3.2.1. Instantaneous flow configuration

Instantaneous impressions of the temperature fields are provided in figure 7 to illustrate the resulting STBLI topology. Solid lines indicate instantaneous (yellow) and mean (white) contours of zero streamwise velocity, and show that the adverse pressure gradient imposed by the incident shock is strong enough to cause substantial (mean-)flow separation in all cases. For the higher Reynolds cases, that is, cases $\mathcal {B}_2$ and $\mathcal {B}_3$, the separation shock emanates from deep within the TBL; for the low Reynolds case $\mathcal {B}_1$, in turn, the compression fan generated at the leading edge of the separated flow region is only coalescing into a shock wave well outside the TBL. This is directly linked to the height and curvature of the sonic line (from which compression waves emanate), which is Reynolds number dependent and dictates the intensity of the separation shock footprint on the surface (Dolling & Murphy Reference Dolling and Murphy1983).

Figure 7. Instantaneous temperature fields: (a) case $\mathcal {B}_1$; (b) case $\mathcal {B}_2$; (c) case $\mathcal {B}_3$. Solid lines indicate instantaneous (yellow) and mean (white) isocontours of zero streamwise velocity, and $\blacktriangle$ indicates mean separation (S) and reattachment (R) locations.

At the separation shock foot, the flow is strongly decelerated and starts to detach. Turbulent structures from the upstream TBL are seeded into the free shear layer, and grow in size as they move away from the wall. The upper limit of the separation bubble is eventually established as a result of the interaction between the detached shear layer and the incident-transmitted shock, which turns the flow back towards the surface and initiates the reattachment process. All STBLI cases exhibit a very mild concave streamline curvature near reattachment, which translates into a weak compression fan instead of a coalesced reattachment shock. We also note that the intersection between the incident and the separation shock occurs approximately $2.5\delta _{0}$ above the wall in all cases.

The organization of the investigated STBLI flows is further examined by employing the swirling strength criterion $\lambda _{ci}$ to visualize instantaneous turbulent structures. This criterion is based on local spiralling motion, which in the vicinity of vortex cores translates into a real eigenvalue $\lambda _{r}$ plus a pair of complex conjugate eigenvalues $\lambda _{cr}\pm i\lambda _{ci}$ of the velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). In figure 8, coherent vortical structures in the investigated STBLIs are visualized by plotting an isosurface of the magnitude of the imaginary part of the complex conjugate eigenvalue $\lambda _{ci}$. In this figure, turbulent structures are also coloured by the local streamwise velocity, and a numerical schlieren visualization is included in the background of each rendering for reference. The Reynolds number increases from figure 8(a) to figure 8(c).

Figure 8. Instantaneous vortical structures visualized with the $\lambda _{ci}$ criterion and coloured by the local streamwise velocity: (a) case $\mathcal {B}_1$ ($|\lambda _{ci}|\delta _{0}/u_{\infty }=2.2$); (b) case $\mathcal {B}_2$ ($|\lambda _{ci}|\delta _{0}/u_{\infty }=3.8$); (c) case $\mathcal {B}_3$ ($|\lambda _{ci}|\delta _{0}/u_{\infty }=5.6$). A numerical schlieren is shown in the background slice for each case, and the streamwise velocity colour bar applies to all renders.

Figure 8 reveals clear differences in terms of size and organization of turbulent structures at low and high Reynolds number. Starting from the upstream TBL, structures resembling hairpin vortices can be observed for the low Reynolds case $\mathcal {B}_1$, see figure 8(a). The higher Reynolds cases $\mathcal {B}_2$ and $\mathcal {B}_3$, in contrast, exhibit structures of much smaller size that do not conform to the canonical hairpin vortex. This is in agreement with the observations made by Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011b) on a $M_{\infty }=2.0$ TBL flow at ${\textit {Re}}_{\tau }\approx 1100$.

At the leading edge of the interaction, where incoming turbulence is compressed and the shear layer detaches from the wall, spanwise variations become more pronounced at high Reynolds number, see figure 8(b,c). Based on the colour of the structures in the detaching shear layer, which indicates the magnitude of the local streamwise velocity, such variations are associated with alternating regions of low- and high-momentum fluid. Furthermore, within and above the high-speed streaks, larger vortical structures emerge in the detached shear layer as it moves away from the wall. These structures then break at the bubble apex, where the overall turbulence intensity is reduced. Past the interaction region, the boundary layer has clearly thickened and all cases exhibit more turbulent structures in the outer region compared with the upstream (undisturbed) TBL.

The background schlieren visualizations emphasize the origin of the separation shock deep within the TBL at high Reynolds number, as seen when comparing figures 8(a) and 8(c). Furthermore, the schlieren visualization in figure 8(b) captures the precise instant when the separation-shock front for the moderate Reynolds case $\mathcal {B}_2$ is instantaneously deformed by the passage of outer-layer bulges. These structures, of order $\delta _{0}$ in span, contribute to the wrinkling of the separation-shock foot at both moderate and high frequencies, as discussed in § 3.1.

We have emphasized above that, despite the shift towards lower energetic frequencies, the energetic frequency scales of the incoming turbulence for case $\mathcal {B}_3$ remain too high to significantly impact the low-frequency dynamics of STBLI. However, these structures, associated with larger streamwise wavelengths, still play a crucial role in modulating the flow organization. This is illustrated in figure 9, which shows instantaneous streamwise velocity fluctuations for case $\mathcal {B}_3$ at the same time instance as in figure 8(c).

Figure 9. Instantaneous streamwise velocity fluctuations for case $\mathcal {B}_3$ at the same time instance as figure 8(c). Isosurfaces in (a) correspond to $u^{\prime }/u_{\infty }=-0.12$ (blue) and $u^{\prime }/u_{\infty }=0.12$ (red), and panels (b–g) illustrate streamwise-normal cuts of (a) at $(x-x_{imp})/\delta _{0}=\{-8.4,-5.6,-4.2,-2,1,5\}$, respectively. A numerical schlieren visualization is shown in the background slice of (a) for reference while solid lines in (b–g) indicate the instantaneous isocontour of zero streamwise velocity (yellow) and pressure gradient $|\boldsymbol {\nabla } p|\delta _{0}/p_{\infty }=4$ (grey).

Three-dimensional isosurfaces of positive (red) and negative (blue) streamwise velocity fluctuations are shown in figure 9(a), while figure 9(b–g) depict two-dimensional contours on $yz$ slices at $(x-x_{imp})/\delta _{0}=\{-8.4,-5.6,-4.2,-2,1,5\}$. These visualizations reveal clear correlations between spanwise variations of the separation shock (grey lines) and the edge of the reverse-flow bubble (yellow lines) with alternating regions of low- and high-momentum fluid in the incoming TBL. At the leading edge of the interaction, they strongly modulate the separation-shock foot, see figure 9(c), where the passage of high-speed fluid brings the shock foot closer to the wall and also delays separation right at the surface. Moreover, as these velocity structures traverse the interaction, they reorganize and progressively exhibit much stronger spatial coherence. This is evident when comparing figure 9(b), representing the upstream TBL, with figure 9(g), corresponding to a location downstream of the interaction region. Spanwise autocorrelation functions at these two locations (reported in Appendix B) clearly highlight this increase in coherence, which is observed across all investigated Reynolds numbers.

