5.3.1 Evolution of perturbation field to statistically stationary state

Figure 6. Contours of the vertical velocity perturbation field ( $v^{P}$ ), on a representative spanwise-normal ( $x$ – $y$ ) plane, as it evolves to a statistically stationary state (e) in time, superposed with contours of the base-flow pressure field for reference. Perturbations influenced by reflected shock motion and shedding structures in the downstream boundary layer are evident (e). Spatial coordinates are scaled by boundary layer thickness. (a) Shortly after initialization $(\unicode[STIX]{x1D70F}=0.05)$ : perturbations localized to forcing location. (b) Before perturbations pass through interaction $(\unicode[STIX]{x1D70F}=1.0)$ : relatively isotropic evolution. (c) Immediately after perturbations pass through interaction $(\unicode[STIX]{x1D70F}=3.0)$ : non-isotropic evolution. (d) Further in development $(\unicode[STIX]{x1D70F}=10)$ : prominent acoustics in near field, amplification begins in interaction. (e) Statistically stationary state $(\unicode[STIX]{x1D70F}=50)$ : perturbations highlight the reflected shock and shedding structures downstream of the separation bubble.

The evolution of the vertical velocity perturbation field (case 7) to a statistically stationary state is depicted in figure 6; the base-flow unsteady pressure field is superposed for reference. In the initial stages, the evolution is somewhat isotropic. However, once the perturbations enter the separated region, their evolution becomes more directional in the downstream direction. By panel (c), the outer envelope starts to approximate the reflected shock location. After about $10\unicode[STIX]{x1D70F}$ , the perturbations have filled the separated region, and waves with acoustic features are evident downstream. These are also strongly directional, and do not permeate upstream of the reflected shock. Figure 6(e) provides a typical representation of the instantaneous perturbation field under statistically stationary conditions. While the base flow may take many flow-through times to reach statistical stationarity, the perturbations reach stationarity quickly, since they are a very small deviation from the base flow, and their growth is dominated by the absolute instability. The constrained perturbation field is localized to the region just upstream of the reflected shock foot and follows the characteristic along the reflected shock to the far field, demonstrating that with variable attenuation, the perturbation field protrusion upstream is limited. In contrast, for an unconstrained case, the perturbations travel everywhere, including upstream through the subsonic boundary layer, as they grow to nonlinear magnitudes. However, linear-perturbation growth is much stronger in the interaction region and farther downstream than upstream, so these regions of dominant perturbation growth are highlighted when the perturbations are constrained, since weakly growing upstream perturbations are damped relative to strongly growing perturbations in the interaction region and downstream; the magnitude of the latter remain approximately constant due to variable attenuation.

Key features of the statistically stationary perturbation field include convective structures that are shed downstream from the separation region. Additional fine-scale perturbation structures highlight vortices present in the base-flow post-reattachment boundary layer, around which sharp gradients in the flow promote perturbation growth. Also prominent are relatively large magnitude perturbations along the reflected shock wave, demonstrating the influence of perturbation dynamics of the separation region on the reflected shock. Generally, the constrained perturbation field evolves to highlight turbulent features with high potential for linear growth, away from the base flow, which is a novel way to visualize the instantaneous character of a turbulent flow.

Figure 7. Properties of normalized pressure perturbations at a representative location in the post-reattachment boundary layer for different target perturbation magnitudes (cases 14–20). The perturbations are normalized by $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}}$ such that collapse of the trajectories demonstrates linearity. (a) Time history of normalized pressure perturbation evolution. Magnified insets demonstrate the minimal cumulative effect of nonlinear error on the trajectories at early and advanced times; this error manifests through slight divergence of the trajectories. (b) Cumulative nonlinear error of each of the trajectories compared to case 19 ( $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}}=1\times 10^{-8}$ ). The error is low-pass filtered to aid in discerning trends. The root-mean-square (r.m.s.) level of the normalized pressure perturbations is included to provide perspective for the error magnitude.

5.3.2 Linearity of constrained perturbation field

Several of the following conclusions rely on the claim that the LC-SLES procedure represents a reasonable approximation for the evolution of constrained linear perturbations as discussed throughout § 3. Notably, any method for propagating linear perturbations through a chaotic nonlinear flow incurs errors that accumulate over time, due to sensitivity of the flow to initial conditions. In this section, we quantify the magnitude of nonlinear errors introduced through the combined effects of the basic SLES procedure (i.e. calculating the evolution of linear perturbations as the difference between two evolving fully nonlinear systems), the variable attenuation procedure, and finite decimal precision: the major contributing causes of nonlinearity. For this purpose, we analyse cases 14–20 that are all initialized with a perturbation impulse taken from a statistically stationary instant of case 11. Each of cases 14–20 proceeds with a distinct perturbation target magnitude, separated by an order of magnitude as described in table 4; the initial impulse for these cases is also scaled with the target perturbation magnitude relative to case 11. Through this premise, linear perturbations would evolve identically in cases 14–20, subject to the relative scaling of the target perturbation magnitude and initial impulse; thus, nonlinear effects may be observed through divergence of the trajectories in time. These simulations evolve for a duration of $100\unicode[STIX]{x1D6FF}_{0}/U_{\infty }$ , to observe the accumulation of nonlinearity over a time scale corresponding to the order of low-frequency unsteadiness.

The effects of nonlinearity are described quantitatively in figure 7. Figure 7(a) describes the evolution of pressure perturbations for cases 14–20, at a representative location in the outer layer of the post-reattachment boundary layer; the greatest perturbation magnitudes are observed in this vicinity. The trajectories remain similar over the time window of interest, demonstrating that linearity of the method is maintained. As time progresses, finer-scale features of the signal are distorted for large magnitude perturbations due to accumulation of nonlinearity, as well as for the smallest magnitude perturbation ( $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}}=1\times 10^{-9}$ ) due to lack of decimal precision. Figure 7(b) shows the level of accumulated nonlinear error with time for each case with respect to case 19 (this case has the smallest target perturbation magnitude for which errors due to lack of decimal precision are not evident and is therefore considered to be the most accurate linear solution). The error is low-pass filtered in time to aid in discerning trends between the level of error and the target perturbation magnitude, and the root-mean-square (r.m.s.) level of the normalized pressure perturbations is included to provide perspective for the error magnitude. Concerning cases in which the lack of decimal precision does not introduce error, the errors initially grow from machine precision, with faster growth rates corresponding to larger perturbation target magnitudes, as anticipated. Near $\unicode[STIX]{x1D70F}\approx 35$ , an event is encountered which introduces substantial nonlinearity into all perturbations, after which the same trend in growth is recovered. Encouragingly, over the duration of these simulations, the cumulative nonlinear error for the target perturbation magnitude of $1\times 10^{-6}$ (employed in cases 5, 7, 9–13, 17) remains less than ${\approx}10\,\%$ of the r.m.s. pressure perturbations. Maintaining linearity at this level of precision over such long duration in a chaotic system is not trivial, and these observations attest to the effectiveness of the LC-SLES method.

5.3.3 Time-mean perturbation field and global absolute linear instability

We now discuss the behaviour of the linearly constrained perturbations in the context of a global absolute linear instability. The self-sustaining nature of the perturbations (cases 11–20) indicates the existence of constructive feedback through the recirculation region. This results in large perturbation growth rates in the recirculation region that overwhelm convective growth downstream as well as any additional forcing of the interaction, as discussed further in § 5.3.4. This is consistent with the presence of a global absolute linear instability, as discussed by Chomaz (Reference Chomaz2005). Discussion in this section focuses on the time-mean behaviour of this absolute instability, but it is relevant to note that the time-local behaviour of this absolute instability is not always consistent with its time-mean behaviour, as addressed in § 5.3.6.

