4.1. Qualitative flow organisation

To first understand the qualitative flow organisation, figure 5 reports an instantaneous visualisation of the turbulent and shock structures in the flow field using an isosurface of the swirling strength coloured by the streamwise velocity and an isosurface of the shock sensor $\theta$. A video of the flow, generated using in situ visualisation at runtime (Bnà et al. Reference Bnà, Colombo, Crivellini, Memmolo, Salvadore, Bernardini, Ghidoni and Noventa2023), is linked in the supplementary material. In addition to the shocks associated with the uncontrolled SBLI, another shock system is visible in correspondence with the microramp, which has been characterised extensively in previous works (Della Posta et al. Reference Della Posta, Blandino, Modesti, Salvadore and Bernardini2023). In particular, an almost-planar shock wave is generated at the leading edge of the ramp, while a conical shock wave surrounds the wake from the trailing edge. These weak shocks interact with the main impinging shock far from the wall. The turbulent structures show, instead, how the incoming flow is first captured at the sides of the microramp, thus generating the primary vortex pair and then the arch-like vortices around the ramp wake. However, the most notable differences with the traditional uncontrolled SBLI take place in correspondence with the foot of the reflected shock wave, whose shape is altered significantly by the incoming flow. The lower part of the shock is completely disrupted at the centre of the domain, where the ramp wake hits the interaction, and bulges are periodically generated when the K–H vortices around the wake pass through the shock. The spanwise alteration of the separation region, which is, however, still present as it can be observed from the presence of reversed flow between the shocks, also affects the flow downstream of the interaction region, where the compression wave after the separation is no longer homogeneous in the spanwise direction.

Figure 5. Instantaneous visualisation of the turbulent and shock structures. Isosurface of the swirling strength coloured by the streamwise velocity component ($\lambda _{ci} L_{sep}^u / U_\infty = 60$), isosurface of the shock sensor in pink ($\theta = 0.9$), numerical schlieren on the $xy$ and $yz$ slices in the background. Insets show a enlarged view of the microramp (a) and interaction (c) regions.

The differences along the span of the microramp-controlled SBLI are also visible in the instantaneous temperature contours at the symmetry and lateral $xy$ planes (figure 6). While the lateral slice reminds of a traditional 2-D SBLI, only with the addition of two extra shocks upstream, the billows of the ramp wake completely alter the incoming flow (as observed in Bo et al. Reference Bo, Weidong, Yuxin, Xiaoqiang and Chao2012), the interaction region and the downstream flow at the symmetry plane, with a significantly thicker boundary layer.

Figure 6. Instantaneous temperature on $xy$ planes at (a) $z^* = -0.3$ and (b) $z^* = 0.0$. The white lines indicate $u/U_\infty = 0$, while the yellow lines indicate points with $M = 1$. Enlarged views of the separation region are shown in the insets. An arrow indicates a sample billow in the microramp wake.

As anticipated, the presence of the MVG radically affects the incoming boundary layer, which shapes the interaction region in turn. Typical near-wall streaks, observable in the velocity contours on $xz$ planes in figure 7, are now overlapped to the large-scale trace of the ramp wake. On the one hand, the low-momentum wake decelerates the flow at the centre of the domain, corresponding to an increase in the reversed flow extent. On the other hand, the vortical motion of the primary vortices and the transversal mixing associated with the arch-like vortices promote local accelerations of the flow, corresponding to a decrease in the reversed flow. Before and even after the interaction, the meandering motion of the ramp wake and the spanwise alteration of the separation generate the formation of alternated regions of accelerated and decelerated flows, similar to large-scale streak structures.

Figure 7. Instantaneous streamwise velocity on $xz$ planes at (a) $y^+ = 1$, (b) $y/h = 0.5$ and (c) $y/h = 1$. The white lines indicate $u/U_\infty = 0$.