Previous works have identified the resulting large-scale velocity structures beyond reattachment as the imprint of Görtler-like vortices in the interaction region (Priebe et al. Reference Priebe, Tu, Rowley and Martín2016; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017; Hu et al. Reference Hu, Hickel and van Oudheusden2021). While these studies also argue that such structures could play a pivotal role in driving the low-frequency unsteadiness of STBLIs, their actual relevance remains unclear. We attempted to extract any large-scale vortex from the turbulent background by filtering the three-dimensional snapshot data following the approach of Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017). However, the large-scale circulation is so weak compared with the small-scale turbulence in the present interactions that it was not possible to discern Görtler-like vortices in the filtered instantaneous snapshots. Their visualization is, however, possible with modal decomposition techniques applied to the full time series, as will be discussed in § 3.3.3.

3.2.2. Characteristics of the reverse-flow region

The spatial distributions of the average skin-friction coefficient, see figure 10(a), exhibit a large and connected separation region characteristic of strong interactions. The figure also shows that the location of the global minimum in $\langle C_f \rangle$ is not affected by the Reynolds number and occurs approximately $1\delta _{0}$ upstream of the inviscid impingement point $x_{imp}$. The overall skin-friction distribution within the reverse-flow region, on the other hand, is noticeably different at low and high Reynolds number. The low Reynolds case $\mathcal {B}_1$, indicated with a dash–dotted line in figure 10(a), exhibits a $W$-shaped distribution characteristic of low Reynolds interactions (Priebe, Wu & Martin Reference Priebe, Wu and Martin2009; Aubard, Gloerfelt & Robinet Reference Aubard, Gloerfelt and Robinet2013; Pasquariello et al. Reference Pasquariello, Grilli, Hickel and Adams2014), with a second negative peak right after separation. Conversely, the higher Reynolds cases exhibit a skin-friction plateau in the first half of the separation bubble that precedes the global minimum on the second half.

Figure 10. (a) Time- and spanwise-averaged skin-friction evolution, and (b) probability of reverse-flow: (- ${\cdot }$ - ${\cdot }$) case $\mathcal {B}_1$; (- - -) case $\mathcal {B}_2$; (——) case $\mathcal {B}_3$. Separated regions in (a) are shaded in red and the grey lines denote the corresponding skin-friction distribution for the undisturbed TBL. Distributions in (b) denote the probability of reverse-flow at the wall (black) and its maximum value in the wall-normal direction (blue).

Despite the aforementioned differences, the separation length $L_{sep}$, i.e. the distance between mean separation and reattachment points, does not appear to change noticeably in the investigated Reynolds number range, see also table 3. This finding is in agreement with the experimental work of Souverein et al. (Reference Souverein, Dupont, Debieve, Dussauge, van Oudheusden and Scarano2010) involving interactions at ${\textit {Re}}_{\theta }\approx \{5000, 50\,000\}$ and the comparative numerical study of Morgan et al. (Reference Morgan, Duraisamy, Nguyen, Kawai and Lele2013) over the range ${\textit {Re}}_{\theta }=[1500, 4800]$, where $L_{sep}$ was also found insensitive to the Reynolds number.

Table 3. Topological properties of the interaction region.

Figure 10(a) also includes the skin-friction distribution of the corresponding undisturbed TBLs (grey lines) in order to better appreciate the upstream influence of the STBLI flow. Inspired by Settles et al. (Reference Settles, Bogdonoff and Vas1976), we define the upstream influence parameter $\Delta x_{ui}=x_{imp}-x_{ui}$ for impinging STBLIs as the distance between the inviscid impingement point $x_{imp}$ and the streamwise location where the skin-friction falls by $1\,\%$ with respect to the undisturbed TBL distribution, denoted by $x_{ui}$. The resulting values for this parameter are also reported in table 3, and they confirm the Reynolds number dependency of the upstream influence that was previously observed in compression ramp flows, i.e. a reduced upstream influence with increasing ${\textit {Re}}_{\tau }$. We find that the logarithmic fit $\Delta x_{ui}/\delta _{0}=-0.357\ln {{\textit {Re}}_{\tau }}+9.689$ approximates the present data with an $R^{2}$ value of $0.995$.

A final remark on figure 10(a) concerns the skin-friction distribution downstream of the interaction. As observed, all curves overshoot the corresponding values of the undisturbed TBLs beyond the reattachment point. This is due to the presence of the expansion fan that originates at the trailing edge of the (virtual) shock generator.

The probability of reverse-flow $\chi$ is shown in figure 10(b) to further illustrate Reynolds number effects in the shock-induced flow separation. Black lines indicate the probability at the wall, i.e. the fraction of the total time when $C_f<0$, and blue lines consider the maximum value in the wall-normal direction, i.e. the fraction of the total time when $u<0$ at any point above the wall.

Initially, both probability distributions overlap for the lower Reynolds cases $\mathcal {B}_1$ and $\mathcal {B}_2$, see figure 10(b), until $\chi \approx 0.60$ and $0.45$, respectively. This overlap ends with a local peak in $\chi$ at the wall that reveals the occurrence of partial flow reattachment near the leading edge of the interaction. Interestingly, the location of the local peak for case $\mathcal {B}_1$, found at $(x-x_{imp})/\delta _{0}\approx -5.6$, coincides with the local minimum in skin-friction near separation in figure 10(a). This correspondence suggests that the characteristic $W$-shaped distribution of $\langle C_f \rangle$ in low Reynolds interactions is a result of partial flow reattachment right after separation.

For the high Reynolds case $\mathcal {B}_3$, both probability distributions in figure 10(b) only overlap until $\chi \approx 0.25$. Moreover, all cases exhibit a maximum in reverse-flow probability at $x-x_{imp}=-\delta _{0}$, where $\chi$ is almost unity for cases $\mathcal {B}_2$ and $\mathcal {B}_3$ when considering the maximum value in the wall-normal direction (see the corresponding blue lines). The location of this peak is also associated with a negative peak in $\langle C_f \rangle$, as shown in figure 10(a), in this case representing the global minimum.

The effect of Reynolds number on the recirculation bubble topology is further analysed using a set of topology parameters defined in figure 11(a) and reported in table 3. These parameters include the aspect ratio of the bubble, represented by its height $h_{rev}$ normalized by $L_{sep}$, and the bubble area $A_{rev}$, which corresponds to its volume per unit span. The latter is normalized by the area of the triangle formed by the separation point, bubble apex and reattachment point locations to highlight the degree of compliance with a perfect triangular shape (with area $h_{rev}L_{sep}/2$).

Figure 11. (a) Relevant definitions concerning the interaction topology, and (b) close-up view of the recirculation region. Dividing streamlines are marked with solid black lines in (b), and blue lines are streamlines passing through $y/\delta _{0}=\{0.05, 0.3\}$ at $x-x_{imp}=-7\delta _{0}$. Dotted grey lines indicate isocontours of reverse-flow probability ($\chi =0.01$ and $0.8$), and dashed magenta lines show the free stream flow deflection.

For the higher-Reynolds cases $\mathcal {B}_2$ and $\mathcal {B}_3$, the normalized $A_{rev}$ is very close to unity, see table 3, which indicates that the corresponding bubbles closely resemble triangles. Conversely, the recirculation bubble for the low Reynolds case $\mathcal {B}_1$ has a smaller normalized $A_{rev}$ of $0.78$. A close-up view of the different recirculation regions in figure 11(b) reveals that the reversed flow is initially confined to the wall for this case, over a distance of approximately $2\delta _{0}$ before the shear layer moves away from the wall. This is consistent with the presence of a compression fan instead of a coalesced separation shock at the leading edge of the interaction, and explains the greater deviation of the bubble shape from a perfect triangle.