The conclusion that self-sustaining constrained perturbations are indicative of a global absolute instability can be confirmed using the framework discussed by Huerre & Monkewitz (Reference Huerre and Monkewitz1990) and Schmid & Henningson (Reference Schmid and Henningson2001) for classifying the nature of instability based on properties of the fundamental solution operator (referred to, in these works, in a less general form: a Green’s function). Since our method is global (Theofilis Reference Theofilis2011) and non-modal (Schmid Reference Schmid2007), including effects from the entire domain, with no restriction on perturbation form, apart from the requirement that they remain linear with respect to the expansion in (3.11), instabilities identified through this method are global and non-modal in nature; the fundamental solution operator propagates the complete global state space (discrete) or global phase space (continuous) in time.

For time-dependent base flows, we generalize the classification of instabilities as follows. Consider the discretized form of (3.7), without forcing and without constraining, which propagates linear perturbations through the flow:

A flow is classified as globally unstable if

where $\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{LN}$ is the Lyapunov norm maximized over all initial impulses; conceptually, this is the maximum possible gain of the system represented by (5.1). A globally unstable flow is additionally classified as absolutely unstable if

A globally unstable flow is additionally classified as convectively unstable if

where $i_{0}$ and $j_{0}$ represent time-independent indices for the considered degree of freedom, non-boundary sets do not include $i_{0}$ on the spatial domain boundary when considering finite domains and $t_{0}$ represents the initial time of impulse perturbation to the base flow, which is arbitrary and independent of the other variables; $\Vert {\mathcal{G}}_{i_{0}j_{0}}(t,t_{0})\Vert _{\infty }$ represents the max-norm of the fundamental solution operator as a function of the current, $t$ , and impulse, $t_{0}$ , times, over the finite set of $i_{0}$ and $j_{0}$ . Naturally, the finite, non-boundary sets of degrees of freedom discussed in (5.3)–(5.4) represent spatially localized regions of the flow. In this sense, equation (5.3) states that the flow is absolutely unstable if the location of maximum perturbation growth is confined to a single local region of the flow. Similarly, equation (5.4) states that the flow is convectively unstable if the location of maximum perturbation growth is not confined to any single local region of the flow. The above use of ‘local’ does not imply the use of any local profile in the analysis, since the full, spatially three-dimensional, ‘global’ flow is analysed. In this context, when perturbation forcing is turned off after an initial impulse, self-sustaining constrained perturbations arise in the STBLI interaction region: the perturbations are perpetually growing, hence the need for constraint, which indicates that the condition, $\limsup _{t\rightarrow \infty }\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{LN}\rightarrow \infty$ , is satisfied for some $t_{0}$ , consistent with a global instability. Furthermore, the constrained perturbations, which reveal the most rapid non-modal growth, remain concentrated in the interaction region, which indicates that the condition $\limsup _{t\rightarrow \infty }\Vert {\mathcal{G}}_{i_{0}j_{0}}(t,t_{0})\Vert _{\infty }/\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{LN}\in (0,1]$ is satisfied for some non-boundary set of $i_{0}$ , $j_{0}$ , $t_{0}$ , in which the $i_{0}$ correspond to degrees of freedom in the interaction region, consistent with a global absolute instability. (Alternatively, for a turbulent boundary layer, the constrained perturbations are not self-sustaining and convect downstream, indicating that the condition $\limsup _{t\rightarrow \infty }\Vert {\mathcal{G}}_{i_{0}j_{0}}(t,t_{0})\Vert _{\infty }/\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{LN}=0$ is satisfied, for all non-boundary sets of $i_{0}$ , $j_{0}$ , $t_{0}$ , consistent with a global convective instability.)

Figure 8. Filled contours of the time-mean linearly constrained streamwise velocity perturbation superposed with contours of the time-mean base-flow pressure field for reference. The time-mean perturbation field is consistent with a time-mean linear tendency (time-mean behaviour of the absolute instability in the unsteady environment) for upstream motion of the reflected shock. All plots have the same contour levels and spatial coordinates ( $x$ – $y$ plane) scaled by boundary layer thickness. (a) Case 11: nominal grid (D) with $9^{\circ }$ shock generator, moderately separated interaction. (b) Case 12: refined grid (B), spanwise averaged, with $9^{\circ }$ shock generator, moderately separated interaction. (c) Case 13: refined grid (B), spanwise averaged, with $11^{\circ }$ shock generator, massively separated interaction. (d) Schematic describing how the time-mean perturbations indicate a time-mean linear tendency for upstream motion of reflected shock. Notably, the distinction between the twin-flow and base-flow shocks in the mean is spatially diffuse due to base-flow unsteadiness.

We can quantify the effect of this global absolute instability in the STBLI by examining the time mean of the self-sustaining linearly constrained perturbation field, $\overline{\unicode[STIX]{x1D731}^{\boldsymbol{P}}}$ , which we designate as the time-mean linear tendency of the base flow. We interpret the non-zero time-mean linear tendency as the time-mean behaviour of the absolute instability in the unsteady environment. This is clarified by noting that the time mean of the perturbation field between two statistically identical turbulent realizations (twin and base) will tend toward zero with time. However, for this STBLI, the absolute instability present in the base flow manifests as a time-mean linearly constrained perturbation field that does not tend toward zero with time, since the instability biases linear-perturbation growth in the unsteady environment. Indeed, as noted earlier, the constraints of § 3.3 effectively highlight this absolute instability at the expense of other more weakly growing (convective) instabilities, which are damped. Since the twin-flow simulation solves the forced and linearly constrained Navier–Stokes equations, it represents a linear departure from the base-flow simulation. As such, their time means (twin and base) can be slightly different – the magnitude of the difference being determined by the perturbation target magnitude ( $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}}$ ) – and is reflective of the time-mean behaviour of the underlying absolute instability. The underlying bias in linear-perturbation growth in the unsteady environment, leading to a non-zero time-mean linear tendency, can be thought of as analogous to an unstable zero-frequency (non-oscillatory) mode, which can be found in global IVP and EVP analyses of steady base flows that do not satisfy the steady Navier–Stokes equations, including time-mean STBLIs (Touber & Sandham Reference Touber and Sandham2009; Pirozzoli et al. Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010; Waindim et al. Reference Waindim, Bhaumik and Gaitonde2016; Nichols et al. Reference Nichols, Larsson, Bernardini and Pirozzoli2017).

The time-mean linear tendency is described in figure 8, visualized by the time-mean streamwise velocity perturbation, for cases 11–13, in panels (a–c) respectively. We note that the mean perturbation field converges at a much slower rate than the mean base flow, since the range of the perturbation field extends several orders of magnitude. Thus, convergence of the mean is sensitive to the behaviour of large magnitude perturbations. As such, discrepancies between the nominal (a) and refined (b) grids are expected and observed, and they are attributable not only to differences in spatial refinement, but also to differences in initial STBLI realizations between different cases. Sensitivity to averaging duration can also occur, even with sufficiently long duration simulations which allow for accurate convergence of the time-mean flow field; results on the nominal grid (a) are best converged in this sense, since case 11 has the longest simulation duration. This comparison also provides an indication of the level of uncertainty in the time-mean perturbation quantities and sensitivity of these quantities to the base flow. Nonetheless, all cases, including the interaction with stronger separation (c, case 13), exhibit similar time-mean linear tendencies, with equivalent physical significance.

The time-mean behaviour of the absolute instability depicted in figure 8 is clarified by noting that, for all cases, in the time-mean sense, the reflected shock position observed in the linearly constrained twin flow is upstream from its position in the base flow. This results in a negative time-mean streamwise velocity perturbation in the region of the reflected shock as illustrated in figure 8(d); i.e. the mean linearized shock position is not coincident with the mean shock position, but is rather located slightly upstream. Notably, the unsteadiness of the base flow inhibits a sharp indication of the position of the linearized and base-flow shocks in the mean, since the difference in position between the shocks, at any given instant, is determined by the perturbation target magnitude, $\mathit{O}(10^{-6})$ , which is certainly much less than the magnitude of unsteady shock displacement in the base flow, $\mathit{O}(1)$ ; however, the spatially diffuse negative time-mean streamwise velocity perturbation in the region of the spatially diffuse time-mean reflected shock reveals this behaviour. In conjunction with the time-local analysis of § 5.3.6, we can furthermore conclude, regarding low-frequency motion, that when the reflected shock is located near its mean position, the corresponding time-local linear tendency promotes upstream shock motion, increasing the size of the separation region, consistent with the observed time-mean linear tendency of the interaction. These findings are not inconsistent with those of Touber & Sandham (Reference Touber and Sandham2009), Pirozzoli et al. (Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010) and Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017), who employ the time-mean base flow to describe a linearly unstable non-oscillatory global mode. However, the current analysis shows that the tendency for upstream motion is definite; there is not an arbitrary nature to the sign (direction) of the absolute instability, it is not symmetric as suggested in previous modal mean-flow analyses, and this observation has physical significance. Indeed, the current findings are consistent with the asymmetry of shock motion discussed by Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009), in which upstream motion occurs more rapidly than downstream motion. We discuss this in detail in § 5.3.5, followed by an analysis of how this absolute instability interacts with the nonlinear nature of the base flow in § 5.3.6.

