHz ($St_L \approx 0.4$). The overall magnitude of the PSD is noticeably higher than the intermittent-region PSD, which is caused by the elevated pressures in the reattachment region. Furthermore, the peak Strouhal number of the reattachment-region PSD is approximately 20 times higher than the intermittent-region peak Strouhal number. This ratio is similar to the value quoted by Estruch-Samper & Chandola (Reference Estruch-Samper and Chandola2018) (${\approx }33 \times$) on a forward-facing step SBLI. The reattachment-region PSD also exhibits excellent agreement over the different azimuthal locations presented. The greatest difference occurs at $\phi = -45^\circ$, where the PSD is lower by about 40 % compared with that at $\phi = 0^\circ$. This reduction is believed to be caused by a combination of azimuthal flow motions and the mean azimuthal undulations in the reattachment region that were also noticed in the surface streakline patterns (figure 4b).

3.3.2. Streamwise evolution of the PSD

The streamwise evolution of the frequency-premultiplied pressure fluctuation PSD is presented along $\phi = 0^\circ$ in figure 8. The incoming boundary layer is captured over an extent of $7.5 \delta \ (-18 \delta \leqslant x \leqslant -10.5 \delta )$ and the PSD contours show nearly horizontal colour bands along the streamwise direction. These horizontal bands show that the PSD is essentially identical and maintains the same peak frequency, which was not resolved in the measurements. The intermittent region is seen as the region of elevated PSD in the low-frequency bands over $-10.5 \delta \leqslant x \leqslant -8.0 \delta$. The length scale of the presence of the low-frequency bands can provide another measure of the length scale of shock motion; this is determined to be 10 mm ($2.5 \delta$ or $0.3 \times L_{sep}$). Together with the peak Strouhal number of 0.019, the characteristic shock velocity is estimated at $7.4\ {\rm m}\ {\rm s}^{-1}$ or 1.2 % of the free-stream velocity. This estimate is lower than the typical 2 % that is reported in the literature; however, lowering of the separation shock velocity with increasing separation size is consistent with Estruch-Samper & Chandola (Reference Estruch-Samper and Chandola2018).

Figure 8. Streamwise evolution of frequency-premultiplied pressure fluctuation PSD obtained at $\phi = 0^\circ$. The abrupt occurrence of the PSD elevation at $St_L \approx 0.2\unicode{x2013}0.4$ is indicated by an arrow.

Within the separation bubble, the PSD shows that the individual colour bands show a linear downward tilt in frequency with downstream distance. Noting that $p_{rms}/p_w$ remains constant along the separation bubble and invoking the self-similar nature of the frequency-multiplied PSD within the separation bubble reported by Estruch-Samper & Chandola (Reference Estruch-Samper and Chandola2018), this downward tilt in the PSD bands shows a downward shift of the peak frequency with downstream distance along the separation bubble. Furthermore, noting that the pressure fluctuations within the separation bubble are predominantly caused by the shear layer above the separation bubble, the shift in the peak frequency demonstrates the growth of the characteristic eddy length scale with downstream distance. These findings have been documented in the literature for different shock-induced separation units. It should be commented that our pressure fluctuation PSD does not resolve the peak frequency of the separation bubble from its inception downstream of the separation shock until the ramp leading edge.

For the compression ramp interactions, the available experimental data on peak frequency extend only until the ramp leading edge because of the difficulty with incorporating the transducers on the compression ramp. In this regard, the present work provides a new insight into the pressure dynamics downstream of the compression corner. Figure 8 shows that a new band of PSD elevation in the Strouhal number range $St_L = 0.2\unicode{x2013}0.4$ occurs just downstream of the compression ramp leading edge (indicated by an arrow). This new band occurs rather abruptly and sharply departs from the linearly decreasing trend of the peak Strouhal number of the separation bubble PSD that occurred until the ramp leading edge. Downstream of the ramp leading edge, the $St_L = 0.2\unicode{x2013}0.4$ band maintains its strength but shifts to slightly higher frequency until mean reattachment; the strength of the band quickly dissipates downstream of the reattachment. It should be noted that the relaxing boundary layer has not recovered to a canonical boundary layer PSD until the ramp elbow located $3.5 \delta$ downstream of the mean reattachment line.