4.2. A 2-D view: a classic 2-D SBLI analysis

Although the qualitative flow description confirmed the literature findings about the increased geometrical complexity of the 3-D flow field for the controlled SBLI, figure 8 shows that the streamwise distribution of the wall-pressure rise is homogeneous in the span, as also observed in the experimental measurements of Babinsky et al. (Reference Babinsky, Li and Ford2009), although tiny differences across the span are visible in the interaction region where the streamwise pressure gradient is minimum. The figure reports the wall pressure at three notable sections in the span, which are: the symmetry plane $z^* = 0$, the lateral plane $z^* \approx - 0.30$ and the spanwise section corresponding to the minimum streamwise extent of the separation, $z^* = -0.0513$ (see § 4.3). The curves show that, compared with the USBLI case, the presence of the microramp induces a downstream shift in the separation onset (indicated with circles) and a slight upstream shift in the reattachment (indicated with squares), which thus entails an overall shorter separation along the span. The wall-pressure standard deviation $\sigma _{p_w}$ at the same sections shows instead that wall-pressure fluctuations are stronger for CSBLI. As in the case of non-adiabatic wall conditions (Bernardini et al. Reference Bernardini, Asproulias, Larsson, Pirozzoli and Grasso2016) or transitional SBLIs (Quadros & Bernardini Reference Quadros and Bernardini2018), a reduction (increase) of the separation length corresponds to an increase (decrease) in the intensity of the wall-pressure fluctuations. The CSBLI curves also present another peak in the rear part of the separation and, most importantly, vary considerably with the span (see § 4.3). In particular, the standard deviation distribution in the interaction region is minimum at the symmetry plane and maximum at the lateral plane, despite having similar values upstream of the separation onset. Indeed, as the increase in $\sigma _{p_w}$ induced by the microramp wake decays quite fast (faster at the symmetry plane) before the interaction region, it is reasonable to believe that the different behaviour for the three sections is more related to spanwise modifications of the shock surface and the separation bubble, or even to 3-D effects, rather than to the vanishing effect of the microramp wake itself (see § 4.3). After the separation, the decay rate of $\sigma _{p_w}/p_\infty$ is larger in CSBLI than in USBLI, and rather homogeneous in the span.

Figure 8. Streamwise distribution of (a) mean wall pressure and (b) wall-pressure standard deviation. Spanwise average of the uncontrolled case (solid black line), controlled case at $z^* = -0.2986$ (solid orange line), $z^* = -0.0513$ (solid light blue line) and $z^* = 0.0$ (solid green line). Symbols indicate the location of the separation point (circles), the reattachment point (squares), and the point with minimum streamwise pressure gradient (diamonds) for the curve of the corresponding colour.

Figure 9 reports instead the distribution of the time-averaged streamwise skin friction component $\overline {C_{f,x}}$ at different sections in the span. As shown by the spanwise-averaged curve of the CSBLI case, the extent of the region with negative $\overline {C_{f,x}}$ is shortened significantly compared with the USBLI case. However, we also note in this case that conditions vary considerably with the span when the microramp is present, suggesting that the streamwise skin friction component alone may provide misrepresentative information regarding the extent and the nature of the separation, because of the non-negligible contribution of the spanwise skin friction component.

Figure 9. Streamwise distribution of the streamwise skin friction coefficient. Spanwise average of the uncontrolled case (solid black line), spanwise average of the controlled case (dashed-dotted blue line), controlled case at $z^* = -0.2986$ (solid orange line), $z^* = -0.0513$ (solid light blue line) and $z^* = 0.0$ (solid green line).

The significant spanwise modulation of the flow is also observed in the $xy$-slices of figure 10, which compares the Favre-averaged streamwise velocity component for the USBLI case (spanwise-averaged) and for the three above-defined spanwise sections of the CSBLI case. To consider compressibility, Favre averages are used, which are defined as $\tilde {\phi } = \overline {\rho \phi }/\bar {\rho }$ for a generic variable $\phi$. The reversed flow region is highlighted and hints that for the two inner sections, the separation is reduced relevantly. However, it is important to note that especially for strongly 3-D flow, as in the case under consideration, the reversed flow is not indicative of the actual separation taking place.

Figure 10. Favre-averaged streamwise velocity on $xy$ planes: (a) USBLI, spanwise-averaged, and CSBLI at (b) $z^* \approx -0.3$, (c) $z^* = -0.05$ and (d) $z^* = 0$. Yellow lines indicate points with $\widetilde {M_\infty } = 1$, white lines indicate points with $\tilde {u}/U_\infty = 0$. Enlarged views of interaction regions are given at the top left corner of each subfigure.

In conclusion, the wall-pressure standard deviation, the streamwise skin friction component, and the $xy$ slices of the mean velocity proved that for many features, but not all, the flow organisation is fully 3-D, especially close to the wall and in the interaction region. The analysis of single streamwise slices of the streamwise components of vectorial quantities may thus be insufficient or even misleading in trying to understand the overall interaction topology and dynamics.