We conclude this section by postulating that for any flow admitting a global absolute linear instability, the time-mean linear tendency of the unsteady turbulent base flow should be qualitatively consistent with traditional stability analyses applied to the time-mean turbulent flow. This postulate is confirmed in the current case of the STBLI, since the time-mean linear tendency is consistent with the behaviour of the absolutely unstable, non-oscillatory, global mode of the time-mean STBLI. Compared to stability analyses of the time-mean turbulent flow, however, the dynamic linear response yields additional insight into the underlying mechanisms that sustain the unsteadiness, particularly in addressing time-local perturbation dynamics, as discussed in § 5.3.6.

Figure 9. Contours of root-mean-square vertical velocity perturbations ( $v_{rms}^{P}$ ) for cases 7 and 9–11, subject to external forcing at different locations, superposed with mean pressure contours of the base flow for reference. All plots have the same contour levels and spatial coordinates ( $x$ – $y$ plane) scaled by boundary layer thickness. (a) Case 7: stochastic forcing in separated shear layer; location, $(53.78,0.79)$ . (b) Case 9: stochastic forcing in upstream subsonic boundary layer; location, $(50.89,0.041)$ . (c) Case 10: stochastic forcing in recirculation region; location, $(56.37,0.041)$ . (d) Case 11: self-sustaining with no forcing; most similar to forcing in recirculation region (case 10).

5.3.4 Effects of stochastic forcing on perturbation field

The linearly constrained perturbation field of the STBLI shows relatively small sensitivity to the specific nature of forcing, as anticipated given the absolute nature of the instability. Results with stochastic forcing are therefore similar to those shown earlier. This is consistent with the discussion by Chomaz (Reference Chomaz2005): in absolutely unstable flows, the internal growth of perturbations overshadows the response to external forcing. On the other hand, the linearly constrained perturbation field of the turbulent boundary layer, without an impinging shock, requires continuous stochastic forcing to sustain its convectively unstable perturbation field, consistent with the common understanding (Chomaz Reference Chomaz2005).

To demonstrate the selective nature of the STBLI perturbation field, the effects of forcing location on the root-mean-square (r.m.s.) vertical velocity perturbation fields are displayed in figure 9. Apart from a local area of influence around each forcing location, the character of the perturbation field is very similar between the different cases. For instance in case 9, the forcing location provides a perturbation source upstream, but these perturbations are attenuated relatively quickly, and the major area of perturbation activity remains in the separation region and downstream boundary layer. Likewise, for case 7, the forcing effects are localized to the separated shear layer. For case 10, in which a perturbation source is added in the recirculation region, the results are quite similar to case 11, containing no forcing. This suggests that the self-sustaining perturbation field of case 11 is akin to a perturbation source originating in the recirculation region; i.e. the absolute instability is sustained by internal perturbation growth within the recirculation region. Additionally, it demonstrates that the STBLI is more sensitive to perturbations in the recirculation region than the upstream region; i.e. all attempts to stochastically force perturbations outside of the recirculation region are overshadowed by unforced perturbation growth in the recirculation region. These observations support the finding that the absolute instability is sustained by constructive feedback through the recirculation region and is not receptive to external forcing. The self-sustaining perturbation field, driven by the absolute instability, is the most natural perturbation field to analyse, since it removes any arbitrariness due to the presence of forcing, and is therefore the subject of much of the analysis below.

5.3.5 Spectral content of base-flow and perturbation fields

We now analyse the unsteady dynamics of the base-flow and perturbation fields using several different time series. These series are chosen to describe a variety of unsteady features: the shock footprint indicated by wall pressure fluctuations, indicators of the size of the separation bubble, as well as the near-field shock location. Observations of the base flow are first summarized to provide a foundation for the subsequent discussion of the accompanying perturbation dynamics. Overall, the dynamics of the base flow agrees well with those described in the literature, providing a measure of base-flow validation necessary for the perturbation analysis. The discussion focuses on several prominent frequency bands, each described qualitatively with a corresponding physical process: low-frequency content $(St_{L}\sim 0.03)$ , corresponding to separation bubble breathing and large-scale shock motion, low–mid-frequency content $(St_{L}\sim 0.1)$ , corresponding to bubble oscillations, high–mid-frequency content $(St_{L}\sim 0.5)$ , corresponding to Kelvin–Helmholtz induced shear layer phenomena, including shedding and flapping of the reflected shock and high-frequency $(St_{L}\gtrsim 1)$ jitter, corresponding to the passage of fine-scale turbulent fluctuations.

In all cases, the time series are sampled at intervals of $20\unicode[STIX]{x1D6FF}_{0}/U_{\infty }$ , and the power spectral density (PSD) is obtained using Welch’s windowed overlapping segments estimator method, by splitting the signal into eight segments with 50 % overlap and windowing with the Hamming function. The PSD is pre-multiplied by the corresponding Strouhal number, based on the time-mean separation length, to effectively represent the power distribution on a logarithmic scale. Further, the PSD is normalized, such that the integral of pre-multiplied power over all resolved frequencies is unity. In some cases, the PSD is filtered using Konno–Ohmachi (KO) smoothing, which ensures constant bandwidth smoothing on a logarithmic scale (Konno & Ohmachi Reference Konno and Ohmachi1998). The base-flow and perturbation fields were evaluated, in multiple simulations, for over five cycles of the low-frequency $(St_{L}\sim 0.03)$ unsteadiness. For the former, adequacy was established by comparing with the known spectra from well-documented results for this configuration (Dupont et al. Reference Dupont, Haddad and Debiève2006; Touber & Sandham Reference Touber and Sandham2009; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013). With regard to the perturbations, these reach statistical stationarity quickly, and short duration simulations are observed to be adequate to describe their time-local behaviour. For example, in principle, the band-isolation study (§ 5.3.6) only requires one cycle of the low-frequency unsteadiness; results were nonetheless confirmed by comparing with the full data set. Therefore, the resolution of low-frequency unsteadiness is reasonable to support the conclusions drawn in this work.

Figure 10. Base-flow reflected shock location, reverse flow mass-per-span and separation bubble mass-per-span signals, with corresponding power spectral densities. (a) Base flow: reflected shock location signal, with evident asymmetry. Positive location corresponds to upstream displacement from the time-mean position. Displacement is normalized by the time-mean separation length. (b) Base flow: reverse flow mass-per-span signal, with time-mean subtracted. (c) Base flow: separation bubble mass-per-span signal, for flow satisfying $u<0.15u_{\infty }$ , with time-mean subtracted. (d) Pre-multiplied and normalized power spectral densities comparing base-flow shock location, reverse flow and separation bubble mass-per-span signals.

Notably, asymmetry is observed in the shock motion, whereby large-displacement upstream motion occurs relatively quickly, followed by slower, less coherent, downstream motion, at intervals representative of the low-frequency $(St_{L}\sim 0.03)$ cycle. This is evident in the base-flow near-field reflected shock location signal (figure 10 a), which describes the streamwise location of the maximum value of velocity dotted with pressure gradient (constrained to the near-field region of the reflected shock), at a height of $y=4\unicode[STIX]{x1D6FF}_{0}$ , in the spanwise centre of the domain. As noted earlier, this asymmetry is consistent with the experimental observations of Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009). Further analysis of the genesis of this asymmetry is provided in § 5.3.6.