A similar abrupt appearance of the $St_L = 0.2\unicode{x2013}0.4$ band was also observed downstream of the compression corner in large-eddy simulations of Grilli et al. (Reference Grilli, Schmid, Hickel and Adams2012) as well as in an impinging SBLI configuration downstream of the theoretical incident shock impingement location reported by Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017). This, however, contradicts the observations made in the forward-facing step configuration by Estruch-Samper & Chandola (Reference Estruch-Samper and Chandola2018) where the authors did not report a separate band of frequencies that is well below the shear layer frequency. More discussions on the distinguishing flow features are presented in the subsequent sections. Overall, the PSD evolution along the SBLI unit shows excellent commonalities with two-dimensional SBLI units and provides experimental support for the abrupt occurrence of the $St_L = 0.2\unicode{x2013}0.4$ band in the reattachment region that was discovered earlier by Grilli et al. (Reference Grilli, Schmid, Hickel and Adams2012) and Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017) in two-dimensional SBLI units.

3.4.1. Zero time lag cross-correlation and cross-coherence mapping

Zero time lag cross-correlation analysis was performed to determine the regions of the SBLI whose pressure fluctuations have appreciable positive or negative correlations with the reference location, chosen as the intermittent-region peak $p_{rms}/p_w$ location $x = -8.5 \delta$ at $\phi = 0^\circ$. This reference location is also maintained for the subsequent analysis. Figure 10 shows the contour of the zero time lag cross-correlation ($Cor (x,s)$) capturing both streamwise and azimuthal domains. A few select isocontour traces are included to emphasize the topology of the correlated region. It can be observed that the pressure fluctuations within the intermittent region exhibit very low correlation with the incoming boundary layer. This low correlation is expected with large separation sizes, wherein the separation shock is minimally impacted directly by the incoming boundary layer momentum fluctuations as stated by Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014) and others. Stepping into the intermittent region, figure 10 shows a region of positive correlation surrounding the reference location that stretches across $s = \pm 8 \delta$; the extent of $Cor (x,s) > 0.5$ extends between $s \approx \pm 4\delta$. The streamwise extent of the region of $Cor (x,s) > 0.5$ is approximately $2.5 \delta$, which is similar to the intermittent-region length obtained from $p_{rms}/p_w$ values.

Figure 10. Two-dimensional contour of the zero time lag cross-correlation across the SBLI unit with the intermittent region as the reference location.

Within the separated flow, a narrow passage of very low correlation can be observed in the early portions of the separated flow corresponding to where the shear layer is in its initial growth phase. Further downstream into the separated flow, appreciable negative correlation emerges, and the occurrence of the negative correlation statistically affirms the similar observation made with the instantaneous snapshots of the pressure fluctuations made in the earlier section. The negative correlation intensifies with downstream distance well into the reattachment region and the relaxing boundary layer. In fact, the highest negative correlation occurs within the relaxing boundary layer just downstream of the mean reattachment. This peak negative correlation location still lies more than $3 \delta$ upstream of the ramp shoulder where the expansion process can impact the correlations (Loginov et al. Reference Loginov, Adams and Zheltovodov2006). Figure 10 shows that the region of negative correlation ($Cor (x,y) < -0.2$) in the separated flow and downstream is spread over a wide azimuthal region ($-4\delta < s< +4\delta$) even though the highest negative correlation is confined to a small spread of azimuth.

