3.1. Baseline flow field characterization

The baseline flow field characterization was made at tare conditions (without back pressure) using a combination of planar laser scattering (PLS) for off-surface shock train imaging and surface pressure field. Over five test runs at the set point conditions were conducted and the baseline flow field for each run were highly repeatable with one another and with those of Leonard & Narayanaswamy (Reference Leonard and Narayanaswamy2021). Figure 5 shows the off-body shock train, visualized using PLS imaging, overlaid on the mean surface pressure field (figure 5 a) and r.m.s. surface pressure field along the isolator (figure 5 b) at tare condition. The shock waves that were discerned from the PLS images are annotated as dashed lines in both figures. An excellent agreement was observed between the shock foot locations obtained by interpolating the PLS images and from the mean and r.m.s. surface pressure fields. The off-surface shock train also evidences a modest shock angle, which suggests that the shock legs are too weak to cause any flow separation at tare condition. This suggestion is corroborated by the mean pressure jump across the shock foot at $x/D = 7$ , which is substantially lower than the empirical estimate of the pressure jump required for flow separation. Furthermore, the $p_{\textit{rms}}$ field evidences an absence of flow separation at $x/D = 7$ and other shock intersection locations between $x/D = 4.1{ - }8.5$ . Overall, the isolator did not exhibit any inherent shock induced separation at tare conditions that may further complicate the analysis of the back pressure configurations.

Both the mean and r.m.s. pressure fields, presented in figures 5(c) and 5(d), respectively, exhibit notable azimuthal uniformity. Examining the shock foot locations reveals a slight curvature that is symmetric about the centre azimuth; the end-to-end variation in the streamwise location of the shock foot across the azimuthal direction is 0.02D. There is also a slight inflation ( $\!\lt\! 10 \,\%$ ) in the mean pressure along $\phi = 0^\circ$ compared with off-centre azimuths. Interestingly, the $p_{\textit{rms}}/p_w$ at the $\phi = 0^\circ$ is slightly lower than the off-centre azimuths; the overall end-to-end variation in $p_{\textit{rms}}/p_w$ at a shock foot location is less than $10 \,\%$ of the peak value. It is likely that these slight variations are caused by a combination of fluorescence light refraction through the curved walls of the isolator, slight image defocusing along the off-centre azimuths and residual image distortions that were left uncorrected from data processing. Another possibility for the azimuthal variation could also be the non-uniformity in the ultraviolet illumination due to the light focusing effects through the curved isolator wall which could cause higher intensity along $\phi = 0^\circ$ and a lower intensity away from the centre azimuth.

Figure 5. Characterization of the surface and off-surface flow field quantification of the isolator at tare condition. (a) Overlay of the mean surface and (b) r.m.s. pressure fields, respectively, and the corresponding off-surface shock structure visualization using planar scattering. Panel (c) presents the mean surface and panel (d) the r.m.s. pressure fields, respectively.

3.2. Surface streakline imagery of back pressured isolator

Surface streakline imagery is first presented for the back pressured isolator to obtain a grasp of the global flow field structure within the isolator. The surface streakline imagery is particularly suited to visualize the shock foot, separation and reattachment loci, and dominant vortices that may be present within the flow unit. The surface streakline field for a jet total pressure of $p_j = 1.75\,{\rm MPa}$ is presented in figure 6. Examining the field along the downstream direction first reveals two shock foot at $x/D \approx 5.2$ and $x/D \approx 7.1$ , respectively, which correspond to the isolator baseline shock system. Interestingly, the location of these shocks remains unaltered compared with the tare case, which shows that the information about the presence of shock train has not propagated into the baseline shocks. Downstream of the isolator baseline shocks is the leading shock of the shock train. The footprint of this leading shock appears broader than the isolator baseline shocks due to the large amplitude unsteady motions executed by this shock. The boundary layer separates downstream of the leading shock, as evidenced by the accumulation of the pigment. The separated flow is observed to be dominated by the presence of a counter-rotating vortex pair (CVP) (see figure 6). These CVPs are likely the main source of fluid exchange between the viscous region and the core flow. The downstream edge of the CVPs marks the reattachment location, where convergence of the streakline traces is observed. No other major feature is discerned downstream of the reattachment locus. Interestingly, no imprints of the succeeding (trailing) shocks of the shock train are present in the streakline imagery. Two likely reasons for the absence of their footprints are that the succeeding shock foot are located farther downstream of the measurement domain for the pressure setting used in figure 6 and/or their footprint is masked by the thick viscous flows beneath the trailing shocks. It should be noted that the trailing shock feet become visible in the r.m.s. pressure field at elevated back pressures and are discussed in the subsequent sections. A spanwise average location of the separation and the reattachment were obtained and the corresponding separation length, $L_{\textit{sep}}$ , was determined to be approximately 38 mm (or $\approx 1.0\,D$ ) for figure 6. The estimated $L_{\textit{sep}}$ will be compared with other estimates made using the peak Strouhal number of wall pressure fluctuation power spectra in § 3.5.

Figure 6. Surface streakline visualization showing the flow structure during back pressured operation.

3.3. Mean and r.m.s. pressure fields of back pressured isolator

The ensemble mean pressure fields of the back pressured isolator over different back pressure settings are presented in figure 7. A sharp increase in the mean pressure caused by the shock train within the isolator can be observed across all the back pressure settings. The location of this increase corresponds to the leading shock of the shock train (henceforth termed simply as ‘leading shock’). For the $p_b/p_\infty = 6.8$ case considered in figure 7(a), the leading shock is located at $x/D = 7.6$ , which is downstream of the two baseline shocks located within the measurement region. For $p_b/p_\infty = 8.0$ presented in figure 7(b), the leading shock is located at $x/D = 6.0$ , which is upstream of the baseline shock foot at $x/D = 7.1$ and considerably downstream of the baseline shock foot at $x/D = 5.2$ . For $p_b/p_\infty = 8.8$ considered in figure 7(c), the leading shock foot is located at $x/D = 5.1$ . For the $p_b/p_\infty = 10.0$ (cone injection) case shown in figure 7(d), the leading shock is upstream of the measurement region. This is observed by noting that the surface pressure even at the most upstream location substantially exceeded $p_w/p_\infty \approx 2.5$ that are observed just upstream of the shock train in figure 7(a–c). It is important to note that even the highest back pressure setting (figure 7 d) did not unstart the inlet, i.e. the isolator shock train was not disgorged from the inlet. This can be discerned from noting that the pressure ratio $p_w/p_\infty$ at the start of the measurement domain ( $x/D = 4.0$ ) is approximately 5.0. This $p_w/p_\infty \approx 5.0$ can be observed to occur within one isolator diameter from the beginning of the shock train for $p_b/p_\infty \leqslant 8.0$ and $p_b/p_\infty \leqslant 8.8$ shown in figure 7(b,c). In fact, a more precise estimation of the leading shock location was obtained to be $x/D = 3.75$ in the following paragraphs.

Figure 7. Mean surface pressure field within the isolator at different back pressure setting: (a) $p_b/p_\infty = 6.8$ ; (b) $p_b/p_\infty = 8.0$ ; (c) $p_b/p_\infty = 8.8$ ; (d) $p_b/p_\infty = 10.0$ (cone injection).

Figure 8. Mean centreline surface pressure field within the isolator at different back pressure settings: (a) surface pressure along the entire isolator normalized by free stream, and (b) surface pressure only within the pseudoshock region normalized by pressure just upstream of the pseudoshock.

Figure 9. The r.m.s. surface pressure field within the isolator setting at different back pressure settings: (a,b) $p_b/p_\infty = 8.0$ ; (c,d) $p_b/p_\infty = 8.8$ ; (e,f) $p_b/p_\infty = 10.0$ . The r.m.s. fields were normalized by the free stream pressure upstream of the inlet in (a), (c) and (e), and by the local mean surface pressure in (b), (d) and (f).