Two additional signals are analysed to elucidate the dynamics of the separation region. A reverse flow mass-per-span signal, comprising the total fluid mass per span in the reversed flow region, is presented in figure 10(b), and a separation bubble mass-per-span signal, of the total fluid mass per span satisfying $u<0.15u_{\infty }$ , is presented in figure 10(c). The reverse flow mass-per-span signal (figure 10 b) is dominated by $St_{L}\sim 0.1$ content, corresponding to bubble oscillations, with a minor side lobe at $St_{L}\sim 0.03$ , corresponding to low-frequency bubble breathing. On the other hand, the separation bubble mass-per-span signal: $u<0.15u_{\infty }$ (figure 10 c) shows a smaller contribution from $St_{L}\sim 0.1$ content, and a larger contribution from $St_{L}\sim 0.03$ content, resulting in a better indication of shock displacement. This trend holds across other thresholds defining the ‘turbulent separation bubble’: larger thresholds on streamwise velocity result in a better indication of low-frequency $(St_{L}\sim 0.03)$ large-scale shock motion, while smaller thresholds result in a better indication of low–mid-frequency $(St_{L}\sim 0.1)$ bubble oscillation dynamics. As § 5.3.6 elaborates, the dynamics in these separate frequency bands can be described relatively independently. Priebe & Martín (Reference Priebe and Martín2012) similarly observe significant $St_{L}\sim 0.1$ content corresponding to streamwise oscillation of the low-pass filtered separation point in the DNS of a Mach 2.9 compression ramp STBLI. In fact, $St_{L}\sim 0.1$ content is found in the separation region power spectra of many studies (Dupont et al. Reference Dupont, Haddad and Debiève2006; Touber & Sandham Reference Touber and Sandham2009; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013), but it is often not discussed in detail.

Figure 11. Comparison of base flow and perturbation wall pressure power spectral densities at streamwise locations surrounding the interaction region. The Strouhal number is based on the incoming free stream velocity and the mean separation length. Streamwise distance has been normalized by the mean separation length, and the mean separation point is located at $x^{\ast }=0$ . Spectrum normalization is performed independently at each streamwise location. (a) Base-flow wall pressure pre-multiplied and normalized power spectral density. (b) Perturbation wall pressure pre-multiplied and normalized power spectral density.

We also observe a significant lagged correlation between the reflected shock motion and the dynamics of the separation bubble. The maximum correlation coefficient between the shock location and the reverse flow mass-per-span signals is 0.38 with a lag of $18\unicode[STIX]{x1D70F}$ . A stronger instantaneous correlation of 0.53 is found with a lag of $15\unicode[STIX]{x1D70F}$ , when correlating the shock location and separation bubble mass-per-span: $u<0.15u_{\infty }$ signals. These correlations of shock motion with the size of the separation, in which the shock motion lags in response to changes in separation, are consistent with previous observations by Morgan et al. (Reference Morgan, Duraisamy, Nguyen, Kawai and Lele2013), confirming that a causal relationship could exist, whereby large-scale shock motion responds primarily to growth or contraction of the separation bubble. This causal relationship (sign of the lag) is independent of the height at which the correlation with the reflected shock is taken, though the magnitude of the correlation and lag vary.

In figure 11, the base flow and perturbation wall pressure PSDs are compared at streamwise locations around the interaction region. The PSD at each streamwise location is normalized, such that the integral of pre-multiplied power over all resolvable frequencies is unity. The base-flow wall pressure dynamics agree well with previous studies (Dupont et al. Reference Dupont, Haddad and Debiève2006; Touber & Sandham Reference Touber and Sandham2009; Agostini et al. Reference Agostini, Larchevêque, Dupont, Debiève and Dussauge2012; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013). Low-frequency content $(St_{L}\sim 0.03)$ , corresponding to shock motion and bubble breathing, is prominent near the time-mean separation and reattachment locations; normalizing the PSD independently at each streamwise location facilitates the identification of prominent low-frequency dynamics at reattachment; these fluctuations are less intense than the low-frequency fluctuations at separation, but they are intense compared to the pressure fluctuations across the full spectrum at reattachment. Low–mid-frequency content $(St_{L}\sim 0.1)$ , corresponding to bubble oscillations, is present near the separation and reattachment points, as well as downstream of the interaction region. High–mid-frequency content $(St_{L}\sim 0.5)$ , corresponding to Kelvin–Helmholtz shedding, is present downstream of the interaction region. High-frequency content $(St_{L}\gtrsim 1)$ , typical of fine-scale features in a turbulent boundary layer, is prominent upstream of the interaction region; it persists throughout the interaction, reducing in frequency as its convective speed slows and its outer scale increases, eventually being overtaken in amplitude by a dominant contribution from $(St_{L}\sim 0.5)$ shed structures downstream of the interaction.

Necessarily, the unsteady dynamics in each of these frequency bands will not collapse with $St_{L}=fL/U_{\infty }$ for all flow conditions. Rather, $St_{L}$ is chosen to describe the unsteadiness since the prominent low-frequency unsteadiness collapses with this Strouhal number for a variety of flow conditions (Dussauge et al. Reference Dussauge, Dupont and Debiève2006), providing a unified scale on which to compare all of the described bands with respect to the low-frequency unsteadiness for these particular flow conditions. Indeed, the high-frequency band associated with fine-scale turbulent fluctuations of the incoming boundary layer must collapse with $St_{\unicode[STIX]{x1D6FF}}=f\unicode[STIX]{x1D6FF}_{0}/U_{\infty }=1$ , independent of the STBLI separation length. It is still to be determined for which Strouhal numbers the two described mid-frequency bands might collapse, since these frequency bands have not been characterized in as extensive detail in many studies.

The wall pressure PSD for the perturbation field is calculated in the same manner as for the base flow. Downstream of the interaction, high frequencies typical of a turbulent boundary layer $(St_{L}\gtrsim 1)$ are identified, consistent with the observation that the perturbation field highlights regions of high-potential linear growth of the base-flow turbulent structures. Mid-frequency $(St_{L}\sim 0.06-0.1)$ content is found near the time-mean separation point, corresponding to perturbations of the reflected shock. This suggests that changes in the linear tendency of the reflected shock are more significantly influenced by mid-frequency mechanisms than low-frequency mechanisms. The wall pressure perturbation PSD upstream of the time-mean separation point requires special care in its interpretation, since the perturbation field does not always penetrate much upstream of the time-mean separation point. For instance, if a particular upstream low-frequency motion of the reflected shock does not reach its greatest upstream extent, the perturbation field will also not be carried farther upstream. The effect of this is to skew the PSD toward higher frequencies. Nonetheless, this analysis illustrates that while the perturbation field is a strong function of the base flow, there are significant differences in the spectral properties of the base-flow and perturbation fields.

To demonstrate the linearity of forcing and attenuation on perturbation field spectral content, wall pressure spectra are presented in figure 12, for cases 6–8, in which forcing and attenuation are applied with different magnitudes, separated by one order of magnitude, as described in table 4. In these spectra, the PSD is normalized by $(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}})^{2}$ , under which the spectra should collapse if linearity is maintained. KO smoothing has been applied to reduce noise at higher frequencies. At the time-mean separation point (a), cases 7 and 8 collapse, while case 6 undershoots the peak. The collapse of cases 7 and 8 is anticipated; however, the deviation of case 6 appears to demonstrate nonlinearity in the scaling of forcing and attenuation target magnitudes between cases. This apparent nonlinearity can result from differences in the specific realization of stochastic forcing, which is unique to each case; it is not indicative that higher-order terms in the expansion of (3.11) are non-negligible, since the results § 5.3.2 demonstrate that the scaling of attenuation alone is linear. Similarly, the results of Unnikrishnan & Gaitonde (Reference Unnikrishnan and Gaitonde2016) demonstrate that the method is linear with respect to scaling of deterministic forcing. Regardless, at the downstream location (b), all cases collapse indicating linearity. The results presented herein regarding the self-sustaining linearly constrained perturbation field are obtained with the attenuation factor of case 7, reaffirming that the perturbations remain linear with respect to the base flow.