4.3. More than 2-D: the controlled SBLI separation

Given the three-dimensionality of the separation, we must resort to more sophisticated tools for its analysis. Following the work of Legendre & Werlé (Délery Reference Délery2001; Délery Reference Délery2013), the topology of the separation can be examined by studying the organisation of the skin friction lines at the wall. Skin friction lines can be viewed as the limit towards the wall of the streamlines and, if a mean flow is well-defined, they can provide insightful information about the actual location of the separation and reattachment.

Figure 11 reports the skin friction lines for the CSBLI case on half of the domain, taking advantage of the flow case symmetry (see Appendix A for a discussion on this assumption), and highlights the formation of curved separation and reattachment lines along the span, where skin friction lines converge to and depart from, respectively. Critical points ($C_{f,x} = C_{f,z} = 0$) are also highlighted according to their nature: saddle points indicated as light green circles, nodes as light blue squares, and foci as orange diamonds. For the sake of completeness, we remind the reader that only two skin friction lines run through a saddle point, while all the others avoid it adopting the shape of a hyperbolic curve, that all the skin friction lines have a common tangent at a node, except for one of them, and that skin friction lines end at a focus after spiralling around it. Considering also the symmetric half not shown in the figure, the periodic boundary conditions, and the two nodes at infinite upstream and downstream, our configuration for the interaction region (figure 11c) has a total of five nodes ($n=5$), four foci ($\,f = 4$) and seven saddle points ($s=7$), which satisfies the necessary topological condition $n+f-s = 2$ (Délery Reference Délery2013). Our arrangement has one node, two foci and three saddle points more than that in Grébert et al. (Reference Grébert, Jamme, Joly and Bodart2023), as the separation line is split into more than one single segment in our case. As a result, the scenario describes a more complex arrangement of the trailing vortices after reattachment, which suggests a difference in intensity, height and development of the primary vortices after the interaction, likely due to the different flow conditions of the incoming flow considered. Moreover, it is possible to see that close to the symmetry plane, the organisation of the main reattachment line is not as clear as in the rest of the plane. The elusive location and nature of the critical points suggest that, despite the long integration period considered, the mean skin friction lines may have not reached here a proper steady-state representative of the flow behaviour above the wall, which thus asks for further research.

Figure 11. Skin friction lines overlapped to a contour of the wall-pressure standard deviation on the $xz$ plane for the CSBLI case: (a) overall domain, (b) enlarged view of the ramp region and (c) enlarged view of the separation region. Half of the domain is shown for symmetry. Saddle points are indicated with circles in magenta, nodes with squares in light blue and foci with diamonds in orange. The main separation and reattachment lines are highlighted in white, other relevant critical lines in blue, whereas the foremost peak of $\sigma _{p_w}$ is indicated with the dashed black line.

The focus, already observed also in oil-flow visualisations (Babinsky et al. Reference Babinsky, Li and Ford2009), indicates the presence of a tornado-like structure which captures the flow close to the wall and pushes it upwards. A visualisation of the streamlines associated with the tornado in figure 12(b) shows that this structure tilts outwards to join the wall region closer to the symmetry plane and the external sides of the separation bubble. Indeed, its axis is first normal to the wall in correspondence with the focus and ends up being aligned with the spanwise direction towards the side.

Figure 12. Streamlines in the separation region of the CSBLI case: (a) qualitative edge of the recirculation bubble (streamlines coloured by mean vertical velocity), (b) the tornado-like structure and (c) the internal structure of the bubble. Vertical slices report the Favre-averaged Mach number $\tilde {M}$, yellow lines indicate $\tilde {M} = 1$ and blue lines indicate $\tilde {u}/U_\infty = 0$ on the vertical slices at the symmetry and lateral planes. Half of the domain is shown for symmetry.

Compared with the classical 2-D recirculation bubble, the region of separated flow is now more complicated and not identified by the simple condition of negative $\overline {C_{f,x}}$. Moreover, reversed flow is now even less indicative of the separation, which makes it difficult to identify the edge of the recirculating bubble. Figure 12(a) uses streamlines to show the qualitative shape of the bubble, which presents a characteristic saddle shape with a higher separated region at the symmetry and the lateral planes and a lower separation region in correspondence with the minimal streamwise separation extent (see figure 14). The trace of the tornado is also visible in the mean wall pressure at the interaction region in figure 13, where the minimum associated with the vortex core is superposed to the stronger pressure rise induced by the separation shock, resulting in the mild irregularity in the span previously observed in figure 8.

Figure 13. Wall pressure in the interaction region. Skin friction lines in the separation region are indicated in white. Half of the domain is shown for symmetry.