Overall, the zero delay correlations provide very interesting pointers to the different interactions that potentially occur in the SBLI units investigated. First and foremost, the regions within the separation unit that are most correlated with the intermittent region occur near the ramp leading edge, reattachment region and just downstream. Second, the streamwise-elongated topology of the regions of high negative correlation indicate a potential role of the streamwise vortices that emanate in the vicinity of the reattachment and convected along the relaxing boundary layer. Such a role and accompanying mechanism of these Görtler-like vortices were also suggested by earlier researchers and the differences between those works and the current findings are elaborated in § 4. Third, the azimuthal spread of the negatively correlated regions shows that the separation shock at a given location is impacted by the events that occur over a significant azimuthal distance within the separated flow, reattachment region and the relaxing boundary layer. Juxtaposing this comment with the large azimuthal spread of the positively correlated region within the intermittent region raises two possible scenarios for how the pressure fluctuations within the reattachment region can influence the separation shock: (1) the pressure fluctuations originating at an azimuthally distanced location in the separated flow diffuse through the separation region and reach the reference location and/or (2) the pressure fluctuation originating at an azimuthally distanced location in the reattachment region impacts a different azimuthal location in the intermittent region, which then propagates in the azimuthal direction to the reference location. The flow processes that lead to these scenarios are nonlinear and are addressed in the subsequent sections.

Before the impact of the different possible regions on the separation shock dynamics is explored further, it is essential to learn if these regions couple with the intermittent region at the relevant frequencies of interest. This is delineated using cross-coherence analysis, which provides a normalized coupling coefficient of the pressure fluctuations between any two locations across the frequency spectrum. Whereas a coherence coefficient ($Coh (f)$) of one corresponds to a linear coupling between the given location and the reference location at that particular frequency, $Coh (f)$ of zero corresponds to no coupling at that frequency. Figure 11 shows the cross-coherence spectra along the streamwise direction for the baseline configuration with the reference location chosen at $x = -8.5\delta$ and $\phi = 0^\circ$. It can be observed that $Coh (f)$ within the incoming boundary layer is quite minor and does not exceed 0.15 across all frequencies. By contrast, the region surrounding the downstream part of the separated flow ($x > -4\delta$), the reattachment region and the relaxing boundary layer exhibit substantial levels of $Coh (f)$. Importantly, $Coh (f)$ exceeding 0.5 can be observed over the Strouhal number range up to 0.05, which is the range corresponding to the low-frequency pulsations of the separation shock foot. This confirms that the pressure fluctuations that occur at the downstream regions of the separated flow and beyond do couple with the separation shock motions at low frequencies and can impact the low-frequency pulsations of the separation shock foot.

Figure 11. Streamwise evolution of cross-coherence spectra across the SBLI with the reference location at the intermittent region.

3.4.2. Two-point cross-correlation analysis

Two-point cross-correlation was obtained over the entire measurement domain to learn how the pressure fluctuations are temporally organized within the SBLI unit with respect to the reference location at the intermittent region ($x = -8.5\delta$ and $\phi = 0^\circ$). Learning which regions of the SBLI lead and lag the intermittent region provides a building block to uncover the mechanisms that drive the separated shock motion. The results presented are an average 40 randomly sampled data snippets that were 1000 samples (25 ms) long. The uncertainty in the cross-correlation quantified as the 99 % one-sided confidence bound from the 40 sample data snippets is also presented.