Additionally, each panel in figure 11(b) includes the dividing streamline (black line) and two other streamlines above the separation region (blue lines). For the low Reynolds case $\mathcal {B}_1$, these streamlines exhibit a much larger curvature radius around the leading edge of the separation bubble. The postshock flow deflection $\vartheta _{sep}$ is slightly larger than $12\,^{\circ }$ in all cases and essentially independent of the Reynolds number. This deflection is also indicated in figure 11(b) and reported in table 3, along with the corresponding theoretical shock angle $\phi _{sep}$.

Furthermore, table 3 provides additional parameters that assess the symmetry between the leading and trailing portions of the bubble. These parameters include the corresponding distance between the separation point and the streamwise location of the bubble apex, denoted $L_u$ following the nomenclature used by Giepman (Reference Giepman2016) for laminar and transitional interactions (see also figure 11a). The corresponding volume of the bubble above this segment, $A_u$, is also reported and highlighted with a red shade in figure 11. Interestingly, both symmetry parameters reveal that the separation bubble for case $\mathcal {B}_2$ exhibits the highest degree of symmetry in shape, with $L_u=0.56L_{sep}$ and $A_u=0.53A_{rev}$. This symmetry is visually evident upon inspecting figure 11(b). The reverse-flow probability within the bubble, however, is clearly asymmetric in all cases, as indicated by the $\chi =0.8$ isocontour.

The isocontour of $\chi =0.01$ is also included in figure 11(b) to illustrate the extent of the instantaneous reverse-flow. According to Simpson's terminology for TBL separation, $\chi =0.01$ marks the condition for incipient separation (Simpson et al. Reference Simpson1989). This isocontour appears as an essentially straight line at the leading edge of the bubble in all cases, see figure 11(b), and closely follows the postshock deflection of the outer flow. Notably, the maximum height above the wall of the $\chi =0.01$ isocontour is approximately $0.9\delta _{0}$, which is two to three times higher than the mean bubble height $h_{rev}$. Moreover, at the trailing edge of the bubble, the contour line is curved and eventually meets the wall approximately $2\delta _{0}$ downstream of the reattachment point. We use the streamwise distance over which the reverse-flow probability is higher than $\chi =0.01$ to define the reverse-flow-based length scale $L_{\chi }$. This parameter could serve as an alternative to $L_{sep}$ in cases involving weakly separated STBLIs with partitioned bubbles. For the present strong interactions, $L_{\chi }$ remains approximately $11\delta _{0}$ and shows little sensitivity to the Reynolds number (see table 3).

3.2.3. Mean and fluctuating wall pressure

After discussing the corresponding differences in recirculation-bubble topology, we now consider the impact of Reynolds number on wall-pressure characteristics of STBLIs. Figure 12(a) shows the mean wall-pressure distribution for the investigated interactions, all exhibiting the typical features of strong STBLIs with substantial flow separation (Matheis & Hickel Reference Matheis and Hickel2015; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017). Characteristic inflection points associated with separation, the onset of reattachment and the reattachment compression are discernible on each curve (Délery, Marvin & Reshotko Reference Délery, Marvin and Reshotko1986). The low Reynolds case $\mathcal {B}_1$, however, presents a more gradual pressure rise at the leading edge of the interaction as a result of the diffuse separation-shock foot, while the higher Reynolds cases $\mathcal {B}_2$ and $\mathcal {B}_3$ exhibit a sharper pressure increase. The pressure plateau is not fully established in any of the cases, and its magnitude, in close agreement with the predictions by Zukoski (Reference Zukoski1967), appears insensitive to the Reynolds number.

Figure 12. Time- and spanwise-averaged (a) wall pressure, and (b) wall-pressure fluctuation intensity. For the line legend, see caption of figure 10. Mean separation and reattachment points are indicated with a marker ($\bullet$), and the plateau pressure computed according to the empirical formula by Zukoski (Reference Zukoski1967) is also shown in (a).

The corresponding wall-pressure fluctuation intensities are presented in figure 12(b). In contrast to the low Reynolds case $\mathcal {B}_1$, which lacks a discernible peak at the leading edge of the interaction, indicative of attenuated wall-pressure intermittency (Priebe et al. Reference Priebe, Wu and Martin2009), distinct peaks are clearly visible for the higher Reynolds interactions due to the stronger separation-shock footprint on the surface. Notably, the peak intensity is almost twice as large for case $\mathcal {B}_3$ compared with $\mathcal {B}_2$. Additionally, case $\mathcal {B}_3$ exhibits the highest wall-pressure fluctuation intensities throughout the interaction, while the lowest values are observed for case $\mathcal {B}_2$. This observation further evidences the non-monotonic effect of Reynolds number on the interaction dynamics. Despite varying magnitudes, the curves in figure 12(b) share common features in good qualitative agreement with previous works on impinging STBLIs (e.g. Dupont et al. Reference Dupont, Haddad and Debieve2006; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017). Specifically, all distributions show an increased level of fluctuation intensity postseparation, attributed to the flapping motion of the shear layer (Dupont et al. Reference Dupont, Haddad and Debieve2006). This is followed by a slight drop in pressure fluctuation intensity within the separated region before rising again due to the reattaching shear layer vortices.

Moreover, the wall-pressure statistics in figure 12 indicate that mean separation occurs downstream of the excursion domain of the separation shock (i.e. beyond the pressure fluctuation intensity peak in figure 12b) for the interactions at higher Reynolds number. We attribute this to the moderate incident-shock strength in the investigated STBLIs, which results in a deeper penetration of the high-momentum flow into the interaction region before the formation of the separation bubble.

3.2.4. Reynolds stresses

The effect of Reynolds number on second-order velocity statistics is analysed next. Figure 13 shows contours of the Favre-averaged streamwise Reynolds stress $\tau _{11}=\widetilde {u^{''}u^{''}}$. Qualitatively, all cases bear similarities in terms of the location of increased fluctuation intensity, which is mainly confined to the shear layer at the leading edge of the separated region. This is in good agreement with previous works (Dupont et al. Reference Dupont, Piponniau, Sidorenko and Debiève2007; Pirozzoli et al. Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017). Additionally, $\tau _{11}$ is highly damped by the strong convex curvature at the bubble apex in all cases. As pointed out by Sandham (Reference Sandham2016), such curvature effects are strongest for this stress component.

Figure 13. Time- and spanwise-averaged streamwise Reynolds stress $\tau _{11}=\widetilde {u^{''}u^{''}}$ for (a) case $\mathcal {B}_1$, (b) case $\mathcal {B}_2$ and (c) case $\mathcal {B}_3$. Solid line colour legend: (yellow) $\langle u \rangle = 0$; (grey) dividing streamline; (purple) $\langle M \rangle = 1$; (black) $95\,\%$ of $\max (\tau _{11})$. Green dashed lines mark selected streamlines, and $\square$ and $\triangle$ indicate the locations of $\max (\tau _{11})$ and max$(\mathcal {P}_{11})$, respectively.

One notable observation concerns the disparity in the location of maximum $\tau _{11}$ across the different cases, as indicated by the $\square$ symbols in figure 13. It is evident that in figure 13(a,b), for the lower-Reynolds cases $\mathcal {B}_1$ and $\mathcal {B}_2$, this point is found near the separation-shock foot, where the production of $\tau _{11}$, i.e. $\mathcal {P}_{11}$, is highest. This point is indicated with a $\triangle$ symbol. Conversely, for the high Reynolds case $\mathcal {B}_3$ in figure 13(c), the streamwise stress distinctly peaks within the detached shear layer, which is no longer correlated with $\text {max}(\mathcal {P}_{11})$ (see also the isocontours of $95\,\%$ of $\max (\tau _{11})$ shown as black solid lines).