The mean pressure profiles along the isolator length measured at $\phi = 0^\circ$ for different back pressures are presented in figure 8. The profiles along the entire length of the isolator are provided in figure 8(a) and the profiles are normalized by the free stream pressure upstream of the inlet. Furthermore, the pressure profiles with $p_b/p_\infty = 10.0$ (cone injection) was scaled by half to make the values comparable to the other profiles. The pressure profiles presented in figure 8(b) were normalized by the corresponding mean pressure just upstream of the leading shock ( $p_f$ ); no scaling was made on $p_b/p_\infty = 10.0$ (cone injection) case. The x = 0 in figure 8(b) starts at the leading shock of the shock train and ends where the field of view terminates. For example, the back pressure $p_b/p_\infty = 6.8$ has the leading shock at x/D = 7.5 while that for $p_b/p_\infty = 8.0$ and $p_b/p_\infty = 8.8$ occurs at x/D = 6.1 and x/D = 5.1, respectively. However, the pressure fields for all cases terminate at x/D = 8.7. As a result, $p_b/p_\infty = 6.8$ case has a much less measured extent downstream of the leading shock and seems incomplete. With this input, it is observed from figures 8(a) and 8(b) that all the profiles within the shock train exhibit an initial steep increase in the mean pressure over the first 0.4D that is followed by a more gradual increase farther downstream. Comparing the streamwise extent of the initial increase and the $p_{\textit{rms}}$ field shown in figure 9, it is observed that this initial pressure rise region coincides with the intermittent region over which the leading shock oscillates. This confirms that the initial pressure rise is indeed caused by the leading shock, which is also consistent with the past observations (e.g. Carroll et al. Reference Carroll, Lopez-Fernandez and Dutton1993). It can be also observed in figure 8(a) that the slope of the initial pressure rise is very similar across the different back pressures, which portrays similar leading shock strengths across different back pressure settings. This again is consistent with the literature on back pressured supersonic ducts (Hunt & Gamba Reference Hunt and Gamba2018) that examined the pressure fields over different back pressures. Furthermore, the pressure jump across the leading shock is determined to be sufficient to cause shock induced separation for all back pressure settings presented, based on the empirical correlations (Babinsky & Harvey Reference Babinsky and Harvey2011). The gradual pressure rise downstream of the first 0.4D corresponds to the separated flow generated by the leading shock. Unlike Hunt & Gamba (Reference Hunt and Gamba2019) who noticed undulation in the mean pressure field from the trailing shocks of the shock train, such undulations are not observed in this work; however, the trailing shock footprints are observed in the $p_{\textit{rms}}$ fields that will be presented subsequently.

A comparison of the measured pressure profiles was made with the empirical profile of Waltrup & Billig (Reference Waltrup and Billig1973). The measured boundary layer thicknesses at $x/D = 6.33$ were used for the inflow parameters required for this profile. Furthermore, the approach Mach number was computed based on the measured static pressure ratio just upstream of the leading shock foot of the shock train and the free stream pressure. Given the modest dependence of the empirical profile on the inflow boundary layer properties, the approximations made towards computing the boundary layer thickness are expected to minimally impact the pressure trends. Figure 8(b) presents the empirical pressure profile as a dashed curve. The profiles of the $p_b/p_\infty = 10.0$ (cone injection) case were shifted in $x$ direction until an agreement was obtained with other profiles. It can be observed that the initial pressure rise until $x/D \approx 0.4$ is similar between the empirical and experimental pressure profiles across the different back pressure settings. A stronger deviation in the experimental and empirical profiles occurs at more downstream locations, wherein the empirical profile overpredicts the pressure values compared with the experiments. Overall, the best agreement between the empirical and experimental profiles occurs with $p_b/p_\infty = 10.0$ (cone injection) case, while $p_b/p_\infty \leqslant 8.8$ cases exhibit noticeable departures, especially for $x/D \gt 0.4$ . Between the different back pressures, whereas the pressure profiles with $p_b/p_\infty = 10.0$ , $p_b/p_\infty = 8.8$ and $p_b/p_\infty = 6.8$ exhibit a nearly identical evolution over $x/D \leqslant 0.6$ , the profile with back pressure $p_b/p_\infty = 8.0$ exhibits a departure to a lower value for $x/D \gt 0.3$ . While the reason for this departure is not clear, we posit that the leading shock foot and the initial part of shock train possibly interacts with the expansion fan emanated within the isolator from the upstream undisturbed flow at this back pressure setting. This is observed from the mean pressure profile of figure 8(a) where the leading shock with $p_b/p_\infty = 8.0$ is located is in the region of pressure decrease (expansion region), contrasting the other back pressures where the leading shock is located near zero pressure slope. It can also be observed from figure 8(b) that the first data point within the measurement location for $p_b/p_\infty = 10.0$ (cone injection) occurs at $x/D = 0.25$ ; this places the location of the leading shock for the $p_b/p_\infty = 10.0$ case at $x/D = 3.75$ .

Figure 9 presents the r.m.s. pressure fields within the isolator section for two different jet injection cases, $p_b/p_\infty = 8.0$ (figure 9 a,b), $p_b/p_\infty = 8.8$ (figure 9 c,d) and $p_b/p_\infty = 10.0$ (figure 9 e,f). Whereas the r.m.s. fields of figure 9(a,c,e) are normalized by the free stream pressure, the r.m.s. fields of figure 9(b,d,f) are normalized by the local mean surface pressure. Moving along the isolator length for $p_b/p_\infty \leqslant 8.8$ , the initial regions of the isolator upstream of the shock train exhibit a uniform $p_{\textit{rms}}/p_\infty$ field. Minor elevations are observed corresponding to the baseline isolator shock that were not perturbed by the shock train; however, these undulations ( $\approx 0.02$ ) are not visible with the present colourmap settings. A strong increase in the $p_{\textit{rms}}/p_\infty$ can be observed beneath the leading shock foot and the $p_{\textit{rms}}/p_\infty$ reaches its maximum value along the isolator length beneath the leading shock foot. The $p_{\textit{rms}}/p_\infty$ subsequently decreases to a relative plateau (observed as a near uniform contour colour between $x/D \approx 6.6 - 7.3$ in figure 9 a, for example) and is followed by a gradual increase in the downstream regions. Multiple minor peaks in the $p_{\textit{rms}}/p_\infty$ can be identified with $p_b/p_\infty = 8.8$ in figure 9(c) that correspond to the trailing shock feet. These peaks are diffuse due to the presence of thick viscous layer beneath the shock foot that dissipates the shock into weak compression waves. Interestingly, the strengths of the individual peak $p_{\textit{rms}}/p_\infty$ are different from one another. Whereas $p_{\textit{rms}}/p_\infty$ local peak exhibits a decreasing trend between the shocks labelled ‘1’ to ‘3’, subsequent shock legs ‘3’ to ‘5’ exhibit an increasing trend. The relative strength of the $p_{\textit{rms}}$ with respect to the local mean pressure (shown in figure 9 d) also exhibits this trend and reiterates a weakening and strengthening of the shock feet unsteadiness within the shock train. The trailing shock train feet are not clearly defined with $p_b/p_\infty = 8.0$ and $p_b/p_\infty = 6.8$ (not presented). However, a broad increase in $p_{\textit{rms}}/p_\infty$ was observed with a modest peak at $x/D \approx 8$ for $p_b/p_\infty = 8.0$ (see figure 9 a,b).

The $p_b/p_\infty = 10.0$ (cone injection) case in figure 9(e,f) reveal an overall elevation in the $p_{\textit{rms}}/p_\infty$ and a appreciably similar $p_{\textit{rms}}/p_w$ values within the shock train when compared with the $p_b/p_\infty = 8.8$ case. Similar to the $p_b/p_\infty = 8.8$ case, figure 9(e,f) reveal the presence of three identifiable trailing shock feet. These shock feet are notably closer to one another for compared with $p_b/p_\infty = 8.8$ of figure 9(c,d). An extensive region of the flow downstream of the shock train is devoid of any noticeable shock foot and this region likely corresponds to the subsonic mixing process of the pseudoshock (more confirmation in § 3.4). Finally, the $p_{\textit{rms}}/p_\infty$ fields show noticeable variations across the azimuthal direction. The variations are especially stark in the trailing edge shock feet; whereas the trailing edge shock feet is clearly delineated for $\phi \gt 0^\circ$ , the imprints of the shock feet are not visible for $\phi \lt -30^\circ$ . To elaborate on a few possible causes of this asymmetry, it should be noted that the basic architecture of the wind tunnel converging–diverging nozzle has its curvature occur only along the top wall, while the bottom wall is straight. This differential curvature may cause the boundary layer thickness be different from the top and bottom walls, which in turn may cause a different mass displacement from the top and bottom walls. This in turn can cause the inflow to the inlet to have a very small angle. This minor flow incidence angle did not impact the isolator baseline flow symmetry to a measurable level. When the shock waves continued to process the inlet flow, the asymmetry possibly got amplified with increasing distance within the isolator. This can be seen in figure 5(d) when one compares the baseline shock at x/D = 5.2 and 8.7. When the back pressuring jet is fed by this mildly asymmetric flow, the resulting shock train further possibly amplifies the asymmetry to measurable levels. This asymmetry also compounded by the differential illumination of the PSP and other secondary effects.