Significant 3-D effects are also visible in the distribution of the wall-pressure standard deviation $\sigma _{p,w}/p_\infty$ in figure 11. Close to the microramp, we can notice regions with relevant pressure fluctuations associated with (i) the impingement of the flow of the primary vortices at the sides of the ramp towards the wall and (ii) the reattachment of the flow after the trailing edge and the following formation of the parallel, primary vortex pair. In the interaction region instead, we observe a first, stronger peak after the separation onset, whose streamwise position follows the separation front and whose magnitude is greater close to the lateral boundaries. The distribution is compatible with the observation that the shock is disrupted at the centre of the domain and, hence, weaker on average (the pressure jump induced by the separation shock at the symmetry plane is 13 % weaker than that at the periodic boundary). As a result of the reduced shock intensity, the amplification of the pressure fluctuations induced by the shock wave is weaker in the central region of the domain. A second peak, with a spanwise-modulated amplitude as well, takes place at approximately half of the separation, in correspondence with the peak height of the bubble. After the second peak, the standard deviation decays, maintaining a non-uniform spanwise distribution even far from the interaction.

Given the separation and reattachment line in figure 11, it is possible to estimate the extent of the time-averaged streamwise separation $\overline {L_s}$ along the span reported in figure 14. The spanwise average $\langle \overline {L_s} \rangle /\delta _{shk}$ shows that the mean separation in the presence of the microramp shrinks by 26.47 % the value observed in the USBLI case. Moreover, the distribution shows that the separation is largely modulated in the span, $(\overline {L_s}_{max} - \overline {L_s}_{min})/ \langle \overline {L_s} \rangle = 26.31\,\%$ along the span, with a minimal separation at $z^* \approx -0.05$ ($\overline {L_s}_{min}/\delta _{shk} = 3.12$),

Figure 14. Spanwise distribution of the time-averaged separation length with respect to the boundary layer thickness at the shock impingement location. Spanwise average of the uncontrolled case (solid black line), local controlled case (solid red line) and spanwise average of the controlled case (dashed red line).

To understand the relationship between the spanwise modulation of the separation and the addition of momentum by the primary vortex pair, figure 15 reports the distribution of the compressible added momentum in the $xz$ plane. First defined in its incompressible version by Giepman et al. (Reference Giepman, Schrijer and van Oudheusden2014), the compressible added momentum is defined as

(4.1)\begin{equation} E_{add} = \int_{0}^{\overline{y^*}} \frac{\overline{\rho u^2}_{CSBLI} - \overline{\rho u^2}_{USBLI}}{\rho_\infty U_\infty^2} \, \mathrm{d} y \end{equation}

and tracks the addition of streamwise momentum towards the wall. Following Giepman et al. (Reference Giepman, Schrijer and van Oudheusden2014), the upper bound of integration $\overline {y^*}$ is taken equal to $0.43\,\delta$, as the separation bubble was shown to be mostly sensitive to the momentum flux at $y$ lower than this threshold. The distribution of $E_{add}/h$ along the entire wall-parallel plane allows us to appreciate that the primary vortex pair brings fresh momentum towards the wall in a large spanwise range, even quite far from the symmetry plane. The peak added momentum just before the interaction is at $z^*\approx -0.1$, just at the side of the minimum separation, which thus suggests that the onset of the separation, and its local extent, follows strictly the addition of momentum associated with the primary vortex pair helical motion.

Figure 15. (a) Normalised compressible added momentum $E_{add}/h$ along the $xz$ plane and (b) enlarged view of the separation region. Skin friction lines in the separation region are indicated in grey. Half of the domain is shown for symmetry.

After a transitory region following the interaction, we can observe that the trace of the primary streamwise vortices becomes once again more visible, indicating that these are able to withstand the shock waves and that their signature lasts for a long streamwise distance (figure 15a). Even far downstream of the separation, we thus have non-uniform conditions as highlighted by figure 16 reporting the spanwise distribution at $x^* = 2$ of the incompressible shape factors and the streamwise skin friction coefficient for the USBLI and CSBLI cases. The shape factor denotes a fuller and healthier boundary layer even after the interaction at the sides of the symmetry plane, which however corresponds to increased skin friction. We point out that the associated increased drag and the potential consequences of a long-lasting, non-uniform flow for the engine components following the inlet may limit the benefits of a reduced separation and thus that these aspects should be addressed carefully when considering MVGs for real applications.

Figure 16. Spanwise distribution at $x^* = 2$ of (a) incompressible shape factor and (b) streamwise skin friction coefficient: USBLI (dashed black line) and CSBLI (solid red line).