Figure 12(a) presents the evolution of the two-point cross-correlation along the streamwise direction of the SBLI unit spanning the incoming through relaxing boundary layers. The cross-correlation field at any given time delay is qualitatively identical to the zero delay cross-correlation field (figure 10) in terms of the regions that correlate to greater or lesser extents with the reference location. Figure 12(b–d) presents the cross-correlations at select locations along $\phi = 0^\circ$ in the form of line plots to provide certain noteworthy details. Beginning with the cross-correlations at the incoming boundary layer presented in figure 12(b), it can be observed that spike corresponding to the shock jitter due to the passage of the turbulent structures could not be captured by the PSP technique; this limitation is imposed by the camera exposure time of $25\ \mathrm {\mu } {\rm s}$ and the inherent attenuation of high frequencies by the paint. With the jitter blocked out, the cross-correlation exhibits two broad negative peaks ($Corr \approx -0.15$), one where the separation shock leads the incoming boundary layer (which is unphysical) and a slightly stronger peak where the separation shock lags the incoming boundary layer by 0.2 ms. Remarkably, the negative correlation persists over $20 \delta$ upstream of the separation shock. Further, the correlation peak at $x = -29.5 \delta$ exhibits a marginal increase in delay and a slight reduction in the peak negative correlation magnitude compared with $x = -14.3 \delta$. A very similar correlation magnitude was also observed in earlier studies with similar separation scales (e.g. Chandola, Huang & Estruch-Samper (Reference Chandola, Huang and Estruch-Samper2017) in forward-facing step, Priebe & Martín (Reference Priebe and Martín2012) in compression ramp and Brusniak & Dolling (Reference Brusniak and Dolling1994) in blunt fin interactions). Furthermore, Poggie & Porter (Reference Poggie and Porter2019) demonstrated that the peak correlation between the incoming boundary layer velocity fluctuation and separation shock location occurs several characteristic boundary layer convection times earlier, consistent with the present measurement. The negative correlation extends over a time scale greater than 1 ms (see figure 12b), which is similar to the separation shock low-frequency pulsation frequency range. The significance of these observations in the context of the separation shock pulsations is expounded in § 3.4.3.

Figure 12. Two-point cross-correlation of different regions within SBLI with reference to the intermittent region. (a) Contour of streamwise evolution of the cross-correlation. Selections of cross-correlation plots from representative regions: (b) incoming boundary layer, (c) intermittent region and (d) separated flow, reattachment locus and relaxing boundary layer. Here ‘S’ and ‘R’ denote the curves corresponding to the mean separation and reattachment locations that were determined from the surface streakline image of figure 4(b). Ref., reference; Sep., separated.

We turn our attention to the intermittent region until the mean separation locus where a positive correlation was observed with the intermittent region at zero lag. Figure 12(c) shows the occurrence of two positive correlation peaks of similar magnitudes corresponding to the reference location leading and lagging, respectively. Tracking the progression of the time delay of the peaks enables computing the average propagation velocity of the pressure perturbations and their propagation direction. For example, the peaks corresponding to the positive time delay (‘reference location lags’) occur at increased delay with downstream distance, which shows that this corresponds to an upstream-propagating perturbation; the corresponding propagation velocity is determined to be ${\approx }0.15 \times u_\infty$ ($= 90\ {\rm m}\ {\rm s}^{-1}$). Earlier works by Gonsalez & Dolling (Reference Gonsalez and Dolling1993) and others used a boxcar approach that tracked the pressure fluctuations above a threshold within the intermittent region over multiple transducers to compute separation shock foot velocity. The cross-correlation is essentially a similar approach that takes both positive and negative pressure fluctuations at the limit of zero threshold. The propagation velocity from the cross-correlation analysis is substantially higher than the separation shock foot velocity of $(0.02\unicode{x2013}0.05) \times u_\infty$ reported in the literature (Gonsalez & Dolling Reference Gonsalez and Dolling1993; Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014). Furthermore, the propagation velocity from the cross-correlation analysis is also higher than the shock velocity obtained in § 3.3.1 based on the intermittent region and characteristic separation shock time scale ($0.015 \times u_\infty$). The peak corresponding to the negative time delay (‘reference location leads’) is determined to propagate downstream at an average velocity of $0.45 \times u_\infty$ ($\approx 260\ {\rm m}\ {\rm s}^{-1}$), once again being substantially greater than the shock propagation velocity.