It should be reiterated that the isolator duct of the present work cannot be considered narrow. The narrowness is determined by $\delta ^*/D$ , which is 0.048 for the present isolator duct at a location just upstream of the leading-edge shock (see figure 2). This means that the displacement effect of the viscous region is felt only within $10\,\%$ of the flow ( $2\delta ^*/D$ overall). Therefore, the upstream shock is interacting with a rather thin boundary layer compared with the duct height. In fact, the RANS simulations, presented subsequently, show that a substantial portions of the individual shocks (leading and trailing) of the shock train are straight, which suggests that a major portion of the isolator flow within the shock train can be considered as inviscid. The natural development of shock train culminates with a viscous mixing region that engulfs the entire duct. This is true for all isolator cross-sections and blockage ratios, narrow and broad; for narrow ducts, this region occurs earlier along the isolator length. For the present study, the mixing region occurs only for $p_b/p_\infty \geq 8.8$ , which again reiterates that the duct may not be considered narrow. Finally, we also note that the pressure rise curve matches remarkably well with the empirical fit of Waltrup & Billig (Reference Waltrup and Billig1973) (figure 8 b) that were based on experimental datasets obtained in a relatively low blockage duct. This is yet another proof that the present isolator duct upstream of the shock train is not viscous flow dominated.

3.4. Estimated effective back pressure and shock train structure

The streamwise locations of the leading shock foot from the mean and r.m.s. pressure fields were used to map the effective back pressures by comparing the corresponding shock train leading-edge locations of RANS simulations. This effective back pressure should be differentiated from the pressure downstream of the shock train that is measured and presented in figure 7. The effective back pressure provided in this section represents a uniform exit (back) pressure of the duct that generates the leading shock at the measured location for the given inflow Mach number and a given blockage. Figure 10 presents the variation of the leading shock foot location ( $x_{\textit{leading}\,\textit{shock}}$ ) at various uniform back pressures imposed at the isolator exit in RANS simulations. Square symbols were added in the plot to identify the leading shock foot locations at different jet and cone injection settings and obtain an estimate of the corresponding effective back pressures. The RANS simulations show that a minimum back pressure of $p_b/p_\infty \approx 2.0$ is needed to set the shock train within the isolator. The leading shock foot location moves upstream rather modestly for $p_b/p_\infty \lt 5.0$ and exhibits a much stronger upstream movement for higher back pressures. Juxtaposing the experimental leading shock location of the shock train with the simulated values estimates the back pressure ratios at $p_b/p_\infty \approx 6.8$ (for $p_j$ = 1.37 MPa), $p_b/p_\infty \approx 8.0$ ( $p_j$ = 2.04 MPa), $p_b/p_\infty \approx 8.8$ ( $p_j$ = 2.74 MPa) and $p_b/p_\infty \approx 10.0$ (for cone injection).

Figure 10. Computed leading shock foot location at different back pressures (black dot curve) and the experimental leading shock foot location overlaid on the plot to estimate the effective back pressures for different injection settings.

The isolator shock train structure is investigated using the velocity divergence fields at the duct centre plane from the RANS simulations at different set point effective back pressures that nearly correspond to the respective jet injection and cone injection settings. A qualitative representation of the divergence fields for various set point conditions are presented in figure 11(a–d). The leading shock of the shock train are labelled in all the figures. It can be observed that the isolator duct is interwoven by the baseline shock train, whose intersection locations with the isolator surface make a good comparison with the figure 5(d). Consistent with the experiments, baseline shocks are very weak and do not exhibit any boundary layer separation.

The shock train from back pressure application also exhibits agreement in the spacing between the leading shock and the successive trailing shocks. Further, the diffusion of the trailing shock near the isolator surface can also be observed across all of figure 11(a–d), which caused the smearing of the pressure gradient observed in the $p_{\textit{rms}}/p_w$ fields. The shock trains also show interesting similarities and differences between one another with increasing back pressure. The leading and trailing shocks across all set points exhibit a short region around $r/D = 0$ where they are normal and they become oblique away from $r/D = 0$ . The boundary layer downstream of the leading shock is separated for all back pressures presented, and thick viscous regions where the shock is dissipated can be observed near the isolator surface extending along the entire downstream duct length. Interestingly, while the trailing shocks can be traced until the isolator exit in figure 11(a,b) that correspond to $p_b/p_\infty = 6.8$ and $p_b/p_\infty = 8.0$ , the trailing shock is dissipated into the viscous mixing region in figure 11(c,d) that correspond to $p_b/p_\infty = 8.8$ and $p_b/p_\infty = 10.0$ , respectively. Even between figures 11(c) and 11(d), figure 11(d) exhibits a noticeably larger streamwise extent of the mixing region.

Figure 11. Computed velocity divergence field from RANS simulations at different back pressure settings that approximately corresponds to (a) $p_b/p_\infty = 6.8$ , (b) $p_b/p_\infty = 8.0$ , (c) $p_b/p_\infty = 8.8$ and (d) $p_b/p_\infty = 10.0$ .

3.5. Wall pressure PSD

The frequency content of the wall pressure fluctuations at different locations upstream and within the shock train are presented next. The PSD was computed for each data set that spanned 1.5 s of run time (12 000 data samples per run) using Welch’s algorithm with a 512 sample for each block. A Hanning window with 50 % overlap was employed while computing the PSD. This resulted in a frequency resolution $\Delta f = 7.8\,{\rm Hz}$ which is sufficient to delineate the PSD across the entire canonical shock wave boundary layer interactions (SBLI) with acceptable noise within the PSD.

As a baseline, the pressure fluctuation PSD for the isolator without back pressure is first presented. Figure 12 shows the frequency premultiplied PSD beneath the shock foot of the baseline (tare) isolator located at $x/D = 5.2$ ; the PSD was averaged over $-30^\circ \leqslant \phi \leqslant +30^\circ$ . Figure 12 also presents the corresponding frequency premultiplied PSD of the incoming boundary layer averaged over $4.2 \leqslant x/D \leqslant 4.7$ and $-30^\circ \leqslant \phi \leqslant +30^\circ$ . The premultiplied PSD is a standard practice to show the relative contribution of the PSDs spanning several decades of frequency. The linear y-scale is the best representation to visualize the relative contributions and has been extensively used in the SBLI literature. The PSD beneath the boundary layer shows a monotonically increasing trend over the entire measured frequency range. This PSD trend is expected since the peak energy of the upstream boundary layer occurs at the characteristic boundary layer frequency ( $u_\infty / \delta$ ) that is over an order of magnitude higher than the Nyquist frequency.

The frequency premultiplied PSD beneath the isolator shock foot shows a broadband spectrum with a modest peak at a very low frequency $f \approx 70 \,{\rm Hz}$ . This peak is followed by a slight minimum at $f \approx 120\,{\rm Hz}$ and a subsequent monotonic increase. Despite there not being a shock-induced flow separation, the isolator shock foot exhibits low frequency pulsations; the cause for these pulsations has to do with the dampening of the shock foot jitter due to the boundary layer turbulent structures passage by viscous forces (detailed in Leonard & Narayanaswamy Reference Leonard and Narayanaswamy2021). Finally, it should be noted that the PSD amplitude beneath the different shock feet along the isolator section are within the same order of magnitude as the boundary layer PSD. Overall, the unsteady shock oscillations are observed to be quite weak without the application of back pressure.

Figure 12. The PSD of the isolator surface pressure fluctuation beneath the shock foot (black) and the upstream boundary layer (blue) for the tare configuration.

Figure 13. (a) The PSD of the leading shock intermittent region surface pressure fluctuations for the different back pressure settings, and (b) evolution of the leading shock foot zero crossing frequency, velocity and Strouhal number based on these parameters across different jet injection pressures. The horizontal arrows point to which y-axis the given plot will correspond to.

The corresponding frequency premultiplied PSD beneath the leading shock is presented in figure 13(a) for different back pressure settings; the PSDs are appropriately scaled to make suitable comparisons. It can be observed that the PSDs for all back pressure settings exhibit broadband aperiodic motions with a broad peak at its most dominant frequency. This peak frequency decreases with increasing back pressure from approximately 800 Hz at $p_b/p_\infty = 6.8$ to 240 Hz at $p_b /p_\infty = 8.8$ ; concomitantly the frequency spread of the PSD also decreases by over a decade between $p_b/p_\infty = 6.8$ and $p_b /p_\infty = 8.8$ . The magnitude of the PSD increases by over a factor of 10 by increasing the back pressure from $p_b/p_\infty = 6.8$ to $p_b/p_\infty = 8.8$ .