Previous numerical studies, such as Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020) and the recent work of Kang & Lee (Reference Kang and Lee2024), which have examined turbulence amplification across canonical STBLIs, consistently identified peak locations of $\tau _{11}$ and turbulence kinetic energy beneath the separation-shock foot and near the point of maximum stress production. While these findings are consistent with our observations in the lower-Reynolds cases, the observed disparity in stress peak locations for case $\mathcal {B}_3$ suggests that the widely accepted $\tau _{11}$ amplification mechanism proposed in the literature, which is associated with the deceleration of the mean flow and the near-wall cycle of the TBL, may not be the primary source of stress at high Reynolds number.

To clarify this, we analyse the corresponding transport budgets along two mean-flow streamlines crossing the interaction. The first streamline, labelled $s_{a}$ in figure 13, intersects the point of $\text {max}(\mathcal {P}_{11})$ in all cases, which is consistently located in the near-wall region beneath the separation-shock foot. As shown in the inset panels of figure 13(a,b), this point is indeed in close proximity to the point of maximum $\tau _{11}$ for the lower-Reynolds cases $\mathcal {B}_1$ and $\mathcal {B}_2$. The second selected streamline, labelled as $s_b$, cuts through the shear layer as shown in figure 13. Notably, this intersection highlights the misalignment between the mean flow and the track of maximum stress in the detached shear layer. For the high Reynolds case $\mathcal {B}_3$, $s_b$ is selected such that it intersects the location of maximum $\tau _{11}$ over the interaction (i.e. the $\square$ symbol in figure 13c).

The distributions of the $\tau _{11}$ transport budgets along the above-mentioned streamlines are depicted in figure 14. The various terms involved in the transport equation for the Favre-averaged Reynolds stresses are detailed in Appendix D, and they include production, $\mathcal {P}$, viscous diffusion, $\mathcal {D}^{\nu }$, turbulent convection, $\mathcal {D}^{t}$, pressure transport, $\mathcal {D}^{p}$, pressure–strain correlation, $\varPi$, turbulent mass flux, $\mathcal {M}$, and dissipation, $\epsilon$. The convective term of the material derivative is evaluated as it appears on the left-hand side of (D1) (with the sign unchanged) so that it serves as an indication of net $\tau _{11}$ gain. For clarity, two-dimensional contours of the most relevant terms are also shown in figure 15, and they should be considered alongside the discussion of figure 14.

Figure 14. Transport budgets for $\tau _{11}$ along (a,c,e) streamline $s_a$, and (b,d,f) streamline $s_{b}$: (a,b) case $\mathcal {B}_1$; (c,d) case $\mathcal {B}_2$; (e,f) case $\mathcal {B}_3$. Symbol legend: production $\mathcal {P}_{11}$ ($\blacksquare$, black); viscous diffusion $\mathcal {D}^{\nu }_{11}/\bar {\rho }$ ($\blacktriangleleft$, teal); turbulent convection $\mathcal {D}^{t}_{11}/\bar {\rho }$ ($\blacktriangle$, green); pressure transport $\mathcal {D}^{p}_{11}/\bar {\rho }$ ($\blacktriangleright$, orange); pressure–strain correlation $\varPi _{11}$ (, blue); turbulent mass flux $\mathcal {M}_{11}$ ($\times$, yellow); dissipation $\epsilon _{11}$ ($\blacklozenge$, grey); convection $\tilde {u}_k \partial _k \tau _{11}$ ($\bullet$, magenta). For reference, the evolution of $\tau _{11}$ (normalized by $u^2_{\infty }$) is also indicated with a solid red line.

Figure 15. Contours of selected $\tau _{11}$ transport budgets: (a–c) production $\mathcal {P}_{11}$; (d–f) pressure–strain correlation $\varPi _{11}$; (g–i) pressure transport $\mathcal {D}^{p}_{11}$; (j–l) turbulent convection $\mathcal {D}^{t}_{11}$. Panels (a,d,g,j), (b,e,h,k) and (c,f,i,l) correspond to cases $\mathcal{B}_1, \mathcal{B}_2$ and $\mathcal{B}_3$, respectively, as indicated at top of panels (a,d,g,j). Contour levels are shown from $-0.05 u_{\infty }^3/\delta _{0}$ (dark blue) to $0.05 u_{\infty }^3/\delta _{0}$ (dark red).

We consider first the corresponding budget distributions along $s_{a}$, which are depicted on figure 14(a,c,e). It is clear that the magnitude of the production term at the separation-shock foot is largest for the high Reynolds case $\mathcal {B}_3$ in figure 14(e). However, the amplification of $\mathcal {P}_{11}$ with respect to that of the undisturbed TBL is only $1.06$ , which indicates a minor impact of the mean-flow deceleration in the near-wall production at high Reynolds number. For the other cases, the amplification of this term is much larger; $1.58$ for case $\mathcal {B}_2$ in figure 14(c), and $2.44$ for case $\mathcal {B}_1$ in figure 14(a). In addition, the fact that $\mathcal {P}_{11}$ for case $\mathcal {B}_3$ exhibits a progressive increase in the upstream boundary layer is simply a result of the selected streamline deviating away from the wall, from $y^+\approx 6$ on the left-hand side of of figure 14(e) to $y^+\approx 10$ at the peak production. This deviation naturally alters the value of the production term as well as other terms like turbulent transport $\mathcal {D}^{t}_{11}$, which changes from a positive to a negative value. Thus, an indication of the STBLI onset is where previously inactive terms in the undisturbed TBL, like convection or pressure transport, become active.

Figure 14(c,e) reveal that the convection term, which indicates net $\tau _{11}$ gain, is correlated with the initial amplification of the pressure transport term for the higher Reynolds cases $\mathcal {B}_2$ and $\mathcal {B}_3$. This is caused by the stronger footprint of the separation-shock dynamics at the wall, which leads to large negative values of the pressure$-$streamwise-velocity correlation for these cases and consequently a sharp rate of change in streamwise direction. The streamwise extent of increased pressure transport $\mathcal {D}^{p}_{11}$, including both positive and negative values, thus reflects the streamwise excursion range of the separation shock. For the low Reynolds case $\mathcal {B}_1$, in contrast, the more gradual compression and thus weaker footprint of the separation-shock dynamics at the wall result in negligible pressure-transport contributions. Here, the increase in $\tau _{11}$ is solely attributed to the amplification of $\mathcal {P}_{11}$ driven by the deceleration of the mean flow, see figure 14(a). This corresponds to the amplification mechanism previously reported in the literature (Fang et al. Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020), which this analysis demonstrates to be predominant only at low Reynolds numbers. Furthermore, the increase in pressure–strain correlation $\varPi _{11}$ eventually brings the amplification of $\tau _{11}$ to an end in all cases. The pressure–strain correlation is responsible for the redistribution of energy from the longitudinal stress $\tau _{11}$ towards the transverse stresses $\tau _{22}$ and $\tau _{33}$ (and indirectly to the shear stress $\tau _{12}$) which exhibit large positive values of $\varPi _{ij}$ at the separation shock foot (not shown here).