Next, examining the downstream regions of the separated flow, the reattachment region and the downstream relaxing boundary layer (figure 12d), two distinct negative correlation peaks are observed, corresponding to the positive and negative time delays with respect to the separation shock. Figure 12(d) shows that the positive time delay branch (separation shock lags) has a negative peak at a maximum delay of 0.125 ms, which occurs in the reattachment region. Both upstream- and downstream-propagating disturbances can be observed to originate at this location and the propagation velocity was determined to be approximately $0.3 \times u_\infty$ ($\approx 180\ {\rm m}\ {\rm s}^{-1}$) for upstream and $0.35 \times u_\infty$ ($\approx 203\ {\rm m}\ {\rm s}^{-1}$) for downstream directions. The trajectory of the upstream-propagating branch was further tracked in figure 12(a) and was found to terminate just downstream of the shear layer initial growth region. Figure 12(a) also shows that the perturbations corresponding to the negative time delay branch (separation shock leads) originates just downstream of the shear layer initial growth region and propagates downstream through the separated flow and into the relaxing boundary layer as observed from the white contour marked with a black dashed line in figure 12(a). Once again, the downstream disturbance propagates at a velocity of ${\approx }0.5 \times u_\infty$ (${\approx }280\ {\rm m}\ {\rm s}^{-1}$) and the velocity is nearly constant along the separated flow and downstream, as discerned from the linear evolution of the peak time delay with distance of the negative time delay branch (black dashed line). The mechanisms that generate the different perturbations and the causes for the divergence in the propagation velocity discussed earlier are presented in § 4.

Before progressing to the next section, it is important to comment on the uncertainty in the propagation velocity of the various pressure perturbations. The main cause of uncertainty is the time discretization of 0.025 ms between the successive pressure fields. Because of the limited size of the SBLI unit, the greatest time lag between the correlation peaks of different locations is only a modest multiple of the time discretization. Furthermore, different entities within the SBLI unit offer different spatial extents from which to compute the velocities, which forms another contributing factor of the uncertainty. Hence, an estimate of the error bounds of the propagation velocity was obtained by considering the peak time lag shifted up and down by the time discretization. The resulting values are presented in table 4.

Table 4. Bounds of the propagation velocity of the various pressure perturbation branches identified.

3.4.3. Conditionally averaged pressure field sequence

The analysis performed so far delineated the various regions within the SBLI that are linearly coupled to the separation shock oscillations, as well as the different upstream- and downstream-propagating disturbances that lead and lag the separation shock motions. While these are indeed key insights that are examined further, important shortcomings of this analysis are that: (1) the flow processes within the SBLI are nonlinear that can cause potential deviations in the observations based on the linear analysis and (2) the analysis provides insufficient physical description of the mechanisms that drive the separation bubble pulsations. In this section, the pressure fluctuation fields within the SBLI unit that precede and succeed the separation shock motions are presented in detail to shed light on the nonlinear effects of the interactions as well as the mechanisms that drive the separation shock motion.

The fundamental challenge with analysing the bulk separation shock motions is that the separation bubble pulsations are broadband and are a superposition of different modes of oscillations as demonstrated by Adler & Gaitonde (Reference Adler and Gaitonde2018) and Priebe & Martín (Reference Priebe and Martín2012), among others. The bulk separation shock motion is often overlapped with local rippling caused by boundary layer structures and it is hard to distinguish if the discerned separation shock motion is due to the local rippling or a global pulsation. While many approaches are possible for studying the bulk separation shock motions, such as tracking the spanwise-average iso-pressure (e.g. Wu & Martin Reference Wu and Martin2008), the strategy presented here exploits the fact that the separation shock pressure fluctuations at a given location have strong azimuthal correlation. As a result, when a large number of pressure fields are conditionally averaged based on the local pressure fluctuation threshold, the conditional averaged quantity will iron out the spanwise ripples and reveal the bulk correlated pressure fluctuations across the azimuth. The implementation of this strategy leverages the large amount of data samples available from the experiments to downselect the instances of bulk separation shock motion that extends over a long duration while collecting adequate numbers of pressure field samples for converged results. The following describes the strategy employed.