Figure 13(a) also presents the pressure fluctuation PSD at the most upstream measurement location for $p_b/p_\infty = 10.0$ ( $x/D = 4.0$ ); readers are reminded that the leading shock for this setting is upstream of the measurement domain. It will be shown from the subsequent discussions that the intermittent region of the leading shock extends over 0.5D for $p_b/p_\infty = 8.8$ , which is expected to hold for $p_b/p_\infty = 10.0$ as well, given the close similarity of $p_b/p_\infty$ values. Therefore, the location $x/D = 4.0$ where the PSD is presented in figure 13(a) is $x/D \approx 0.25$ downstream of the pseudoshock leading edge, which is within the leading shock intermittent region. The PSD with $p_b/p_\infty = 10.0$ also exhibits a broadband spectrum with a peak in PSD that occurs at approximately 300 Hz. The near constancy of the peak PSD frequency observed with increasing back pressure between $p_b/p_\infty = 8.0$ and $p_b/p_\infty = 10.0$ is also consistent with a similar observation of the near constant shock stem oscillation frequency with increasing back pressure made by Hunt & Gamba (Reference Hunt and Gamba2019). It should be noted that the strength of the PSD for $p_b/p_\infty = 10.0$ is very similar to $p_b/p_\infty = 8.8$ . However, since the location where the PSD was obtained for $p_b/p_\infty = 10.0$ is situated downstream of the peak $p_{\textit{rms}}$ location, it is expected that the PSD beneath the shock foot for $p_b/p_\infty = 10.0$ will exceed $p_b/p_\infty = 8.8$ , which continues the increasing trend of the PSD with back pressure.

The broadband aperiodic nature of the leading shock oscillations has been observed in other works with supersonic isolators (Xiong et al. Reference Xiong, Wang, Fan and Wang2017; Hunt & Gamba Reference Hunt and Gamba2019). The question is if the leading shock foot motions adhere to SBLI, given the significant complexities that occur downstream of the leading shock. This question is addressed first by examining the features of the separation shock dynamics and subsequently using cross-correlation analysis. Prior works on canonical SBLI had successfully scaled the shock oscillation frequency and shock velocity using the shock zero crossing frequency and the intermittent region length defined as the distance between $5\,\%$ to $95\,\%$ of the shock foot oscillation amplitude; these measurements were made in the prior works using pointwise measurements at a relatively coarse resolution. The 2-D pressure fields of the present study provide a much finer spatial resolution that enables a closer examination of the Strouhal number scaling and an independent examination of the leading shock foot speeds at different back pressure settings. Figure 13(b) presents the shock zero crossing frequency ( $f_c$ ) measured at the peak $p_{\textit{rms}}$ location at different back pressure settings. The $f_c$ exhibits a monotonic decrease with increasing back pressure with an overall decrease of $50\,\%$ between $p_b/p_\infty = 6.8$ and $p_b/p_\infty = 8.8$ . It is interesting to note that the magnitude of $f_c$ is between a factor of two to three higher than the peak frequency of the frequency premultiplied PSD of the pressure fluctuation presented in figure 13(a). The Strouhal number based on $f_c$ and the intermittent length $L_i$ (region over which the intermittency factor lie between 0.05 and 0.95), presented in figure 13(b), exhibits a remarkably tight agreement in their values across the different back pressures spanning a range between 0.026 to 0.033. These values are also in excellent agreement with the Strouhal number (based on separation scale) range of the separation shock oscillations in canonical SBLI, which are reported to occur between 0.02–0.05 (Gonsalez & Dolling Reference Gonsalez and Dolling1993; Dussauge & Piponniau Reference Dussauge and Piponniau2008). Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014) argue that the Strouhal number can also be viewed as the ratio of the separation shock speed to the free stream velocity. While their argument used an estimated shock speed as $2L_i \times f_c$ , the present study allows an independent determination of the shock speed by tracking the leading shock foot motions from the pressure fields. The instantaneous location of the leading shock foot was taken where the local pressure exceeds the inflow static pressure by 20 $\,\%$ , following Poggie & Porter (Reference Poggie and Porter2019). The corresponding ratio of the leading shock to free stream velocity is presented in figure 13(b) across the different back pressures. It can be observed that whereas a coincidence between the Strouhal number and the leading shock-to-free stream speed ratios is not observed, the differences between their values are less than $25\,\%$ across the different settings; furthermore, the ratio $u_s/U_\infty$ lies within the 0.02–0.05 bound observed in other canonical SBLI units (see Gonsalez & Dolling Reference Gonsalez and Dolling1993 for a compilation of the shock speeds).

Evidently, the overall spectral features of the pressure fluctuation PSDs beneath the leading shock sharply resemble the separation shock oscillations in canonical (single shock) 2-D SBLI units. However, a notable difference is observed in the magnitude of the $f_c$ in the present study compared with other 2-D SBLI units. Erengil & Dolling (Reference Erengil and Dolling1991), for example, reported that the shock zero crossing frequency was $0.12U_\infty /\delta$ in their compression ramp interactions, which held constant across different inviscid shock strengths; similar zero crossing frequency was also reported in Gonsalez & Dolling (Reference Gonsalez and Dolling1993). By contrast, the $f_c$ obtained in the present configuration is an order of magnitude lower and shows a decreasing trend with increasing back pressure. The substantially lower $f_c$ is because the leading shock maintains a near-constant velocity during its unsteady motions at a speed of $3\,\%\,U_\infty$ that is very similar to canonical 2-D SBLI units. Since the intermittent length $L_i$ of the leading shock of the present study is significantly larger compared with the canonical SBLI units available in the literature, the zero crossing frequency of the leading shock is consequently lower. Therefore, the observations suggest that the unifying parameter between the leading shock of the present study and the canonical SBLI is the constant shock speed, which is consistent with the arguments made by Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014). For the remaining discussions, these broadband low frequency oscillations will be referred to as ‘SBLI’ modes.

Given the strong similarity in the Strouhal number scaling between the present and the canonical SBLI units, the Strouhal number based on $L_i$ (from figure 9) and the peak frequency of the pressure oscillation PSD (from figure 13 a) were used to estimate the corresponding separation scale, $L_{\textit{sep}}$ , of the leading shock at different back pressures. A strong caveat should be exercised that no additional off-surface visualization or measurements were made to evaluate the calculated $L_{\textit{sep}}$ at the exact jet injection pressures. Furthermore, prior works have shown the existence of multiple separation bubbles within the shock train and have argued that the shock train dynamics exhibit significant deviations from a canonical SBLI (Hunt & Gamba Reference Hunt and Gamba2019). Therefore, in the present work, the $L_{\textit{sep}}$ is treated as an equivalent separation scale that can drive the leading shock if the leading shock unit resembled a canonical SBLI unit; the results presented so far support a strong resemblance. To alleviate the risk of using the estimated $L_{\textit{sep}}$ as a representative separation scale, the separation scale obtained using the surface streakline imagery at two intermediate jet injection pressures are compared with the estimated $L_{\textit{sep}}$ trend. The intermediate pressures were also used to evaluate (and establish) the continuing trend of the separation scales even at the intermediate back pressures other than the ones that are extensively reported in this work. Figure 14 presents the equivalent $L_{\textit{sep}}$ estimates at different back pressures. A remarkable agreement is observed between the estimated $L_{\textit{sep}}$ and the separation scale based on the streakline imagery at intermediate jet injection pressures. The $L_{\textit{sep}}$ exhibits over a four-fold increase in size between $0.47D$ and $2.24D$ over the back pressure range; based on the reference boundary layer thickness measured at $x/D = 6.33$ , the $L_{\textit{sep}}/\delta$ is between $4.16$ and $19.86$ . From Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014), these separation scales belong to the size range where the separation shock motions are driven by the inherent pulsations of the downstream separated flow with only a modest direct contribution from the upstream boundary layer. If the $p_b/p_\infty = 8.8$ case Strouhal number is assumed for $p_b/p_\infty = 10.0$ , the corresponding $L_{\textit{sep}}$ is estimated at 2.2D based on the peak frequency of the wall pressure fluctuation PSD of figure 13(a).

Figure 14. Estimated separation length scale $L_{\textit{sep}}$ for the different jet injection back pressures implemented in the present work. The separation scale is presented both in physical units as well as in terms of the isolator inner diameter. The measured $L_{\textit{sep}}$ from surface streakline imagery at intermediate jet injection pressures are also included in the figure.