The role of pressure transport on the amplification of $\tau _{11}$ at high Reynolds number is further highlighted in the budget distributions along the streamline $s_{b}$, which are shown on figure 14(b,d,f) for all cases. The level of $\mathcal {P}_{11}$ in the upstream TBL is much lower along this streamline, as expected away from the wall. For case $\mathcal {B}_{1}$, the progressive amplification of the production term together with increased turbulent convection $\mathcal {D}^{t}_{11}$ lead to a net gain in streamwise stress, see figure 14(b). A similar scenario is observed for case $\mathcal {B}_{2}$ in figure 14(d), with an additional contribution of pressure transport, albeit very moderate, at the leading edge of the interaction. The role of pressure transport, however, is much more significant for case $\mathcal {B}_{3}$ in figure 14(f). As shown, the evolution of the convection term at the leading edge of the interaction is highly correlated with the pressure-transport budget, especially its initial steep rise and peak value (occurring at $y^+\approx 350$). This results in the largest rate of change of $\tau _{11}$, which also contributes to the rapid amplification of $\mathcal {P}_{11}$. Even though the magnitude of the production term is much lower than the global production peak observed in the near-wall region (see figure 14e), it undergoes a much larger amplification from the free stream value along this streamline. Once the pressure-transport contribution becomes negative, the increased production (also affected by the mean-flow deceleration) and turbulent diffusion terms still enable a net gain in $\tau _{11}$ for over half a boundary layer thickness in the streamwise direction, eventually leading to the global peak stress along the detached shear layer.

The present analysis thus demonstrates that the maximum amplification of $\tau _{11}$ at high Reynolds number is not linked to the near-wall cycle of the approaching TBL and is not so much influenced by the mean-flow deceleration at the leading edge of the interaction. Instead, it is largely driven by the unsteadiness of the (sharp) separation shock, which has a much stronger footprint close to the wall. As a result of the large pressure–streamwise-velocity correlation in the shock-excursion domain, the pressure-transport term acts as an important source of stress that results in the global $\tau _{11}$ peak being located away from the wall and within the detached shear layer. The contours of $\tau _{11}$ production, pressure–strain, pressure-transport and turbulent convection shown at the leading edge of the interaction in figure 15 further illustrate the effect of Reynolds number on the budget terms. In particular, figure 15(g–i) show the $\tau _{11}$ pressure-transport budget $\mathcal {D}^{p}_{11}$, where visible differences in the separation-shock foot region are observed.

For the sake of conciseness, we do not extend the discussion to the other Reynolds stresses, since no apparent Reynolds number dependency was observed in their corresponding amplification. The only other notable difference observed with previous literature is the location of maximum spanwise Reynolds stress $\tau _{33}=\widetilde {w^{''}w^{''}}$, which is found within the first half of the detached shear layer in the present simulations. This finding is not aligned with the LES results of Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017), where the $\tau _{33}$ peak appears approximately $3\delta _{0}$ downstream of the reattachment location and very close to the wall. We observe a moderate $\tau _{33}$ increase at the same location in our LES data, but the stress intensity is substantially lower than in the majority of the shear layer. Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017) attribute the increased fluctuation to the presence of Görtler-like vortices resulting from the concave streamline curvature at reattachment. While this curvature is rather mild in the present STBLIs, which involve a weaker incident shock than that of Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017), it remains above the critical Görtler number. The observed reorganization of velocity fluctuations across all interactions is also consistent with the presence of streamwise-aligned vortices, see § 3.2.1, but their apparently limited impact on $\tau _{33}$, in line with the observed lower vortex strength, perhaps suggests a less pivotal role in the low-frequency dynamics in our cases.

3.3.1. Wall pressure

Figure 16 shows the premultiplied PSD map of the wall pressure for each of the investigated interactions. The corresponding signals were obtained from wall-pressure taps equispaced in the streamwise direction along the domain centreline and sampled at a frequency $f_s \approx 37u_{\infty }/\delta _{0}$. Spectra have been estimated using Welch's algorithm, with Hanning windows and $10$ segments with $65\,\%$ overlap. The segment length exceeds $100 L_{sep}/u_{\infty }$ in all cases.

Figure 16. Frequency-weighted and normalized PSD map of wall pressure at the centreline: (a) case $\mathcal {B}_1$; (b) case $\mathcal {B}_2$; (c) case $\mathcal {B}_3$. Dashed lines indicate mean separation $(S)$ and reattachment $(R)$ locations for each case. Contour levels range from zero (white) to $0.3$ (black).

The wall-pressure PSD maps in figure 16 illustrate one of the most prominent features of STBLIs: the shifting of the frequency of the most energetic fluctuations to much lower values. Consistent with previous works, the separation shock exhibits broadband low-frequency dynamics at a separation-length-based Strouhal number $0.01\lesssim St_{L_{sep}}\lesssim 0.1$. The spatial extent associated with these low-frequency dynamics, indicative of the streamwise excursion domain of the separation shock, is approximately $\delta _{0}$ for the higher-Reynolds cases $\mathcal {B}_2$ and $\mathcal {B}_3$ in figure 16(b,c). However, for the low Reynolds case $\mathcal {B}1$, the low-frequency content extends for almost $2\delta _{0}$ in the streamwise direction but appears much less energetic, as seen in figure 16(a). This lower energy level is expected considering the diffused character of the separation-shock foot. Moreover, the separation point for this case (labelled $S$) occurs within the low-frequency region, while it is found downstream of the separation-shock excursion domain for cases $\mathcal {B}_2$ and $\mathcal {B}_3$. This observation is consistent with the analysis of wall-pressure fluctuations in figure 12(b).

Beyond the region of low-frequency unsteadiness, energetic frequencies of turbulent fluctuations notably decrease, and this trend is preserved well into the reverse-flow bubble in all cases. This observation confirms the formation of larger turbulent structures in the detached shear layer, associated with lower frequencies. Within the recirculation region, a spectral peak around $St_{L_{sep}}\approx 0.1$ is also visible in all cases at $(x-x_{imp})/\delta _{0}\approx -1$, which corresponds to the point where the skin-friction exhibits a global minimum in figure 10(a). Beyond this point, and near the reattachment location (labelled $R$), energetic frequencies are highly broadband in all cases, spanning over two decades of $St_{L_{sep}}$. These frequencies include: (i) $St_{L_{sep}}$ larger than unity, characteristic of small-scale turbulence (but larger than in the upstream TBL); (ii) intermediate frequencies centred around $St_{L_{sep}}\approx 0.5$, related to the reattaching shear layer vortices (Dupont et al. Reference Dupont, Haddad and Debieve2006); and (iii) $St_{L_{sep}}\approx 0.1$, possibly associated with shear-layer dynamics, akin to the behaviour observed in step flows where this frequency is linked to the flapping motion of the shear layer (Hu et al. Reference Hu, Hickel and van Oudheusden2021). Finally, far downstream of the interaction, the spectra relax towards a state characteristic of canonical wall-turbulence in all cases.

The intermittent nature of the wall pressure at the separation-shock foot is illustrated in figure 17. In figure 17(a), the wall-pressure signal of a tap embedded within the separation-shock excursion domain is shown for each STBLI, while figure 17(b) includes the corresponding normalized p.d.f. of the signal. These data clearly illustrate the accentuated intermittency of the wall pressure at high Reynolds number; notably, the signal for the high Reynolds case $\mathcal {B}_3$ in figure 17(a iii) exhibits a strongly bimodal behaviour, alternating between $p_{\infty }$ and the postshock pressure level (Dolling & Or Reference Dolling and Or1985). The signal is also clearly aperiodic, consistent with the broadband nature of the low-frequency unsteadiness. These observations are in very good agreement with previous high Reynolds experiments on the impinging shock configuration (Dolling & Murphy Reference Dolling and Murphy1983; Dupont et al. Reference Dupont, Haddad and Debieve2006; Rabey & Bruce Reference Rabey and Bruce2017).