Figure 12. Wall pressure perturbation power spectral densities demonstrating the linearity of forcing and attenuation for cases 6–8. The PSDs have been normalized by $(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P_{L\,target}})^{2}$ to demonstrate collapse of the spectra. (a) Wall pressure perturbation power spectral density at the time-mean separation point. (b) Wall pressure perturbation power spectral density downstream of the reattachment point $(x^{\ast }=1.1)$ .

The attenuation factor is related to the enforcement of linearity and thus provides insight into the dynamics of the interaction, since it distinguishes instances in time where perturbation growth is more prominent from those where it is less so. Characteristics of the variable attenuation factor for the self-sustaining perturbation field (case 11) are presented in figure 13. The attenuation-factor PSD is characterized by two peaks at approximately $St_{L}\sim 1$ , corresponding to the passage of convecting fine-scale turbulent structures, and $St_{L}\sim 200$ , corresponding to the changing location of the maximum norm of the perturbation field $(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P})$ with the fine-scale features of convecting structures; recall, the perturbation field evolves to highlight turbulent features with high potential for linear growth. Scrutiny of animations of the evolving perturbation and base-flow fields indicates that the maximum norm of the perturbation field $(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P})$ is often associated with a hairpin-like structure downstream of the interaction region. Since the attenuation factor, $\unicode[STIX]{x1D6FC}$ , is directly dependent on $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P}$ , it varies with the coherence time of these turbulent structures, explaining the low-frequency peak. Additionally, the band of frequencies comprising this peak indicates that high-frequency $(St_{L}\sim 1)$ turbulent content and high–mid-frequency $(St_{L}\sim 0.1{-}0.5)$ content have a much stronger tendency for linear growth in the base-flow turbulent environment than low-frequency $(St_{L}\sim 0.03)$ content. It follows that any mechanism for instability acting over low-frequency scales must have a much smaller growth rate than its higher-frequency $(St_{L}\sim 0.1{-}2)$ counterparts. The high-frequency $(St_{L}\sim 200)$ peak corresponds to variation of $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P}$ with short-time evolution of fine-scale features on a passing coherent structure. Since the turbulent structure evolves throughout its time of coherence, the spatial location at which $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P}$ is found will vary with the evolution of that turbulent structure as the location for maximum potential growth changes. The attenuation factor is smoothly resolved by the chosen time step, with the high-frequency variation of $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}}^{P}$ being resolved by $\mathit{O}(10)$ time steps. Note that the discrete attenuation factor $(1-\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}\unicode[STIX]{x1D70F})$ exceeds unity in some instances, leading to brief amplification of the perturbation field. On average, however, the discrete attenuation factor takes a value of 0.860, with a standard deviation of 0.084. It follows that, on average, the absolute instability of the STBLI corresponds to a Lyapunov exponent of approximately $140U_{\infty }/\unicode[STIX]{x1D6FF}_{0}$ , significantly larger than the Lyapunov exponent of $10.3U_{\infty }/\unicode[STIX]{x1D6FF}_{0}$ identified earlier for the shock-free turbulent boundary layer; i.e. the STBLI absolute instability results in an exponential rate of perturbation amplification that is an order of magnitude larger than the shock-free, convectively unstable, turbulent boundary layer.

Figure 13. Discrete variable attenuation-factor signal and power spectral density for the self-sustaining perturbations of case 11. (a) Discrete variable attenuation-factor signal: $1-\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}$ . (b) Pre-multiplied and normalized power spectral density of discrete variable attenuation factor.

5.3.6 Band isolation of base-flow and perturbation fields

In this section, we isolate the flow characteristics associated with the prominent frequency bands observed in the spectral analysis of § 5.3.5. For this purpose, a series of low-pass finite-impulse-response (FIR) window-based filters are applied. Since the low-frequency dynamics are broadband, a sharp cutoff is neither necessary nor desirable. They are therefore implemented through a time domain moving average filter, to optimize smoothing characteristics in the time domain, albeit with a wider transition band. The first low-pass cutoff is chosen at $St_{L}=0.06$ , to isolate low-frequency unsteadiness of bubble breathing around $St_{L}\sim 0.03$ . The low-pass content is then subtracted from the instantaneous signal, and a second low-pass filter is applied with a cutoff at $St_{L}=0.2$ , to isolate the low–mid-frequency bubble oscillations around $St_{L}\sim 0.1$ . The first two modes are then subtracted from the instantaneous signal, and a third filter with low-pass cutoff frequency of $St_{L}=0.7$ is applied, to isolate high–mid-frequency Kelvin–Helmholtz shedding around $St_{L}\sim 0.5$ . The residual contains higher frequencies, typical of convecting fine-scale turbulent structures, discussed earlier in the context of figure 13. This procedure results in a set of band-pass filtered modes that form a complete basis for the time-resolved flow.

This technique allows for band isolation of the flow similar to previous analyses of STBLIs. Several efforts (Pirozzoli et al. Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013; Priebe et al. Reference Priebe, Tu, Rowley and Martín2016; Nichols et al. Reference Nichols, Larsson, Bernardini and Pirozzoli2017) have accomplished this by applying a Fourier decomposition, or DMD, to fluctuations of a stationary flow; the latter approach is equivalent to the former under these conditions (Chen et al. Reference Chen, Tu and Rowley2012). Others (Pirozzoli et al. Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010; Priebe & Martín Reference Priebe and Martín2012) have similarly employed filtering techniques for this purpose. As anticipated, band-isolated modes of the STBLI agree qualitatively well, regardless of the technique used to generate them. For this application, in which the frequencies of interest are broadband (figure 11 a), we employ the filtering approach, from which a single mode reproduces the time-local dynamics more faithfully. The band-isolated modes of the base flow agree well with the trends observed in these previous studies; the low-frequency $(St_{L}\sim 0.03)$ and high–mid-frequency $(St_{L}\sim 0.5)$ modes have been well documented and compare favourably. We emphasize that the primary insight provided in this section is not to validate the observed band-isolated behaviour of the base flow, but rather to band isolate the base flow and perturbations simultaneously, drawing insight from how the band-isolated perturbations are correlated with the base flow at certain phases of each band-isolated cycle. This analysis provides unique information about linear tendencies, which was not available to the aforementioned efforts that analysed the band-isolated base flow.

Figures 14, 16 and 17 display a representative cycle of the band-pass filtered results corresponding to each of these frequency bands. In each, the left column shows the band-pass filtered streamwise velocity base-flow component, with the time-mean streamwise velocity added to aid in visualization, while the right column shows the band-pass filtered streamwise velocity perturbation component. Each cycle is described by snapshots taken at evenly spaced intervals. Contour levels are consistent through each column in all figures, and the spatial coordinates ( $x$ – $y$ plane) are normalized by the time-mean separation length with the streamwise origin located at the time-mean separation point.

Figure 14. Low-pass filtered low-frequency separation bubble breathing and shock motion: $St_{L}<0.06$ , $f<1.1~\text{kHz}$ . (a,c,e,g) Streamwise velocity base flow with arrows indicating base-flow motion. (b,d,f,h) Streamwise velocity perturbation with arrows indicating time-local linear tendency. Black lines are included for separation size reference. (a) $u$ phase 0: bubble at neutral extent, contracting. (b) $u^{P}$ phase 0: bubble at neutral extent, contracting. (c) $u$ phase $\unicode[STIX]{x03C0}/2$ : bubble at smallest extent. (d) $u^{P}$ phase $\unicode[STIX]{x03C0}/2$ : bubble at smallest extent. (e) $u$ phase $\unicode[STIX]{x03C0}$ : bubble at neutral extent, growing. (f) $u^{P}$ phase $\unicode[STIX]{x03C0}$ : bubble at neutral extent, growing. (g) $u$ phase $3\unicode[STIX]{x03C0}/2$ : bubble at largest extent. (h) $u^{P}$ phase $3\unicode[STIX]{x03C0}/2$ : bubble at largest extent.