3.6. Spatiotemporal organization of pressure fluctuations

The spatiotemporal organization of the wall pressure fluctuation PSD is addressed as a first step to understanding the mechanisms that drive the leading shock pulsations. The frequency premultiplied PSDs were normalized by the $p_{\textit{rms}}^2$ to portray the relative PSD contributions at different frequencies within the pseudoshock region. The PSDs were obtained as an average over $-50^\circ \lt \phi \lt +50^\circ$ . Figure 15 presents the PSD evolution along the streamwise direction spanning the entire measurement domain and across different back pressure settings. The upstream boundary layer exhibits a monotonically increasing trend across all back pressures in figure 15(a–c). The unperturbed baseline isolator shock trains are also identified for $p_b/p_\infty = 6.8$ (figure 15 a) and $p_b/p_\infty = 8.0$ (figure 15 b) settings and their PSDs are quantitatively identical to those of the tare conditions discussed in figure 12. Subsequently, the PSD shows a strong departure to dominant low frequency content beneath the leading shock, which was discussed in figure 13. The distance over which the dominant low frequency content occurs is consistent with the intermittent region length across all back pressures.

Figure 15. Evolution of PSD of the pressure fluctuations along the isolator section averaged over azimuthal region $-50^\circ \lt \phi \lt +50^\circ$ . Four different test cases are presented: (a) $p_b/p_\infty = 6.8$ ; (b) $p_b/p_\infty = 8.0$ ; (c) $p_b/p_\infty = 8.8$ ; (d) $p_b/p_\infty = 10.0$ .

The PSDs downstream of the leading shock show an immediate dominance of high-frequency content for $p_b/p_\infty = 6.8$ to $p_b/p_\infty = 8.8$ settings in figure 15(a–c). Such high-frequency content has been reported beneath the separated flow of canonical SBLI units and these frequencies are predominantly contributed by the shear layer eddies that develop over the separation bubble (Chandola et al. Reference Chandola, Huang and Estruch-Samper2017). It should be noted that the present measurements cannot resolve the peak frequency of these pressure oscillations because they occur well above the Nyquist limit of the measurements. The PSD field at $p_b/p_\infty = 8.8$ also reveals distinct bands of elevated frequencies within the shock train; these are annotated as ‘higher shock oscillation modes’ in figure 15(c). The dashed lines that correspond to the peak $p_{\textit{rms}}/p_w$ of the successive trailing shock feet are also presented in figure 15(c) to locate these frequency bands in relation to the trailing shocks. It is observed that successive shock feet exhibit a consistent increase in the frequency bands of oscillations until shock 4. Interestingly, shock 4 and shock 5 exhibit a prominent low frequency band whose frequency aligns closely with the SBLI mode of the leading shock; furthermore, shock 5 exhibits another frequency band that coincides with the higher shock oscillation mode of trailing shock 1. A similar but much weaker reemergence of SBLI mode frequencies of the leading shock oscillations can also be observed in $p_b/p_\infty = 8.0$ at $x/D \approx 8.1$ . Based on the $L_{\textit{sep}}$ estimates, the first three trailing shock feet that were identified in the $p_{\textit{rms}}/p_w$ profiles of $p_b/p_\infty = 8.8$ are located within the separation bubble extent. Further, the location of the reemergence of the SBLI mode frequencies occurs just downstream of the estimated separation bubble scale for both back pressures in both $p_b/p_\infty = 8.0$ and $p_b/p_\infty = 8.8$ cases.

The PSDs of the trailing shocks of the $p_b/p_\infty = 10.0$ case, presented in figure 15(d), reveal interesting similarities and departures from the jet injection cases. The first trailing shock exhibits a broadband bimodal PSD. At the lower frequencies, the PSD content is very similar to the SBLI mode of the leading shock PSD with a broad peak at approximately 300 Hz. This broad peak is followed by a minimum and another sharper peak at approximately 840 Hz. This bimodal structure persists through the trailing shocks 2 and 3, and the contribution from the 840 Hz strengthens with distance while the low frequency band at 300 Hz weakens. Downstream of shock 3, the PSD continues to be broadband, but the frequency content at 840 Hz dominates the remaining frequencies resulting in a more tonal nature of the pressure oscillations over the majority of the measurement domain. Higher resonances of the baseline 840 Hz can also be observed between $7.2 \leqslant x/D \leqslant 8.0$ , but their strengths are substantially lower than the baseline frequency. It is also interesting to note that the 840 Hz peak also penetrates upstream of the trailing shock 1 a little into the intermittent region of the leading shock. In fact, the pressure fluctuation PSD obtained within the intermittent region exhibit a very minor peak at 840 Hz and a notable tonal peak at 1200 Hz, as seen in figure 13(a). The nearly tonal nature of the pressure fluctuations suggests the acoustic origins of these frequency bands. Therefore, this frequency band and its resonances will be referred to as ‘acoustic’ modes. Comparing the higher shock oscillation modes of the $p_b/p_\infty = 8.8$ case (figure 15 c) with the acoustic mode of the $p_b/p_\infty = 10.0$ case in figure 15(d), it can be observed that the fundamental acoustic mode frequency of the $p_b/p_\infty = 8.8$ case at 840 Hz observed in trailing shock 1 corresponds very closely to the fundamental acoustic mode of the $p_b/p_\infty = 10.0$ case, and the higher-order acoustic modes observed in trailing shocks 2, 3 and 4 are also observed in the downstream isolator section with $p_b/p_\infty = 10.0$ ( $x/D \gt 7$ ). Interestingly, the strengths of these modes with $p_b/p_\infty = 8.8$ are much weaker compared with the $p_b/p_\infty = 10.0$ case and the occurrence of multiple acoustic resonances in the $p_b/p_\infty = 8.8$ case contrasts the dominant occurrence of the fundamental mode in the cone injection.

Overall, a rather complex spatial distribution of the wall pressure fluctuations PSD is observed that exhibits both agreements and deviations from the prior works on shock train dynamics in planar isolators. The first point of consistency is the appearance of higher oscillation modes within the shock train that was also reported by Ikui et al. (Reference Ikui, Matsuo, Minoru and Masanobu1974b ) and Hunt & Gamba (Reference Hunt and Gamba2019). The present work reveals a staggered increase in the acoustic mode frequencies for the $p_b/p_\infty = 8.8$ case that is consistent with Hunt & Gamba (Reference Hunt and Gamba2019). However, the re-emergence of the low frequency content at a downstream distance observed in trailing shocks 4 and 5 of $p_b/p_\infty = 8.8$ was not reported in the earlier works, which could be due to inadequate measurement stations at these locations in their works and/or differences in the pressure dynamics in the downstream regions. Second, it is evident that the dynamics of the regions upstream of the leading shock is unaltered by the pseudoshock, which was reported by Xiong et al. (Reference Xiong, Wang, Fan and Wang2017).

The point of debate the present observations raise is the strong adherence of the leading shock dynamics to a canonical 2-D SBLI unit in terms of the Strouhal number scaling and the shock speed. Hunt & Gamba (Reference Hunt and Gamba2019) correctly pointed out that the acoustic modes do not occur in a canonical 2-D SBLI unit. Furthermore, Hunt & Gamba (Reference Hunt and Gamba2019) demonstrated a strong deviation in the peak Strouhal number (based on $L_{\textit{sep}}$ ) of the shock train oscillations compared with a canonical 2-D SBLI; in their study, the pressure fluctuation beneath the leading shock peaked at a Strouhal number of 0.56, which significantly higher than canonical 2-D SBLI units. The present study, however, exhibits strong similarities between the leading shock dynamics and canonical 2-D SBLI dynamics in terms of the peak Strouhal number of the pressure fluctuation PSD. A major differentiating influence in the planar inlets arises from the junction and sidewall separations, which are absent in the present axisymmetric isolators. Therefore, the fundamental question is if the shock train dynamics are indeed governed by mechanisms similar to a canonical 2-D SBLI when it is devoid of the junction and the sidewall influences. It should be noted that Hunt & Gamba (Reference Hunt and Gamba2019) indeed point to the influence of the primary separation towards driving the leading shock pulsations in planar isolators; the question is if the primary separation dynamics take a greater dominant role when the junctions/sidewall contributions cease to exist.

Figure 16. Two-dimensional maps of zero-lag cross-correlation coefficient of the pressure fluctuations within the isolator section with intermittent region as the reference location. Three different test cases are presented: (a) $p_b/p_\infty = 8.0$ , jet injection; (b) $p_b/p_\infty = 8.8$ , jet injection; (c) cone injection.