The overall goal of the strategy is to isolate the positive and negative pressure excursion events in the intermittent region (interpreted as separation shock motions) and observe the conditionally averaged sequence of pressure fluctuation fields within the entire SBLI preceding and succeeding the pressure excursion over a time period corresponding to the peak separation shock pulsation frequency. The key requirement of the chosen instances to build the conditional average is that there should be no pressure fluctuations of the opposite sign in the intermittent region which can corrupt the statistical process over the interrogation duration. To this end, the entire pressure field sequence was low-pass-filtered with 2 kHz cut-off to remove the frequency content that substantially exceeds the low-frequency pulsations of the separation bubble. A reference region within the intermittent region was chosen as an average over a $2.4\ {\rm mm} \times 2.4\ {\rm mm}$ region, and the maxima or minima in the pressure fluctuations at the reference location were identified. Next, those maxima (minima) with positive (negative) pressure fluctuations that were separated from the preceding neighbour by more than a threshold of 2.5 ms were identified and downselected (annotated as ‘$B$’ and ‘$A$’ in figure 13). This downselection ensures that only one minimum (maximum) occurs between ‘$A$’ and ‘$B$’ (denoted as $M_{AB}$ in figure 13) and $M_{AB}$ is separated from the nearest positive (negative) pressure fluctuation by a reasonably long period of time. On an average, the nearest preceding and succeeding zero crossing of pressure fluctuation in the reference location was separated by approximately 1.5 ms from $M_{AB}$. Overall, approximately 900 individual realizations of maxima (and minima) were collected from more than 10 000 identified maxima (and minima) spanning four test runs. The run-to-run variations in the sequence were minimal and did not interfere with the results. The time instances of $M_{AB}$ were set as a reference time $\tau = 0$ and a sequence of conditionally averaged pressure fluctuation fields was obtained between $\tau = \pm 1.2\ {\rm ms}$ referenced to $\tau = 0$. The pressure fluctuation fields were normalized by their local $p_{rms}$ to emphasize the relative excursion to the r.m.s. value, which made a more equitable representation of both low-$p_{rms}$ and high-$p_{rms}$ regions within the SBLI. It should be noted the low-pass filtering at 2 kHz did not impact the overall conditional averaging sequence.

Figure 13. Schematic illustration of the relative temporal layout of the surrounding peaks ‘$A$’ and ‘$B$’ in relation to the local maxima/minima ‘$M_{AB}$’ that is referenced as $\tau =0$ for the conditional averaging of the pressure field time sequence.

Figure 14(a) presents the time sequence of $p'/p_{rms}$ at specific locations of interest and figure 14(b–e) shows a sequence of conditional average $p'/p_{rms}$ fields at different time delays corresponding to a positive pressure fluctuation in the intermittent region (separation shock moving upstream of its mean location). Supplementary movie 2 provides the sequence from which figure 14(b–e) was extracted. Figure 14(a) shows that $p'/p_{rms}$ even at $\tau = -2\ {\rm ms}$ exhibits a mild positive pressure value in the intermittent region of $+0.1$, which suggests that the pressure excursion of the intermittent region begins appreciably earlier than the instance of its peak. Correspondingly, the incoming boundary layer (measured $-2 \delta$ upstream of the intermittent region) and regions within the separated flow exhibit a modest negative $p'/p_{rms}$ at this time instant, as shown in figure 14(b) for $\tau = -0.45\ {\rm ms}$. Between about $\tau = -2\ {\rm ms}$ and $\tau = -0.3\ {\rm ms}$, there is a strong decrease in $p'/p_{rms}$ within the incoming boundary layer, as observed from figure 14(a), to reach a minimum of $p'/p_{rms} = -0.5$ at $\tau = -0.3\ {\rm ms}$. Figure 14(c), corresponding to $\tau = -0.35\ {\rm ms}$, shows that there is a groundswell of negative $p'/p_{rms}$ (purple contour) over the entire incoming boundary layer region within the measurement domain that extended over $-12 \delta$ upstream of the intermittent region and across the entire span of the test article. The corresponding $p'/p_{rms}$ in the intermittent region in this duration increased steadily to 0.7. The $p'/p_{rms}$ within the downstream regions of the separated flow and the reattachment region decreased to $-0.5$, which did not correspond to a minimum. Interestingly, the slope of $p'/p_{rms}$ curves in the downstream region is nearly identical to that of the incoming boundary layer, as seen in both figures 14(a) and 14(f). A closer inspection of figure 14(b) reveals an earlier onset of the negative pressure fluctuation in the reattachment region compared to upstream locations within the separated flow, as observed from a pale blue contour of negative pressure fluctuation that extends from the mean reattachment location and into the downstream relaxing boundary layer. This negative pressure fluctuation region can also be seen to spread into the separated flow upstream of the ramp leading edge by $\tau = -0.35\ {\rm ms}$ (figure 14c). Subsequently, whereas the reattachment location has started a steep decline in its pressure fluctuations starting at $\tau = -0.5\ {\rm ms}$, the location just upstream of the ramp leading edge (blue curve in figure 14a) has started to exhibit a steep decline in the pressure fluctuations a little earlier at $\tau = -0.4\ {\rm ms}$.