Just as the time-mean linearly constrained perturbation quantities represent the time-mean linear tendency of the base flow, the filtered (short-time-mean) linearly constrained perturbation quantities represent the time-local linear tendency of the base flow in the corresponding frequency band. This ability to probe the time-local linear tendency of the base flow is one of the primary advantages of the LC-SLES method over perturbation analyses of the time-mean flow. Provided the filter cutoff frequency is moderately lower than any turbulent noise (i.e. ensuring the flow evolves long enough for ergodic averaging to smooth out turbulent noise), the resulting time-local linear tendencies can provide relatively clean indications of the time-local character of flow stability. For instance, one may examine the time-local linear tendency associated with the base-flow reflected shock at an upstream position and infer, at that instant, the direction in which linear mechanisms are driving the base-flow reflected shock. In higher-frequency bands, the time-local linear tendencies are somewhat more contaminated by high-frequency turbulence due to a limited duration for ergodic averaging, as well as flow history effects (i.e. due to the hyperbolic – wave-like – nature of the perturbations in time, the past history of the perturbations becomes more significant, and the observed linear tendencies will lag behind the causal mechanisms in the base flow at higher frequencies), which must be factored into the analysis.

Breathing of separation bubble $(St_{L}\sim 0.03)$ . Snapshots from the low-pass filtered $(St_{L}<0.06)$ flow field are shown in figure 14. The results are discussed by considering four phases of the breathing cycle at frequency $St_{L}\sim 0.03$ .

$\mathit{Phase}=0:$

The bubble volume is at a neutral extent and in the depletion phase of the low-frequency cycle. Negative $u^{P}$ perturbations along the separated shear layer and in the region of the reflected shock indicate the linear tendency for retardation of the flow, upstream motion of the shear layer/shock and bubble growth. Nonetheless, the bubble continues to contract due to nonlinear mass-depletion mechanisms discussed below.

$\mathit{Phase}=\unicode[STIX]{x03C0}/2:$

The bubble volume is at its smallest extent. Like phase 0, negative $u^{P}$ perturbations indicate the linear tendency for bubble growth, and the magnitude of (negative) $u^{P}$ in this phase is larger than for phase 0, suggesting a greater potential for growth than before. The bubble now begins to grow.

$\mathit{Phase}=\unicode[STIX]{x03C0}:$

The bubble volume is at a neutral extent and in the growth phase of the low-frequency cycle. Negative $u^{P}$ perturbations similarly indicate the linear tendency for growth, and the bubble continues to grow.

$\mathit{Phase}=3\unicode[STIX]{x03C0}/2:$

The bubble volume is at its largest extent. Now positive $u^{P}$ perturbations are prominent along the separated shear layer and in the region of the reflected shock, indicating the linear tendency for acceleration of the flow, downstream motion of the shear layer/shock, and bubble depletion. The bubble begins to contract.

The linear-perturbation state (figure 14 b,d,f,h) is consistent with the observed motion of the separation bubble and reflected shock in the base flow (figure 14 a,c,e,g) at phases $\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x03C0}$ , and $3\unicode[STIX]{x03C0}/2$ . This suggests that at these times, the bubble/shock system is responding to the time-local linear tendency of the base flow. However, at phase 0, the perturbation state is contrary to the motion of the bubble, and the bubble continues to contract, while the perturbations indicate that it should grow. This suggests that a nonlinear mechanism is responsible for bubble contraction at this time. This nonlinear depletion process could be due to the mass entrainment mechanism proposed by Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009), in which fluid shedding from the recirculation region accounts for the nonlinear dynamics driving bubble contraction. We further note that while this low-frequency bubble/shock motion occurs at approximately periodic intervals throughout the simulation, it is not generally distributed uniformly throughout each cycle. In fact, the observed motion is similar to the experimental observations of Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009), in that the bubble remains at a small or moderate size through most of the cycle, followed by rapid growth, then a moderate rate of bubble depletion accompanied by the corresponding shock motion. The cycle asymmetry is not necessarily evident in the low-pass phase diagrams, as the sudden growth is smeared from phase $3\unicode[STIX]{x03C0}/2$ into phase $\unicode[STIX]{x03C0}$ , resulting in a larger bubble at phase $\unicode[STIX]{x03C0}$ than would typically be observed in the instantaneous flow. However, the linear-tendency trends represented in this cycle are informative in describing the origin of cycle asymmetry.

In fact, the observed behaviour is consistent with interplay between the base-flow linear tendency and mass-depletion (nonlinear) responses of the bubble. In the initial phases of the cycle, the linear tendency of the base flow promotes bubble growth, while mass depletion promotes contraction. As such, the bubble is suspended in a state of near equilibrium for a time, as mass depletion gradually reduces the bubble size through phases $\unicode[STIX]{x03C0}/2$ – $\unicode[STIX]{x03C0}$ . At this point, mass depletion from the bubble nears completion, so nonlinear forcing promoting bubble contraction diminishes, and the base-flow linear tendency promoting growth overcomes the contraction effect of mass depletion. The bubble then grows rapidly to phase $3\unicode[STIX]{x03C0}/2$ ; i.e. the imbalance, resulting from the reduction in downstream nonlinear forcing relative to upstream linear tendency, allows for the sudden and intense upstream mass flux in the reverse direction, during which the bubble grows to the state observed at phase $3\unicode[STIX]{x03C0}/2$ . However, the growth is abruptly slowed once the character of the base-flow linear tendency changes as shown at phase $3\unicode[STIX]{x03C0}/2$ , at which point the bubble is so large that the base-flow linear tendency now promotes contraction. The upstream motion of the reflected shock is reversed, and the bubble quickly relaxes to a more moderate state. The cycle then resets, and the bubble is suspended in a state of near equilibrium due to the competing linear-tendency and mass-depletion mechanisms. While the essential components of this mechanism have been proposed previously (Plotkin Reference Plotkin1975; Piponniau et al. Reference Piponniau, Dussauge, Debieve and Dupont2009), the current dynamic linear response, obtained using the LC-SLES technique, suggests a synthesis of these concepts, providing the first demonstration of causal dynamics embedded in the time-local linear tendency of the base-flow low-frequency band. It also facilitates the description of a possible mechanism capable of explaining asymmetry in the shock motion cycle, of which the key component is found in the (phase 0) competing mechanisms of base-flow linear tendency and (nonlinear) mass depletion. Notably, the above description may be extended, in the case of an STBLI configuration with different base-flow properties, to explain the opposite asymmetrical bias; e.g. if the base-flow time-mean linear tendency described a downstream tendency for shock motion, yet the interaction retained linear restoring tendencies at extreme shock displacements, then the above explanation may be applied, with inverted directional bias, leading to rapid downstream shock motion and contraction of the separation bubble, followed by moderate upstream shock motion and bubble growth; however, such behaviour is not observed in any of the cases investigated in this work.

Figure 15. Schematic indicating the linear restoring tendencies of the reflected shock at extreme upstream (phase $\unicode[STIX]{x03C0}/2$ ) and downstream (phase $3\unicode[STIX]{x03C0}/2$ ) displacements. The ‘time-mean shock position’ (which has a linear tendency for upstream motion) does not coincide with the ‘time-mean linearly stable shock position’ as first indicated in figure 8.