It should be remarked that without back pressure application, the shock oscillations have a very different signature and spectrum. The shock dynamics of tare operation have been reported and analysed in Leonard & Narayanaswamy (Reference Leonard and Narayanaswamy2021) for the specific case of the absence of shock-induced separation. It was shown that the shock train oscillations were dominated by frequencies that are over an order of magnitude lower than the frequencies with shock induced separation (also see figure 12). Furthermore, the mechanism driving these oscillations was suggested to be the phase lag of shock oscillations induced by the wall shear. By contrast, with back pressure deployment, the dynamical content and the driving interactions of the SBLI with back pressure are distinct from that of the tare operation, as will be expounded subsequently.

3.7. Cross-correlation analysis

3.7.2. Streamwise evolution of cross-correlation

The cross-correlation fields, ${\rm Corr} (x,\tau )$ , across different lags are presented to provide an understanding of how the pressure fluctuations are temporally organized within the pseudoshock region. Knowledge of the regions that lead and lag a given reference location will help identify the occurrence of disturbance nodes that can feed the leading shock oscillations and ultimately shed light on the mechanisms that drive the shock oscillations. The reference location for the jet injection cases is at the peak $p_{\textit{rms}}$ location of the leading shock of the pseudoshock and at $x/D = 4.0$ for the $p_b/p_\infty = 10.0$ case.

Figure 17. Evolution of cross-correlation coefficient of the pressure fluctuations along the isolator section with the intermittent region as the reference location. Three different test cases are presented: (a) $p_b/p_\infty = 8.0$ , jet injection; (b) $p_b/p_\infty = 8.8$ , jet injection; (c) cone injection.

Figure 17 presents the cross-correlation field averaged over $-50^\circ \leqslant \phi \leqslant +50^\circ$ along the measurement domain. Line contours of ${\rm Corr} (x,\tau )$ are included in each figure to delineate the regions of positive, negative and weak correlations. The locations of the trailing shocks are also annotated in figure 17(b,c). The overall cross-correlation field at a given time lag in the vicinity of $\tau = 0\ {\rm ms}$ resembles the zero-lag correlations in terms of the locations that are positively and negatively correlated. Beginning with the ${\rm Corr} (x,\tau )$ in the incoming boundary layer that is observable in figure 17(a), it is seen that the value of ${\rm Corr} (x,\tau ) \approx -0.1$ across all time lags. This suggests only a modest direct contribution from the incoming boundary layer towards driving the leading shock oscillations. This trend is expected to hold for the higher back pressure settings as well. A sharp increase in the ${\rm Corr} (x,\tau )$ can be observed within the intermittent region of the leading shock where the ${\rm Corr} (x,\tau )$ assumes high positive values. Tracking the location of the peak correlation in figure 17(a) reveals that the peak correlation magnitude occurs increasingly earlier with downstream distance; this corresponds to the occurrence of upstream propagating perturbations within the intermittent region. The progression of the time delay of the peak correlation provides a phase speed of $u_\phi \approx 0.07 u_\infty$ within the intermittent region ( $x/D \approx \pm 0.3$ surrounding the reference location) for both $p_b/p_\infty = 8.0$ and $p_b/p_\infty = 8.8$ settings. The phase speed is computed using the standard definition as the inverse of the difference in the time lag between the reference location and a given station of interest divided by the distance between the two locations. This phase speed is approximately a factor of two higher than the shock foot speed, but is very similar to the values reported in other works on canonical SBLI (Jenquin et al. Reference Jenquin, Johnson and Narayanaswamy2023). The upstream propagating perturbation continues to be present within of the separated flow and the propagation speed of the perturbations increases to $u_\phi \approx 0.3 u_\infty$ downstream into the separation bubble (e.g. $x/D \leqslant 6.3$ in figure 17 b). Farther downstream at $x/D \approx 7.5$ in figure 17(b), the pressure perturbation changes into a downstream propagating perturbation with a propagation velocity of $u_\phi \approx 0.3 u_\infty$ ; this downstream propagating perturbation continues until the downstream end of the measurement domain. Within this downstream propagating perturbation region, distinct increases in the ${\rm Corr} (x,\tau )$ can be observed beneath the foot of trailing shocks 4 and 5, which exhibited notable overlap in the PSD with the intermittent region PSD. Overall, the cross-correlation fields reveal that the pressure fluctuations at $x/D \approx 7.0$ within the separated flow lead all other locations within the pseudoshock for $p_b/p_\infty = 8.8$ jet injection case and $x/D \approx 6.5$ for $p_b/p_\infty = 8.0$ jet injection case. This is evidence of the presence of a node in the vicinity of the reattachment location that originates the pressure perturbations that drive the leading shock motions.

The corresponding cross-correlation field for the $p_b/p_\infty = 10.0$ case reveals that the intermittent region assumes a high positive ${\rm Corr} (x,\tau )$ . This positive ${\rm Corr} (x,\tau )$ extends over the bulk of the isolator, including the separated regions of the duct. Tracking the peak correlation once again reveals a downstream propagating perturbation with velocity $u_\phi \approx 0.3 u_\infty$ , which extends along the entire length of the measurement domain. Interestingly, in addition to the strong positive ${\rm Corr} (x,\tau )$ , a weak ${\rm Corr} (x,\tau )$ can also be observed extending along the length of the isolator. This perturbation is also observed to propagate downstream at a similar phase velocity as the positive ${\rm Corr} (x,\tau )$ . Importantly, unlike the lower back-pressure cases, a clear node within the separated flow that emanates upstream propagating perturbations into the leading shock cannot be located in the $p_b/p_\infty = 10.0$ case.

Figure 18. Spectral distribution of the phase angle of the pressure fluctuations within the isolator section with the intermittent region as the reference location. Two different cases are presented: (a) $p_b/p_\infty = 8.8$ , jet injection; (b) cone injection. The corresponding line plots of the phase angle are presented (c) for $p_b/p_\infty = 8.8$ (solid line) and $p_b/p_\infty = 10.0$ (dashed line) for the SBLI and acoustic modes.

3.7.4. Flow asymmetry effects on PSD and statistical coupling

There were obvious azimuthal variations in the $p_{\textit{rms}}/p_w$ field and the positive azimuth exhibited higher $p_{\textit{rms}}/p_w$ compared with negative azimuth. Even though the dynamical content and statistics reported were averaged over $-30^\circ \leqslant \phi \leqslant +30^\circ$ , the results are expected to be biased over the locations of higher-pressure fluctuations. Also, the averaging did not include more outboard locations, especially the weaker pressure fluctuation regions of $\phi \leqslant -30^\circ$ . This challenges the generality of the dynamic content of the shock train and the statistical analysis reported, and the conclusions about the driving mechanics of the shock train oscillations that will be drawn subsequently. Therefore, specific investigations were conducted to determine the azimuthal variations of the PSD of the surface pressure fluctuations and a select collection of important statistical quantities upon which further conclusions are based.

Figure 19. Evaluation of flow asymmetry on shock train dynamics and select statistical results at various locations within the shock train: (a–d) frequency premultiplied PSD beneath shock feet; (e–h) coherence magnitude at various locations within the shock train with reference location beneath the leading shock intermittent region; (i–l) corresponding phase delay spectra at various locations within the shock train. The shock train locations include: (a,e,i) leading shock intermittent region; (b,f,j) trailing shock 1; (c,g,k) trailing shock 4; (d,h,l) trailing shock 5. The results shown correspond to $p_b/p_\infty = 8.8$ back pressure setting.

Figure 19(a–d) presents the frequency premultiplied PSD of the surface pressure fluctuations within the leading shock intermittent region and successive trailing shocks 1, 4 and 5, at multiple azimuthal locations; the corresponding back pressure setting was $p_b/p_\infty = 8.8$ . The PSD for this chosen back pressure setting exhibits the contribution from the SBLI mode as well as acoustic mode beneath the leading and trailing shocks of the shock train. In all figure 19(a–d), the PSD at $\phi = -60^\circ$ was scaled by 1.25 to match the other PSDs in the overall magnitude. It can be observed that the PSDs across all azimuths exhibit remarkable coincidence with one another over the entire frequency range measured. The PSDs of the leading shock intermittent region for all the azimuths exhibit both SBLI and acoustic modes, as shown in figure 19(a). Quantitatively, however, whereas the PSDs are coincident in the low frequency unsteadiness band (SBLI modes) across all azimuths in figure 19(a), the acoustic mode is relatively reduced at $\phi = -60^\circ$ . This mild reduction in the acoustic mode strength is also present in the trailing shock 4 at $\phi = -60^\circ$ (figure 19 c), but is absent in the trailing shocks 1 and 5 (figure 19 b,d), as well as trailing shocks 2 and 3 (not shown).