Figure 17. (a) Instantaneous wall-pressure signal near the separation shock foot, and (b) corresponding normalized probability density distribution (p.d.f.): (a i,b i) case $\mathcal {B}_1$; (a ii,b ii) case $\mathcal {B}_2$; (a iii,b iii) case $\mathcal {B}_3$. Dotted lines in (b) depict a reference Gaussian distribution.

Furthermore, the corresponding p.d.f. of the high Reynolds wall-pressure signal, shown in figure 17(b iii), exhibits high right-skewness (skewness coefficient of $\alpha _3 = 0.56$), with the major apex at $- \sigma _{p_{w}}$ and the minor apex at $+0.6\sigma _{p_{w}}$. As noted by Dolling & Or (Reference Dolling and Or1985), the former peak reflects the high likelihood of encountering a value in a narrow range around $p_{\infty }$, while the latter indicates the probability of observing a broader range of postshock pressure levels.

For the low Reynolds case $\mathcal {B}_1$, in contrast, wall-pressure fluctuations at the separation-shock foot exhibit an almost perfectly Gaussian distribution, see figure 17(b i), which is characteristic of low Reynolds interactions with a diffused separation-shock foot, where intermittency is attenuated (Ringuette et al. Reference Ringuette, Wu and Martin2008). Case $\mathcal {B}_2$ lies midway between cases $\mathcal {B}_1$ and $\mathcal {B}_3$ in terms of intermittent behaviour. Although the wall-pressure signal beneath the separation shock exhibits moderate intermittency, deviations from the free stream pressure level are not as pronounced or distinct as those observed at high Reynolds number.

3.3.2. Separation shock and recirculation bubble

The effect of Reynolds number on the STBLI unsteadiness is further examined in figures 18 and 19, which, respectively, show the time variation of the spanwise-averaged separation-shock location and bubble-volume signals for all cases, alongside their respective spectral content. The instantaneous separation-shock location is extracted from three-dimensional snapshots by searching for peak values of the pressure gradient field $|\boldsymbol {\nabla } p|$ in a wall-normal slice outside the TBL (at $1.5\delta _{0}$ from the wall, before intersecting the incident shock), and then averaging the resulting shock front in the spanwise direction. The instantaneous volume of the separation bubble, on the other hand, is estimated as the volume of reverse-flow (i.e. $u<0$) in the corresponding three-dimensional snapshot.

Starting with the separation-shock location signals in figure 18, it is evident that all signals exhibit noticeable low-frequency unsteadiness, with deviations from the mean location of up to $0.4\delta _0$ for the high Reynolds interaction. The spectra of the signals, which are shown in premultiplied form, highlight those frequencies in the range $0.01\lesssim St_{L_{sep}}\lesssim 0.1$ which make the largest contributions to the signal variance. This observation indicates that there are mechanisms driving the low-frequency dynamics that are fundamentally independent of the Reynolds number.

Figure 18. (a) Time variation of the spanwise-averaged separation-shock location at $y=1.5\delta _{0}$, and (b) corresponding premultiplied and normalized PSD of the signal: (a i,b i) case $\mathcal {B}_1$; (a ii,b ii) case $\mathcal {B}_2$; (a iii,b iii) case $\mathcal {B}_3$.

Despite the spanwise averaging, higher frequencies are found relatively energetic in the shock-location signal for case $\mathcal {B}_1$. This is again attributed to the diffused character of the separation shock at low Reynolds number, which is much more sensitive to turbulent fluctuations. Instantaneous visualizations of the flow for this case (not shown here) demonstrate that these fluctuations induce variations in the separation-shock front that are comparable to the range of motion of the shock at low frequencies.

Figure 19 shows the time evolution of the reverse-bubble volume and the corresponding spectral energy for the investigated interactions. Several similarities and differences can be discerned when comparing the data with that of figure 18. All bubble-volume signals exhibit noticeable low-frequency unsteadiness at $St_{L_{sep}} < 0.1$, which can be associated with expansions and contraction of the recirculation region from both ends. However, these low-frequency dynamics are not the dominant contributors to the signal variance; instead, all PSDs peak at a distinct higher frequency, centred around $St_{L_{sep}}\approx 0.1$–$0.2$ as seen in figure 18(b).

Figure 19. (a) Time variation of the separation-bubble volume, and (b) corresponding premultiplied and normalized PSD of the signal: (a i,b i) case $\mathcal {B}_1$; (a ii,b ii) case $\mathcal {B}_2$; (a iii,b iii) case $\mathcal {B}_3$.

This observation is in agreement with the numerical works of Morgan et al. (Reference Morgan, Duraisamy, Nguyen, Kawai and Lele2013) and Adler & Gaitonde (Reference Adler and Gaitonde2018) on impinging STBLIs, where the most significant bubble oscillations were found around $St_{L_{sep}}\approx 0.1$. As previously pointed out, such frequency is prominent in subsonic detached shear layers (Cherry, Hillier & Latour Reference Cherry, Hillier and Latour1984; Schrijer, Sciacchitano & Scarano Reference Schrijer, Sciacchitano and Scarano2014) and supersonic step flows (Hu et al. Reference Hu, Hickel and van Oudheusden2021), where it is linked to the flapping motion of the shear layer. A similar phenomenon may occur in the investigated STBLIs, for which animations show visible oscillations in shear layer deflection. These oscillations have the potential to influence the entrainment process and reinjection of fluid into the bubble (Wu & Martin Reference Wu and Martin2008; Piponniau et al. Reference Piponniau, Dussauge, Debieve and Dupont2009), which could explain the observed spectral peak at $St_{L_{sep}}\approx 0.1$.

Furthermore, the fact that the spectra in figure 19(b) for the low Reynolds case $\mathcal {B}_1$ peak at a higher frequency, around $St_{L_{sep}}\approx 0.2$, while for the high Reynolds case $\mathcal {B}_3$ the peak is found at $St_{L_{sep}}\approx 0.1$, suggests a possible correlation between these dynamics and the characteristic scales of the incoming turbulence, although this correlation is not yet clear.

A final remark concerns the relationship between the reverse-flow bubble volume and the location of the separation shock. In line with previous studies (e.g. Wu & Martin Reference Wu and Martin2008; Morgan et al. Reference Morgan, Duraisamy, Nguyen, Kawai and Lele2013; Adler & Gaitonde Reference Adler and Gaitonde2018), we find a significant correlation between both signals for a small negative lag, see figure 20(a), which indicates that separation-shock excursions are preceded by bubble-volume variations. In fact, we find that the time lag between both signals is approximately the acoustic propagation time from reattachment to separation; that is, the cross-correlation peak is found at $\tau^* =\Delta t\, a_{w}/L_{sep}\approx 1.0$ in all cases, as shown in figure 20(b), with $a_{w}$ denoting the speed of sound at the wall temperature. This remarkable consistency in the normalized time lag provides compelling evidence for a downstream mechanism being responsible for the motion of the separation shock.

Figure 20. Cross-correlation between the separation-bubble volume and the spanwise-averaged separation-shock location signals, with the time lag normalized by (a) $\delta _{0}/u_{\infty }$, and (b) $L_{sep}$ over the speed of sound at the wall temperature, $a_w$. Line legend: (- ${\cdot }$ - ${\cdot }$) case $\mathcal {B}_1$; (- - -) case $\mathcal {B}_2$; (——) case $\mathcal {B}_3$.