Figure 16. Band-pass filtered mid-frequency separation bubble oscillation: $0.06<St_{L}<0.2$ , $1.1~\text{kHz}<f<3.6~\text{kHz}$ . (a,c,e,g,i,k) Streamwise velocity base flow with time mean added and arrows indicating base-flow motion. (b,d,f,h,j,l) Streamwise velocity perturbation with arrows indicating time-local linear tendency along the wall and along the separated shear layer. (a) $u$ phase 0: bubble downstream $(0.7,0.02)$ moving upstream. (b) $u^{P}$ phase $0$ : bubble downstream $(0.7,0.02)$ moving upstream. (c) $u$ phase $\unicode[STIX]{x03C0}/3$ : bubble midstream $(0.5,0.02)$ moving upstream. (e) $u^{P}$ phase $\unicode[STIX]{x03C0}/3$ : bubble midstream $(0.5,0.02)$ moving upstream. (e) $u$ phase $2\unicode[STIX]{x03C0}/3$ : bubble upstream $(0.2,0.02)$ ; fluid sheds. (f) $u^{P}$ phase $2\unicode[STIX]{x03C0}/3$ : bubble upstream $(0.2,0.02)$ ; fluid sheds. (g) $u$ phase $\unicode[STIX]{x03C0}$ : bubble moves quickly downstream with shed fluid $(0.7,0.02)$ . (h) $u^{P}$ phase $\unicode[STIX]{x03C0}$ : bubble moves quickly downstream with shed fluid $(0.7,0.02)$ . (i) $u$ phase $4\unicode[STIX]{x03C0}/3$ : bubble $(0.9,0.02)$ and shed fluid move downstream. (j) $u^{P}$ phase $4\unicode[STIX]{x03C0}/3$ : bubble $(0.9,0.02)$ and shed fluid move downstream. (k) $u$ phase $5\unicode[STIX]{x03C0}/3$ : bubble downstream $(1.0,0.02)$ ; fluid accelerated out of interaction. (l) $u^{P}$ phase $5\unicode[STIX]{x03C0}/3$ : bubble downstream $(1.0,0.02)$ ; fluid accelerated out of interaction.

This analysis also suggests that the STBLI permits a time-mean linearly stable configuration, in which the reflected shock is slightly upstream from the time-mean location. At the extremes of the low-frequency cycle (phases $\unicode[STIX]{x03C0}/2$ and $3\unicode[STIX]{x03C0}/2$ ), the bubble and shock motion are consistent with a linear tendency for restoration to a more moderate configuration; previously it was demonstrated that the time-mean base flow is linearly unstable, with a time-mean linear tendency for upstream reflected shock motion, similar to the linearly unstable non-oscillatory mode found in the IVP calculations of Touber & Sandham (Reference Touber and Sandham2009) and the IVP/EVP calculations of Pirozzoli et al. (Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010) and Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017). Hence, the linearly stable configuration would consist of a separation bubble slightly larger than the time-mean size and a reflected shock slightly upstream from the time-mean location, as indicated in figure 15. Low-frequency unsteadiness could then be described in terms of nonlinear mechanisms forcing the reflected shock, which at extreme displacements experiences a linear tendency for restoration to more moderate displacements, as is the premise of the ODE model proposed by Plotkin (Reference Plotkin1975) and derived by Touber & Sandham (Reference Touber and Sandham2011). However, we find that the ‘time-mean linearly stable shock position’ and the ‘time-mean shock position’ do not coincide, contrary to the assumption of this ODE model, a correction for which could lead to an improved model. We suggest implementing such a correction in the form of a damped and driven harmonic oscillator model, as opposed to the Langevin model that is currently employed; necessarily, to reproduce the shock motion asymmetry with such a model, the stochastic forcing that drives the oscillator should have a functional dependence on the position and velocity of the shock, reflecting contributions from internal mechanisms; e.g. the downstream forcing, resulting from bubble mass depletion, must diminish as the bubble contracts. It follows that, in addition to common techniques which reduce the size of the separation, concepts for control that reduce the discrepancy between the ‘time-mean’ and ‘time-mean linearly stable’ shock positions may prove effective at muting low-frequency unsteadiness.

Oscillation of separation bubble $(St_{L}\sim 0.1)$ . Figure 16 shows snapshots of the band-pass filtered $(0.06<St_{L}<0.2)$ flow field highlighting oscillation of the separation bubble at frequency $St_{L}\sim 0.1$ . In this band, six phases are employed to describe the cycle.

$\mathit{Phase}=0:$

The bubble (white oval) is downstream (position $(0.7,0.02)$ ) moving upstream. The separated shear layer linearly tends to retard upstream motion (indicated by positive $u^{P}$ ), but along the wall (position $(0.5,0.02)$ ), there is a linear tendency for upstream acceleration of bubble motion (indicated by negative $u^{P}$ upstream of the current bubble location).

$\mathit{Phase}=\unicode[STIX]{x03C0}/3:$

The bubble is midstream (position $(0.5,0.02)$ ) moving upstream. The shear layer linearly tends to retard upstream motion, but along the wall (position $(0.5,0.02)$ ), there is a linear tendency for acceleration of upstream bubble motion (more extreme than during phase 0).

$\mathit{Phase}=2\unicode[STIX]{x03C0}/3:$

The bubble is upstream (position $(0.2,0.02)$ ), and accelerated by the linear tendency along the wall (position $(0.2,0.02)$ ), but decelerated by the linear tendency away from the wall (position $(0.2,0.1)$ ). The bubble splits: low velocity fluid (pink oval) moves away from wall and is shed downstream, accelerated by the shear layer linear tendency (position $(0.2,0.1)$ ). The shed fluid continues moving downstream.

$\mathit{Phase}=\unicode[STIX]{x03C0}:$

The bubble moves relatively quickly downstream with the shed fluid (position $(0.7,0.02)$ ). This motion is retarded by the shear layer linear tendency.

$\mathit{Phase}=4\unicode[STIX]{x03C0}/3:$

The bubble (position $(0.9,0.02)$ ) and shed fluid continue to move downstream. This downstream motion continues to be retarded by the shear layer linear tendency.

$\mathit{Phase}=5\unicode[STIX]{x03C0}/3:$

The bubble reaches its farthest downstream extent (position $(1.0,0.02)$ ). The bubble and shed fluid are now accelerated downstream, out of interaction region, in accordance with the observed linear tendency. Separation size achieves a minimum. A new near-wall kernel with linear tendency promoting bubble growth and upstream motion appears (position $(0.4,0.02)$ ).

This cycle demonstrates the oscillatory dynamics of the separation bubble at $St_{L}\sim 0.1$ . At phase 0, a small negative $u^{P}$ kernel appears near the midpoint of the interaction region. The bubble emerges near this position and has a linear tendency to be driven upstream, along the wall, in the initial phases $0{-}\unicode[STIX]{x03C0}$ , as demonstrated by the local region of negative $u^{P}$ upstream of the current bubble position. During this upstream bubble motion, the linear tendency of the shear layer opposes upstream motion, except in the near-wall, near-bubble region, as demonstrated by a positive region of $u^{P}$ . After traversing upstream to near the time-mean separation point, the bubble splits, and some low velocity fluid moves away from the wall entering the off-wall region of faster moving flow (phase $2\unicode[STIX]{x03C0}/3$ ). The bubble then quickly moves downstream accompanying the shed fluid (phases $\unicode[STIX]{x03C0}{-}4\unicode[STIX]{x03C0}/3$ ). The shear layer linearly tends to oppose this downstream motion, as indicated by a broad region of negative $u^{P}$ . Finally, after traversing downstream to near the time-mean reattachment point, the shed fluid now linearly tends to be accelerated out of the interaction region (indicated by a region of positive $u^{P}$ ), and the bubble contracts. A new kernel of negative $u^{P}$ appears near the interaction region midpoint, and the cycle repeats.