Figure 14. Time sequence of the conditionally averaged $p'/p_{rms}$ fields at various instants of separation shock motions. (a) Line plots of $p'/p_{rms}$ sampled at specific locations of interest corresponding to positive $p'/p_{rms}$ in the intermittent region. Corresponding two-dimensional contours of $p'/p_{rms}$ within the SBLI region at specific instances of time with respect to peak of $p'/p_{rms}$ in the intermittent region: (b) $\tau = -0.45\ {\rm ms}$, (c) $\tau = -0.35\ {\rm ms}$, (d) $\tau = 0.0\ {\rm ms}$ and (e) $\tau = +0.2\ {\rm ms}$. (f) Line plots of $p'/p_{rms}$ sampled at specific locations of interest corresponding to negative $p'/p_{rms}$ in the intermittent region. b.l., boundary layer.

During $-0.3\ {\rm ms} < \tau \leqslant 0\ {\rm ms}$ (instant of peak $p'/p_{rms}$ of the intermittent region), figure 14(a) shows that the boundary layer $p'/p_{rms}$ approaches zero across the entire measurement domain. Concomitantly, there is a continued decrease of $p'/p_{rms}$ in the reattachment region and reaches a minimum at $\tau = -0.05\ {\rm ms}$, as shown in figure 14(a) (and also figure 14f). Within the separated flow, $p'/p_{rms}$ sharply decreases during this time period to reach a minimum value. Figure 14(a) shows that $p'/p_{rms}$ within the separated flow just upstream of the ramp leading edge $x = -0.6 \delta$ reaches its minimum at $\tau = -0.075\ {\rm ms}$, which is slightly earlier than that of the reattachment location; our limited time resolution ($\Delta \tau = 0.025\ {\rm ms}$) does not allow a better accuracy on the instant of the minimum. Interestingly, the minima of $p'/p_{rms}$ attain very similar values (${\approx }1$) at both locations close to ramp leading edge and reattachment. Correspondingly, $p'/p_{rms}$ of the intermittent region sharply increases and reaches its maxima exceeding 1.7 at $\tau = 0\ {\rm ms}$. This rapid increase in $p'/p_{rms}$ evidences that sharp bursts of pressure fluctuations occur in the intermittent region on a considerably shorter time scale when compared with the low-frequency unsteadiness. These bursts of pressure fluctuations are responsible for the high value of upstream-propagating perturbation velocities observed in the two-point correlation analysis of § 3.4.2. The swift motions of the separation shock are also consistent with the observations of Priebe & Martín (Reference Priebe and Martín2012) who demonstrated the growth/burst cycle of the separation bubble that accompanies these motions. The underlying mechanism behind the bursts of separation shock pressure excursion is further expounded in § 4.1.

Supplementary movie 2 also shows that the negative $p'/p_{rms}$ contour moves upstream within the separation bubble and engulfs a large azimuthal region within the separated flow during $-0.3\ {\rm ms} < \tau \leqslant 0\ {\rm ms}$. This strong increase in the size of the negative $p'/p_{rms}$ region in the separation bubble can be observed in figure 14(d) obtained at $\tau = 0\ {\rm ms}$. The propagation speed of the upstream edge of negative $p'/p_{rms}$ over this duration was determined by tracking different negative $p'/p_{rms}$ iso-contours, the values standing at $(0.3\unicode{x2013}0.5) \times u_\infty$. The value range is very similar to the value of $0.45 \times u_\infty$ obtained for the upstream-propagating perturbation within the separated flow in the cross-correlation analysis.