We also note that the reverse-flow region and the separation-shock location are expected to exhibit correlations at various other time scales associated with multiscale turbulent fluctuations. However, the use of the separation-shock location in spanwise-averaged form effectively removes such correlations and only preserves the observed acoustic connection.

3.3.3. Modal analysis

In this section, we aim to establish connections between the most energetic frequencies identified in the already discussed temporal spectra and global flow phenomena through SPDMD (Jovanović et al. Reference Jovanović, Schmid and Nichols2014). This decomposition method is a variant of the standard DMD algorithm described by Schmid (Reference Schmid2010), which seeks to decompose a data sequence into a set of modes that optimally represent its evolution. In the SPDMD framework, the standard DMD solution is additionally sparsified to produce a low-rank representation of the system dynamics. The sparsity-promoting step is controlled by a user-defined regularization parameter that determines the trade-off between accuracy and sparsity. For algorithmic details, the reader is referred to the original publications (Schmid Reference Schmid2010; Jovanović et al. Reference Jovanović, Schmid and Nichols2014).

The database for the SPDMD analysis consists of $8192$ three-dimensional snapshots per case, recorded at a sampling interval of $0.5\delta _{0,i}/u_{\infty }$. Each snapshot includes the instantaneous streamwise velocity, pressure and streamwise vorticity fields to establish coherent links among these variables. All fields are stored in single precision to alleviate memory requirements, and the decomposition algorithm is parallelized following the approach described by Sayadi & Schmid (Reference Sayadi and Schmid2016).

The regularization parameter of the SPDMD algorithm is selected to yield a sparse representation comprising $41$ modes per case. Since the resulting convex optimization problem in the SPDMD method involves regularizing the least-squares deviation between the snapshot matrix and the linear combination of standard DMD modes (see equations (6) and (9) in Jovanović et al. (Reference Jovanović, Schmid and Nichols2014)), each field in the snapshot matrix (velocity, pressure, vorticity) is appropriately rescaled to ensure that it contributes approximately a third of the $\ell _2$-norm. To guarantee the statistical significance of the sparse representation, dynamic modes with oscillation periods exceeding one-third of the total simulation time are excluded from selection by the SPDMD algorithm.

Figure 21 shows the resulting modal amplitudes and frequencies of the standard DMD solution (grey circles) and SPDMD solution (red crosses) for each case. The SPDMD solution exhibits a consistent structure characterized by a dominant mean mode and $20$ complex conjugate pairs primarily concentrated in the low-frequency range ($St_{L_{sep}}<0.1$) and extending into the moderate-frequency range ($St_{L_{sep}}\approx 0.1$ to $0.6$). Notably, as the Reynolds number increases, there is a higher concentration of the retained modes at low frequencies, where modes generally exhibit the largest amplitudes. For instance, in case $\mathcal {B}_1$, the highest frequency in the sparse representation is $St_{L_{sep}}=0.712$, while for case $\mathcal {B}_3$, it is $St_{L_{sep}}=0.375$.

Figure 21. Modal amplitude $|\alpha _i|$ and frequency distribution of the standard DMD solution ($\circ$, grey) and SPDMD solution ($\times$, red) for (a) case $\mathcal {B}_1$, (b) case $\mathcal {B}_2$ and (c) case $\mathcal {B}_3$. Modal amplitudes are normalized with respect to the amplitude of the mean mode in the corresponding SPDMD solution.

We also note that the SPDMD algorithm has excluded dynamic modes with frequencies above $St_{L_{sep}}\approx 0.7$ from all sparse solutions. For this reason, the high-frequency range $St_{L_{sep}}>1.0$ is omitted in figure 21. However, the maximum modal frequency in each standard DMD solution is $St_{L_{sep}}\approx 10$.

The performance loss of the SPDMD solution, i.e. the fraction of energy that is not captured by the retained dynamic modes (see Jovanović et al. Reference Jovanović, Schmid and Nichols2014), is less than $14\,\%$ in all cases. We also find that the SPDMD solution requires a factor $10$ more modes to reduce the performance loss by another $1\,\%$–$2\,\%$, which is consistent with the broadband nature of the flow. This observation aligns with the findings of Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016), who noted the necessity of including all DMD modes within a particular frequency range to accurately reconstruct the variance of the corresponding dynamics.

Despite these considerations, our focus remains on the corresponding flow structures contained in the retained modes, which can provide valuable insights into the physical mechanisms driving the energetic STBLI dynamics.

A representative low-frequency mode from the SPDMD solution of cases $\mathcal {B}_1$ and $\mathcal {B}_3$ are depicted in figures 22 and 23, respectively. The corresponding modal frequencies are $St_{L_{sep}}=0.076$ and $St_{L_{sep}}=0.039$, and the associated modal amplitudes represent the largest and second largest amplitudes after that of the mean mode. Isosurfaces of modal pressure (figures 22a,b and 23a,b), streamwise velocity (figures 22c,d and 23c,d) and streamwise vorticity fluctuations (figures 22e,f and 23e,f) are shown in the figures at two phases. These phases correspond to the instant when the separation shock is most upstream (figures 22a,c,e and 23a,c,e) and a quarter of a cycle afterwards when it moves fastest downstream (figures 22b,d,f and 23b,d,f). For visualization purposes, a top-hat filter with a constant filer width of $\Delta =0.1\delta _{0}$ in all directions has been applied to the vorticity field, since the vorticity includes more small-scale noise than the other variables. In addition, animations of the dynamic modes described are available in our data repository (Laguarda et al. Reference Laguarda, Hickel, Schrijer and van Oudheusden2023) and should be considered in conjunction with the discussion.

Figure 22. Isosurfaces of positive (red) and negative (blue) fluctuation from a representative low-frequency mode ($St_{L_{sep}}=0.076$) in the SPDMD solution of case $\mathcal {B}_1$: (a,b) pressure; (c,d) streamwise velocity; (e,f) streamwise vorticity. There is a phase shift of $\Delta \theta _m = {\rm \pi}/4$ between (a,c,e) and (b,d,f).

Figure 23. Isosurfaces of positive (red) and negative (blue) fluctuation from a representative low-frequency mode ($St_{L_{sep}}=0.039$) in the SPDMD solution of case $\mathcal {B}_3$: (a,b) pressure; (c,d) streamwise velocity; (e,f) streamwise vorticity. There is a phase shift of $\Delta \theta _m = {\rm \pi}/4$ between (a,c,e) and (b,d,f).

The low-frequency modes in figures 22 and 23 reveal that the characteristic large-amplitude excursion of the separation shock accounts for most of the energy in the pressure field, see figures 22(a,b) and 23(a,b). Its upstream motion is linked to negative pressure fluctuations of the reattachment compression, and vice versa, which is consistent with expansions and contractions of the recirculation bubble from both ends. The incident-transmitted shock, on the other hand, exhibits considerable pitching motion that is not consistent with a pure translation of the shock–shock interaction point. This suggests a non-negligible change in separation-shock deflection at low-frequencies. An asynchronous motion between the separation shock and the incident-transmitted shock is also observed, characterized by a phase shift of approximately $\Delta \theta _m = {\rm \pi}/2$. Furthermore, the fluctuating streamwise velocity field in figures 22(c,d) and 23(c,d) show the presence of large-scale streaky structures statistically linked to the low-frequency dynamics of the separation shock. These streaks emerge at the separation-shock foot and convect downstream. Their strength is reduced at the bubble apex, as a result of the expansion, and increases again throughout the reattachment compression.

Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016) identified very similar streaks in their DNS data of a compression ramp flow at Mach $2.9$ by performing DMD of the three-dimensional streamwise mass-flux field. The authors argue that the observed elongated structures emerge as a result of Görtler-like vortices that play a key role in the low-frequency unsteadiness of the interaction, see also Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017). Inspection of streamwise vorticity fluctuations in figures 22(e,f) and 23(e,f) indeed reveals the emergence of counter-rotating, large-scale streamwise vortices near separation and, more prominently, around the reattachment location. This observation confirms, at the very least, a correlation between the vortical activity throughout the interaction and the characteristic longitudinal excursions of the separation shock at the front.

The observed large-scale vortices, successfully isolated from the turbulent background with the employed modal decomposition technique (unlike with traditional averaging methods), exhibit varying strength throughout an oscillation cycle. This variation is accompanied by a translation of the vortex core primarily along the spanwise direction. Moreover, the streamwise vorticity is clearly in phase with the streamwise velocity field at reattachment, where both large-scale streaks and vortices appear and disappear at the same time instants within the oscillation cycle (see also the animation available online). The separation shock is moving fastest when these structures are established, and it reaches either end of its excursion domain when both streaks and vortices weaken and recombine. This observation provides further support for a direct coupling between the separation-shock motion and the flow dynamics at reattachment.

The specific arrangement of velocity streaks and streamwise vortices for the selected low-frequency mode of case $\mathcal {B}_3$ is depicted in more detail in figure 24, which shows contours of streamwise velocity and streamwise vorticity fluctuations on a streamwise-normal slice at $(x-x_{imp})/\delta _{0}=4$, downstream of the interaction. The phase depicted corresponds to that in figure 23(b,df), and the isosurfaces of positive and negative fluctuation employed in the three-dimensional renderings are here indicated with solid and dashed lines, respectively.

Figure 24. Contours of (a) modal streamwise velocity, and (b) modal streamwise vorticity from the selected low-frequency mode of case $\mathcal {B}_3$ ($St_{L_{sep}}=0.039$) at $(x-x_{imp})/\delta _{0}=4$. Isosurfaces of positive and negative fluctuation in figure 23 are here indicated with solid and dashed lines, respectively.

Large-scale structures are clearly visible in both the streamwise velocity and vorticity fields in figure 24. Specifically, two counter-rotating vortex pairs are visible, arranged in a staggered pattern relative to the velocity streaks, which is consistent with a vortex-induced upwash or downwash. Additionally, the streak and vortex spacing is approximately $0.7\delta _{99}$ in span, and their corresponding cores are found at a distance $0.3$–$0.5\delta _{0}$ from the wall.

Figure 25 includes a spanwise-normal slice of the modal vorticity field at $z/\delta _{0}=1.1$ that further highlights the topology of an individual large-scale, anticlockwise rotating streamwise vortex at the trailing edge of the interaction. The figure clearly shows that the vortex does not extend to the surface but is accompanied by a region of clockwise vorticity fluctuation (opposite to that of the vortex) close to the wall, beyond $(x-x_{imp})/\delta _{0}\approx 3$. These observations align with the findings of Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017) regarding the presence of streamwise-aligned vortices in impinging STBLI and the associated $\tau _{33}$ increase in the near-wall region beyond reattachment. However, as noted in § 3.2.4, the increased stress magnitude in this region does not lead to a global peak in the here-investigated interactions. This discrepancy is likely a consequence of the selected aerodynamic parameters, which result in a milder reattachment compression, thereby reducing the impact of the observed large-scale vortices on the spanwise velocity variance relative to other turbulent fluctuations.

Figure 25. Contours of modal streamwise vorticity from the selected low-frequency mode of case $\mathcal {B}_3$ ($St_{L_{sep}}=0.039$) at $z/\delta _{0}=1.1$. Mean zero streamwise velocity and mean sonic lines are indicated with yellow and purple lines, respectively. For additional details, see caption of figure 24.

Our data thus confirms the statistical link between velocity streaks, vortical structures and large-scale oscillations of the separation shock at low-frequencies, which was previously postulated by Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016) and Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017) but only supported by single-field analyses. The general characteristics of the modal shapes discussed above are consistent across the low-frequency range $St_{L_{sep}}<0.1$ in the various sparse representations considered. The main differences in fact lie in the number of low-frequency modes selected by the SPDMD algorithm (three for case $\mathcal {B}_1$, five for case $\mathcal {B}_2$ and nine for case $\mathcal {B}_3$). Flow structures are very similar for all cases, which is why data for case $\mathcal {B}_2$ is omitted here. An animation of a representative low-frequency mode for case $\mathcal {B}_2$ is also available in our data repository (the associated $St_{L_{sep}}$ is $0.047$).

A final note on the low-frequency modes pertains to the dynamics of the bubble. While the separation shock shows spanwise-coherent behaviour, see figures 22 and 23, the characteristic quasi-two-dimensional breathing motion of the reverse-flow bubble becomes apparent only when the corresponding spanwise-averaged modal shape is superimposed on the mean flow. Such animations are also available in our data repository. However, when considering three-dimensional modal shapes, the observed streaks appear to obscure the expected behaviour of the bubble. This observation challenges the traditional conceptualization of the bubble motion at low frequencies and highlights the complexity of the actual dynamics involved.

Given the importance of $St_{L_{sep}}\approx 0.1$–$0.2$ in the spectra of the reverse-flow bubble, we also examined SPDMD modes at these frequencies. These modes exhibit qualitative similarities to the low-frequency modes previously discussed. Consequently, they have been omitted from the paper for brevity (animations are still available online). Overall, a significant fraction of the modal pressure variance at this frequency is still concentrated around the separation shock, which exhibits a shorter excursion domain and slightly more pronounced spanwise variations. Streaky structures are also observed in the corresponding modal streamwise velocity fields, correlated with the separation-shock motion in a similar fashion as in the low-frequency modes. The same remark applies to the modal streamwise vorticity, where large-scale structures persist throughout and beyond the interaction region. These findings, along with further analysis of these modal shapes, suggest that the potential flapping motion of the shear layer at $St_{L_{sep}}\approx 0.1$, manifests as a highly three-dimensional phenomenon. Furthermore, the presence of large-scale streaks and vortical structures is not limited to low-frequency modes; instead, they appear over a broad frequency range, which raises questions about their role in the low-frequency dynamics of STBLIs.

For completeness, contours of modal pressure for a high-frequency SPDMD mode of case $\mathcal {B}_3$ ($St_{L_{sep}}=0.375$) are shown in figure 26 to illustrate the complete SPDMD reduced-order representation (an animation is also available in our data repository). At this frequency, shock oscillations are clearly associated with shear layer vortices that convect downstream, potentially inducing eddy Mach waves in the supersonic region of the flow (Nichols et al. Reference Nichols, Larsson, Bernardini and Pirozzoli2017). This particular modal shape is consistent across all high-frequency modes of the SPDMD representations and is correlated with small-scale velocity and vorticity fluctuations (not shown here).

Figure 26. Pressure fluctuations associated with a high-frequency mode ($St_{L_{sep}}=0.375$) in the SPDMD solution of case $\mathcal {B}_3$. There is a phase shift of $\Delta \theta _m = {\rm \pi}/4$ between (a) and (b).