The spectral coherence between the pressure fluctuations at a reference location within the leading shock intermittent region between $-10^\circ \leqslant \phi \leqslant +10^\circ$ and all other locations in the isolator was computed to evaluate any deviations in the coupling strengths in the azimuthal direction. Figure 19(e–h) presents the coherence within the leading shock intermittent region and the trailing shocks 1, 4 and 5 for $p_b/p_\infty = 8.8$ . Once again, an excellent agreement in the coherence was obtained across all the azimuthal locations and within the shock train. The most deviation was observed at $\phi = -60^\circ$ with a maximum difference of less than 0.2; this occurs in the leading shock intermittent region and trailing shock 1. The remarkable agreement in the coherence suggests that spectral coupling magnitude across the entire azimuthal domain is nearly identical and spans both the SBLI and acoustic modes that are of interest in the current work.

Finally, the phase angle between the pressure fluctuations at the same reference location and at all other locations within the shock train are presented in figure 19(i–l). Figure 19(i–l) presents the phase angle at the intermittent region, trailing shocks 1, 4 and 5, respectively. At the intermittent region, all azimuthal locations exhibit zero phase angle until 1 kHz and measurable excursions in the phase angle at higher frequencies (figure 19 i). It should be noted that frequencies below 1 kHz encompass both the SBLI and acoustic modes. The zero phase angle illustrates that the entire leading shock oscillates in unison along the azimuthal direction, i.e. executes bulk shock motion. Similarly, all other trailing shocks execute in-phase oscillations along its azimuthal spread at a fixed phase delay from the reference location for this frequency range. This reinforces that each trailing shock of the shock train also oscillates in unison, thereby evidencing a bulk motion in the SBLI and acoustic modes of interest. These observations fortify that the dynamics and statistical relationship in the shock train oscillations observed thus far by averaging over $-30^\circ \leqslant \phi \leqslant +30^\circ$ are indeed representative of the entire azimuthal extent of the shock location. The weakening of shock foot at $\phi \leqslant -30^\circ$ and the observed asymmetry in the $p_{\textit{rms}}/p_w$ does not influence the overall dynamical behaviour and coupling of the shock train in both SBLI and acoustic modes that are of interest in this work.

3.8. Discussions

The spectral trends and the corresponding correlation analyses reveal that a common feature of the shock oscillations across all jet injection cases is the occurrence of the SBLI mode of the leading shock that is driven by the separation bubble pulsations, which adheres to the description of a canonical SBLI unit despite the complications of the downstream separated flow within the pseudoshock. A clear node of pressure fluctuations is observed within the separated flow in the vicinity of the reattachment region for the SBLI mode, which is in conjunction with prior studies of canonical SBLI units (Chandola et al. Reference Chandola, Huang and Estruch-Samper2017; Jenquin et al. Reference Jenquin, Johnson and Narayanaswamy2023). A differentiating feature between the different cases is that whereas the acoustic modes are absent in the $p_b/p_\infty = 6.8$ and $p_b/p_\infty = 8.0$ cases, these oscillations occur at a low strength with $p_b/p_\infty = 8.8$ and significantly higher strength with $p_b/p_\infty = 10.0$ . The ensuing discussions will delve deeper into the origins of the acoustic modes, the reasons for their absence at low back pressures and the disparity in strength with changing back pressures.

A significant number of prior works pointed to the acoustic resonance that gets established within the isolator as a mechanism the drives the acoustic modes (Ikui et al. Reference Ikui, Matsuo and Minoru1974a ). An estimate for the resonance frequency has been suggested for transonic diffusers which are considered as an ideal organ pipe. The frequency $f_n$ of a resonant mode $n \in (0, 1, 2, \ldots )$ is given by (Newsome Reference Newsome1984)

(3.1) \begin{equation} f_n = \frac {(2 n + 1)}{4} \frac {c}{L} (1 - M^2). \end{equation}

In the above equation, $c$ is the speed of sound, $L$ is the length of the diffuser and $M$ is the Mach number downstream of the shock. Applying this equation for the present isolator results in the fundamental harmonic frequency $f_0 = 205\,{\rm Hz}$ for a limiting case of $M = 0$ , and the value of $f_0$ reduces further for a better estimate of $M$ . This estimate is significantly lower than the observed fundamental acoustic mode frequency of 840 Hz. Similarly Sugiyama et al. (Reference Sugiyama, Takeda, Zhang, Okuda and Yamagishi1988) suggested that the shock oscillations are driven by the acoustic wave that are set up between the leading pseudoshock and the duct exit. Applying this suggestion to the present study also yields inconsistent results; for example, this suggestion would lead to a substantial decrease in the shock/acoustic mode frequency between $p_b/p_\infty = 8.8$ and $p_b/p_\infty = 10.0$ cases, which was not observed in the present work. One can understand that the origin of the discrepancy is that these suggestions were intended for transonic diffusers where the flow downstream of the terminal shock is fully subsonic.

As such a feedback resonance within the entire duct is not feasible in the present study since the flow within the pseudoshock is not entirely subsonic. In fact, based on the wall pressure trace, the flow Mach number corresponding to the maximum mean static pressure recorded within the isolator remains supersonic for all the jet cases and subsonic only over a part of the isolator section for cone injection. For such situations, acoustic interactions can occur through the subsonic boundary layer region or within the limited cells of inviscid subsonic region when the shock train consists of normal shocks. In this context, the line plots of the phase delay presented in figure 18(c) provide a few interesting pointers. Considering the $p_b/p_\infty = 8.8$ case, the location of peak positive phase delay which corresponds to the node of pressure oscillations at $x/D \approx 6.6$ for the SBLI mode and $x/D \approx 6.8$ for the acoustic mode. Both these locations are in the close vicinity of the reattachment location, which was estimated at $x/D \approx 7.3$ based on $L_{\textit{sep}}$ estimates. Similarly, considering the $p_b/p_\infty = 10.0$ case, the node of pressure oscillations for the acoustic mode is located at $x/D \approx 5.6$ . Going by the $L_{\textit{sep}}$ estimate of 2.2D for the $p_b/p_\infty = 10.0$ case based on the intermittent region PSD, and pinning the leading shock location to $x/D = 3.75D$ , the node of the pressure fluctuation is once again located in the vicinity of the reattachment location of the separated flow downstream of the leading shock. Collating all the back pressure cases, it is clear that the node of the acoustic mode occurs within the separated flow in the vicinity of the reattachment region. However, if it is true that the acoustic modes are driven by the upstream propagating pressure fluctuations within the separated flow, then it stands to reason why acoustic modes were absent until $p_b/p_\infty \leqslant 8.0$ , and only very weak oscillations were observed for $p_b/p_\infty = 8.8$ compared with the $p_b/p_\infty = 10.0$ case. A part of the reason could be the attenuation of the acoustic wave in supersonic boundary layers, as noted by Gawehn et al. (Reference Gawehn, Gülhan, Al-Hasan and Schnerr2010); however, this does not explain the total absence of the acoustic modes at lower back pressures whose magnitudes are still comparable to $p_b/p_\infty = 8.8$ .

Robinet & Casalis (Reference Robinet and Casalis2001) theorized that the shock oscillations in transonic diffusers occur from feedback interactions between the acoustic waves that emanated at the reattachment location and the shock wave. These interactions are inviscid in nature, i.e. the viscous attenuations of the feedback acoustic waves do not change the nature of the outcome. Summarily, Robinet & Casalis (Reference Robinet and Casalis2001) hypothesized that the acoustic waves that are incident on a normal shock wave get reflected at different angles depending on their frequency. However, there exists a frequency for a given distance between the shock and the reattachment point where the reflected wave angle is the same as the incident wave angle; this coincidence results in a feedback amplification that can drive the leading shock oscillations. This angle is unique for a given Mach number upstream of the shock and the acoustic frequency, and is called the ‘critical angle’ in their work. Robinet & Casalis (Reference Robinet and Casalis2001) developed a 1-D stability analysis for a normal shock that is perturbed by an incident acoustic wave and demonstrated an excellent agreement with predicting the terminal shock oscillation frequencies of the transonic diffuser experiments of Bogar et al. (Reference Bogar, Sajben and Kroutil1983).