Beyond $\tau = 0\ {\rm ms}$, the intermittent region $p'/p_{rms}$ shows an initially rapid and subsequently a gentler decay. The incoming boundary layer $p'/p_{rms}$ gradually approaches zero at $\tau = 1\ {\rm ms}$ and stabilizes around this value. Concomitantly, the pressure fluctuations in the separation bubble and the reattachment region show a steady decay towards zero. Supplementary movie 2 shows that the $p'/p_{rms}$ decay first began within the separation bubble and occurred slightly earlier than $\tau = 0\ {\rm ms}$ (at $\tau = -0.075\ {\rm ms}$). The $p'/p_{rms}$ decay onset in the downstream locations such as the reattachment region and relaxing boundary layer occurs at progressively later times than $\tau = -0.075\ {\rm ms}$. This can be observed in figure 14(e), where appreciably negative $p'/p_{rms}$ occurs only in the reattachment region and the relaxing boundary layer. This progression of the onset of $p'/p_{rms}$ decay was seen as a downstream-propagating perturbation in the cross-correlation positive time delay branch (figure 12a). Notably, whereas the cross-correlation delays showed that the separation shock leads the fluctuations, the actual origin of the perturbation was also located in the upstream vicinity of the separation bubble. Furthermore, the downstream propagation velocity of the $p'/p_{rms}$ front corresponded very closely with the downstream propagation velocity obtained from the cross-correlation analysis.

The azimuthal coverage of the $p'/p_{rms}$ elevations and depressions within the SBLI paints an interesting picture of the underlying dynamics. Figure 14(b–e) shows that positive $p'/p_{rms}$ within the intermittent region spreads over the entire azimuthal region (visualized from red contour corresponding to zero) and the elevated values of $p'/p_{rms}$ (exceeding 50 % of peak $p'/p_{rms}$ at a given instance) were maintained over a substantial azimuthal region ($-6\delta < s < +6\delta$) surrounding the reference location ($s = 0$). This shows that the conditional average sequence indeed captured the bulk separation shock motion rather than the spanwise undulations. The corresponding azimuthal spread of negative $p'/p_{rms}$ within the separated/reattachment region is much more restricted compared with the intermittent region to within ($-3\delta < s < +3\delta$). Upon closer inspection of the intermittent region $p'/p_{rms}$ in figure 14(b,c), it can observed that this azimuthal region ($-2\delta < s < +2\delta$) also exhibits very high values of $p'/p_{rms}$ in the intermittent region. These observations reveal a few important details of the separation shock motions. First, the negative $p'/p_{rms}$ of the separated flow, which has a rather confined azimuthal extent, impacted a substantially large azimuthal region of the intermittent region. In other words, the separation shock motion at a given azimuthal location is non-trivially impacted by the pressure fluctuations in the separated/reattachment region at other azimuthal locations. Second, for the chosen azimuthal location, the greatest impact from the separated/reattachment region occurs over a limited region. Remarkably, the azimuthal length scale $\pm 2 \delta$ that showed the greatest negative $p'/p_{rms}$ in the reattachment region and relaxing boundary layer is very similar to the spacing between the Görtler vortices (Floryan Reference Floryan1991; Grilli et al. Reference Grilli, Hickel and Adams2013; Loginov et al. Reference Loginov, Adams and Zheltovodov2006). This observation along with the elongated nature of the pressure fluctuations in figure 9(b–e) and the fact that these $p'/p_{rms}$ streaks are observed earlier than a strong separation shock motion lend support to the postulate of Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016) and Grilli et al. (Reference Grilli, Hickel and Adams2013) that the Görtler vortices may be an important entity in driving the separation shock motions. The corresponding sequence of negative $p'/p_{rms}$ in the intermittent region shown in figure 14(f) is a mirror image of the positive $p'/p_{rms}$ situation presented.