This behaviour is qualitatively similar to several weakly damped oscillatory modes described by Pirozzoli et al. (Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010) and Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017) in their BiGlobal stability analyses of time-mean STBLI flow fields. Specifically, Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017) describes several weakly damped oscillatory modes $(0.06<St_{L}<0.13)$ for a Mach 2.28 STBLI, with an impinging shock generated by a $10.7^{\circ }$ far-field wedge. These modes depict periodic oscillations of the separation bubble and reflected shock, similar to our observations of the band-pass filtered $(St_{L}\sim 0.1)$ flow fields. The ‘type II’ modes of Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017), in which the ‘reflected shock plays a more passive role’, are particularly similar to the phases of the separation bubble described in figure 16. Additionally, $(St_{L}\sim 0.1)$ content is found in the reflected shock signal (figure 10 d), consistent with the ‘type I’ modes, of the BiGlobal analysis, which primarily depict periodic reflected shock motion. However, we suggest a new interpretation of these observations, in conjunction with our recent observations of the dynamic linear response of the time-resolved STBLI: associating the zero-frequency (non-oscillatory) unstable mode with low-frequency $(St_{L}\sim 0.03)$ unsteadiness, as discussed in the context of § 5.3.3 and figure 14, and associating the weakly damped oscillatory modes with low–mid-frequency $(St_{L}\sim 0.1)$ unsteadiness, as discussed in the context of figure 16. Nonetheless, qualitative agreement between the observed oscillatory behaviour in the LES and BiGlobal stability analyses confirm this mechanism to be an important aspect of low–mid-frequency $(St_{L}\sim 0.1)$ STBLI dynamics.

The band-isolated dynamics of the base flow and perturbations are consistent with the action of a mass-depletion mechanism, such as in the conceptual model of Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009); the competition of the accompanying nonlinear downstream forcing and upstream linear tendency are necessary to describe the low-frequency dynamics. However, like Morgan et al. (Reference Morgan, Duraisamy, Nguyen, Kawai and Lele2013) we find that the ‘turbulent separation bubble’ (§ 5.3.5) responds more significantly at higher frequencies $(St_{L}\sim 0.1)$ than predicted by the model $(St_{L}\sim 0.03)$ . Higher-frequency $(St_{L}\sim 0.1)$ content is also prominent in the dynamics of subsonic separation unsteadiness (Kiya & Sasaki Reference Kiya and Sasaki1985), and it is not entirely clear that the difference in mixing layer spreading rate, between the low- and high-speed separated flows, fully accounts for the lower frequency $(St_{L}\sim 0.03)$ content in the STBLI, without additional considerations of shock-induced effects. Indeed, Agostini, Larchevêque & Dupont (Reference Agostini, Larchevêque and Dupont2015) discuss the significance of the recirculation bubble, with effects unique to this STBLI configuration, in governing both low- and mid-frequency shock dynamics for moderately to massively separated interactions, drawing comparison with aspects of incompressible separated flows.

Furthermore, the prominence of this frequency band in our reverse flow and separation bubble spectra suggests that the accompanying dynamics (figure 16) may be the dominant mechanism for mass transport from the recirculation region to the boundary layer. However, this does not eliminate the possibility for entrainment associated with Kelvin–Helmholtz $(St_{L}\sim 0.5)$ shedding to contribute to mass transport from the recirculation region; ultimately, all of these processes are nonlinearly coupled. Notably, this oscillatory mechanism may also be responsible for the ‘low-frequency’ unsteadiness observed by Kiya & Sasaki (Reference Kiya and Sasaki1985). As discussed previously, a potential mechanism for low-frequency $(St_{L}\sim 0.03)$ unsteadiness in the STBLI relies on interplay between the linear tendency of the STBLI base flow and nonlinear forcing associated with this oscillatory $(St_{L}\sim 0.1)$ mass-depletion mechanism. In the case of Kiya & Sasaki (Reference Kiya and Sasaki1985), there is no shock and thus no interplay to produce unsteadiness at $St_{L}\sim 0.03$ , in which case oscillatory bubble mass changes near the frequency $St_{L}\sim 0.1$ are the lowest-frequency unsteady features present in that flow. That is, the linear component (figure 14) of the mechanism proposed to describe STBLI unsteadiness at $St_{L}\sim 0.03$ only pertains to flows with a similar time-mean linear tendency to the STBLI, and may not pertain to incompressible separated flows, since the separation is not shock induced. For these cases, the linear oscillatory mechanism (figure 16) proposed to describe STBLI unsteadiness at $St_{L}\sim 0.1$ may be the primary mechanism of ‘low-frequency’ unsteadiness.

Figure 17. Band-pass filtered mid-frequency Kelvin–Helmholtz shedding: $0.2<St_{L}<0.7$ , $3.6~\text{kHz}<f<12.7~\text{kHz}$ . (a,c,e,g,i) Streamwise velocity base flow with time mean added and arrows indicating base-flow motion. (b,d,f,h,j) Streamwise velocity perturbation with arrows indicating time-local linear tendency. (a) $u$ phase $0$ : pink wave crest peak. (b) $u^{P}$ phase $0$ : pink wave crest peak. (c) $u$ phase $\unicode[STIX]{x03C0}/2$ : white grows, pink recedes. (d) $u^{P}$ phase $\unicode[STIX]{x03C0}/2$ : white grows, pink recedes. (e) $u$ phase $\unicode[STIX]{x03C0}$ : white grows, pink recedes. (f) $u^{P}$ phase $\unicode[STIX]{x03C0}$ : white grows, pink recedes. (g) $u$ phase $3\unicode[STIX]{x03C0}/2$ : white grows, pink recedes. (h) $u^{P}$ phase $3\unicode[STIX]{x03C0}/2$ : white grows, pink recedes. (i) $u$ phase $0$ : white wave crest peak. (j) $u^{P}$ phase $0$ : white wave crest peak.

Kelvin–Helmholtz (K–H) shedding $(St_{L}\sim 0.5)$ . Figure 17 shows snapshots from the band-pass filtered $(0.2<St_{L}<0.7)$ flow field highlighting K–H phenomena at frequency $St_{L}\sim 0.5$ . The base flow clearly demonstrates wave-like motion through the recirculation region, which corresponds to flapping of the reflected shock foot. K–H waves in this frequency band contribute to the origin of shed structures, leading to dominant mid-frequency $(St_{L}\sim 0.5)$ content in the post-interaction wall pressure spectra of figure 11(a). We note, however, as shown in the reverse flow and separation bubble spectra of figure 10(d), that the mass of fluid in the recirculation region does not respond significantly at this frequency, and instead simply rides the waves. This suggests that while K–H $(St_{L}\sim 0.5)$ waves originate in the separation region and account for the increase in convective content around $St_{L}\sim 0.5$ downstream of the interaction, it is insufficient to describe the mid-frequency separation bubble dynamics without noting both the oscillatory behaviour in the $St_{L}\sim 0.1$ band, as well as wave-like behaviour in the $St_{L}\sim 0.5$ band.

Since this band approaches higher frequencies $(St_{L}\sim 1)$ associated with convecting fine-scale turbulence, we observe more significant contamination from turbulence and flow history effects in the corresponding band-pass filtered perturbation field than in previous lower-frequency bands. Still, periodic changes in the perturbation field character near the time-mean separation point and along the separated shear layer coincide with the genesis frequency and convection of K–H waves, suggesting that these may also be driven by a periodic linear mechanism acting throughout the interaction; the separation bubbles are approximately driven along the major gradients of high to low (red to blue) streamwise velocity perturbation, consistent with the observed linear tendency. The clarity of the band-pass filtered perturbation field in this higher-frequency band could be improved through a conditional analysis, but is not considered in the current work.

This sequential low-pass filtering analysis thus aids in understanding the dynamic relation between the base-flow and linearly constrained perturbation fields in different frequency bands. In each frequency band, and at specific phases, it allows for the identification of whether the STBLI is responding consistently with the time-local linear mechanisms identified in the evolving flow, or responding contrary to the identified linear mechanisms, suggesting the increased importance of nonlinear mechanisms. We anticipate that these linear mechanisms are an essential source of unsteadiness in all similar STBLIs, including those that are influenced by low-frequency coherent fluctuations of the incoming flow, though such coherent fluctuations are not observed in the present study; the relative significance of internal to external mechanisms as sources of unsteadiness remain flow-specific questions. Although this analysis does not demonstrate a statistically robust measure of the dynamical behaviour in each frequency band, it does identify clear trends associated with each frequency band that are qualitatively similar throughout many realizations. As such, it successfully illustrates several key unsteady broadband features present in the STBLI and aids in identifying potential linear mechanisms which may sustain these features in the unsteady turbulent environment.