In the present study, we extended this analysis to delineate the predictions of this stability theory for the present experiments. Before the results are presented, multiple simplifications that were made by using this analysis need to be stated. First, the analysis was made for a 1-D flow problem, which is not the case in the present study. The main reason for not doing the stability analysis on the 2-D mean flow is the significantly higher computational cost that will be incurred. However, it is encouraging to observe the success of this analysis on transonic diffusers, where again the flow was not 1-D. Second, while the theory does not require a subsonic core flow downstream of the shock, the previous applications were made for transonic diffusers where the downstream core flow was indeed subsonic. As such there are no prior works that used the 1-D stability analysis on a supersonic isolator flow where the core flow is not subsonic. Third, while the inflow Mach number to the terminal (normal) shock of the transonic diffuser is well defined, it is not clear if we had to use the inflow Mach number upstream of the pseudoshock or the shock normal Mach number for making the estimate for the present case. With these caveats, it should be mentioned that the objective of the analysis is to unravel the underlying interactions that drive the shock train oscillations without seeking an exact quantitative agreement with the measured shock oscillation frequencies.

Figure 20. Most amplified acoustic mode frequency for different separation length scales over three different Mach numbers that correspond to the total and shock normal Mach numbers for $p_b/p_\infty = 8.8$ and $p_b/p_\infty = 10.0$ cases.

Figure 20 presents the leading shock oscillation (acoustic mode) frequency due to feedback shock–acoustic interactions over a range of separation lengths observed in the present experiments. Three different inflow Mach numbers were chosen. First, $M = 2.5$ corresponds to the estimated Mach number upstream of the leading shock based on the isolator surface pressure just upstream of the pseudoshock and using oblique shock relations. Next, $M = 2.0$ corresponds to an ‘effective’ shock normal Mach number based on the pressure at the estimated reattachment location of the $p_b/p_\infty = 8.8$ case and the pressure upstream of the pseudoshock, and $M = 1.75$ corresponds to the above estimate for the $p_b/p_\infty = 10.0$ case. It can be observed from figure 20 that a solution to the feedback amplification does not exist for the separation size below a threshold for a given Mach number. This threshold value is $L_{\textit{sep}}/D \approx 1.6 - 1.8$ for the chosen Mach number range. This threshold occurs because below this separation size the critical angle criterion for any frequency cannot be accommodated within the spatial constraint posed by the distance between the shock wave and the reattachment point. Interestingly, the estimated $L_{\textit{sep}}$ for $p_b/p_\infty = 6.8$ and $p_b/p_\infty = 8.0$ were both below this threshold limit. Beyond the threshold $L_{\textit{sep}}$ , a sharply decreasing trend could be observed in the amplified frequency with increasing $L_{\textit{sep}}$ , where higher Mach numbers yield a higher frequency for a given $L_{\textit{sep}}$ . The divergence in the most amplified frequency across different Mach numbers considered sharply decreases with increasing $L_{\textit{sep}}$ ; for example, the spread is approximately $1.5\,{\rm kHz}$ for $L_{\textit{sep}} = 2.5 D$ . The mean separation scale for $p_b/p_\infty = 8.8$ and with $p_b/p_\infty = 10.0$ were both estimated to be around $2.3D$ ; at this separation scale, the most amplified frequency ranged between $1.2\,{\rm kHz}$ to $3\,{\rm kHz}$ for the different Mach numbers chosen. Given the separation shock oscillates over the intermittent region, which is approximately $0.5D$ , the most amplified frequencies spreads over a range of values causing a spectral spread in the PSD. It should also be noted that for the approximations that were made with extending the 1-D analysis tool, this level of quantitative agreement is quite remarkable.

The analysis presented above makes several qualitatively consistent behaviours with the experimental observations and provides a platform to explain the mechanisms that drive the shock–acoustic interactions. Based on the above arguments, a picture emerges that the shock–acoustic interactions are most likely driven by inviscid interactions between the acoustic perturbations emanated at the reattachment point and the leading shock wave. An important requirement to have a positive feedback amplification is that the separation scales should be large enough to support an acoustic frequency and be incident on the shock wave at its critical angle; this in turn requires a threshold back pressure to generate the shock–acoustic interaction modes. Furthermore, the unsteadiness of the separation bubble causes broadband oscillations of the separation scales and hence a spread in the most amplified frequency from the shock–acoustic interactions. Interestingly, figure 20 shows that this spread in the most amplified acoustic mode frequency shrinks with increasing separation size for a given Mach number and also across different inflow Mach numbers upstream of the shock train leading edge. The manifestation of these features are observed across many other shock train dynamics studies, which was discussed in § 3.3.

It should be remarked that the self-exited oscillations should be understood in the context of the effective pressure ratio influencing the interaction. However, the shock train oscillations are driven by multiply coupled interactions during the pseudoshock processing of the flow field, which was expounded in considerable depth in this work. This challenges the objective quantification of the driving back pressure ratio. We, therefore, performed RANS simulations with different imposed uniform pressure boundary conditions at the isolator exit. The isolator exit pressure that best matched the leading shock foot location of the shock train between computations and experiments was reported as an effective back pressure that drives the shock train structure. It should be noted that this back pressure need not drive the shock train dynamics. As expounded in this work the effective back pressure that influences the interactions depends more directly on the separation scales at the different operation conditions and the occurrence of feedback shock–acoustic interactions. There is indeed an implicit relationship between this driving back pressure of the shock train dynamics determined by the separation scales and the isolator exit pressure.

The analysis presented above shows that there are two major mechanisms, separation bubble pulsations and shock–acoustic interactions, which coexist and mutually compete to drive the shock train oscillations. For low back pressure settings, e.g. $p_b/p_\infty \leqslant 8.0$ in this study, the low frequency separation bubble pulsations are shown to dominate the shock train dynamics as observed from the PSD maps (see figure 15 a,b) and the statistical analyses. With increasing back pressure, the shock–acoustic interactions make can increasing contribution above a certain threshold back pressure ( $p_b/p_\infty \approx 8.8$ in this study, see figure 15 c) and makes a prominent contribution to the shock train oscillations at elevated back pressures, as shown in figure 15(d) obtained at $p_b/p_\infty \geq 10.0$ . It is interesting to note that the computed flow fields of figure 11 obtained at $p_b/p_\infty = 8.8$ and $p_b/p_\infty = 10.0$ reveal the presence of a subsonic mixing layer towards the isolator exit at this back pressures. Based on the dissipation of the trailing shocks, the mixing layer was traced to begin at $x/D \approx 9.0$ for $p_b/p_\infty = 8.8$ and $x/D \approx 8.0$ for $p_b/p_\infty = 10.0$ . While it is not clear if the occurrence of a subsonic mixing layer is a necessary criterion to observe the shock–acoustic interactions, the evidences strongly suggest that the extent of the subsonic mixing layer correlates with the strengthening of shock–acoustic feedback coupling in this study.

3.9. Relationship to other inlet/isolator geometry

The axisymmetric shock train dynamics presented in this article have direct logical relationship and extensions to the shock dynamics of high-efficiency self-starting hypersonic inlets. These inlets are typically axisymmetric in geometry and compress the inflow through an asymmetric shock train system. A number of recent works investigated the flow field and shock field within these high-efficiency inlets across different geometries and reported noteworthy similarities in the shock train oscillations and evolutionary trend with the current study. Johnson et al. (Reference Johnson, Jenquin and Narayanaswamy2022b ) conducted an experimental investigation of the shock train dynamics within a back pressured axisymmetric truncated Busemann inlet with a Mach 4 inflow and observed tonal oscillations of the shock foot leading edge peaking at $\approx 1 {\,\rm kHz}$ overlaid on broadband motions at elevated back pressures. Similar observations were also made by Bustard et al. (Reference Bustard, Noftz, Jewell and Juliano2025) in an oscillating waverider inlet subject to back pressure. Recently, Schram et al. (Reference Schram, Thill and Narayanaswamy2025b ) investigated the axisymmetric truncated Busemann inlet subject to back pressures at negative incidence angle in a Mach 4 infow. Once again, the authors observed a similar progression of the shock train leading-edge oscillations from broadband oscillations to tonal oscillations at $\approx 1\,{\rm kHz}$ . Furthermore, Baccarella et al. (Reference Baccarella, Liu, McGann, Lee and Lee2021) studied the axisymmetric isolator shock train oscillations with back pressure imposed using mass addition and combustion that included two different flow enthalpy settings. The authors observed tonal oscillations of the isolator shock train above certain fuel equivalence ratio and mass addition thresholds, with the peak shock oscillations at a $400\, {\rm Hz}$ – $1\,{\rm kHz}$ range. The consistency of the qualitative trending from broadband to tonal shock train leading-edge oscillations in these applied geometries reinforce that the transition in the driving mechanism pervades across various geometries. The rather close alignment in the tonal frequency across multiple studies spanning multiple Mach numbers and compression ratios supports the observation made from figure 20 that the acoustic mode frequency spread narrows over different inflow Mach numbers at large separation scales.