Moreover, the fluid induces an additional drag force on the microramp itself that may be significant. We quantified this drag by defining an equivalent surface with the same overall drag of the ramp, with a width equal to $s$ and a constant $C_f$ equal to the skin friction at the leading edge of the ramp ($D = 1/2\rho _\infty U_\infty ^2 C_{f,LE} L_x^{D} s$). The length of this surface $L_x^{D}$ is equal to $2.52L_{sep}^u$, meaning that, in terms of drag, the ramp is equivalent to a significantly longer boundary layer upstream of the interaction.

On the other hand, the momentum transfer towards the wall has a slight beneficial effect in terms of total pressure. Indeed, the drop in the total pressure, averaged on the $yz$ plane, between the inflow and outflow surfaces in the presence of the microramp is 0.82 % smaller than that without control.

4.4. Vortical structures: a mean view

An interesting issue regarding the effects of microramps on SBLI is the mutual interaction between the arch-like vortical structures induced by MVGs and the interaction region.

According to the literature and the results in the previous sections, the main effect on the SBLI of the arch-like vortices is that they periodically disrupt the impinging and reflected shock waves, leading to regions of large curvature in the shock surfaces (the ‘bump’ in figure 5). The effect of the shock waves on the arch-like vortices is instead less clear. Some works (Yan et al. Reference Yan, Chen, Lu and Liu2013) suggest that these vortices travel rather undisturbed across the shock, however, quantitative information supporting this statement, regarding for example the effect of SBLI on the trajectory and intensity of the arch-like vortices, is lacking.

It is known from the literature (Jeong & Hussain Reference Jeong and Hussain1995) that vorticity magnitude may be an improper measure to identify coherent vortical structures. However, as the trace of the vortices related to the microramp and the SBLI is strong enough compared to the background shear, at least in the ramp wake and in the interaction region, we consider in the following the behaviour of the Favre-averaged vorticity, defined in this work as the curl of the Favre-averaged velocity ($\tilde {\boldsymbol {\zeta }} = \boldsymbol {\nabla } \times \tilde {\boldsymbol {u}}$, see Appendix B), to assess the streamwise development of these vortices. In particular, we can assume that, at the symmetry plane, the negative peaks of the spanwise component $\widetilde {\zeta _z}$ correspond to the trace of the K–H vortices associated with the arch-like vortices and with the shear layer delimiting the separation (see figure 17).

Figure 17. Contours of the spanwise component of the Favre-averaged vorticity (a,b) and the density-weighted Favre-averaged vorticity (c,d) for the USBLI (a,c) and the CSBLI (b,d) cases. Yellow lines indicate points at $\tilde {M} = 1$, blue and black lines indicate respectively the position of the top and bottom shear layers of the CSBLI case, while red lines indicate the shear layer of the USBLI case.

Locating these minima and recording the magnitude of the vorticity along these loci allows us to track the trajectory of the main vortical structures and investigate the streamwise evolution of their intensity. Figure 18 reports the wall-normal coordinate of the points with minimal Favre-averaged spanwise vorticity for a given streamwise section, whereas figure 19 reports the corresponding value of the mean spanwise vorticity.

Figure 18. Wall-normal position of shear layers at the symmetry plane based on the spanwise Favre-averaged vorticity (a) without and (b) with density weighting: USBLI shear layer (red), CSBLI bottom shear layer (black) and CSBLI top shear layer (blue).

Figure 19. Spanwise component of the Favre-averaged vorticity (a) without and (b) with density weighting along the respective shear layers at the symmetry plane: USBLI shear layer (red), CSBLI bottom shear layer (black) and CSBLI top shear layer (blue).

Observing the last figure, the peaks in correspondence with the shock position suggest an increase in the intensity of the external shear layers that directly intersect the impinging shock in both the USBLI and CSBLI cases (red and blue curves). Whether this increase is associated with the generation of vorticity induced by baroclinic, diffusive, turbulent or other effects is not clear at this stage. However, if we observe the equation for the evolution of the Favre-averaged vorticity (see Appendix B), we obtain that

(4.2)\begin{equation} \frac{\mathrm{D}}{\mathrm{D}t}\left( \frac{\tilde{\boldsymbol{\zeta}}}{\bar{\rho}} \right) = \left(\frac{\tilde{\boldsymbol{\zeta}}}{\bar{\rho}}\right) \boldsymbol{\cdot} \boldsymbol{\nabla} \tilde{\boldsymbol{u}} + \frac{\boldsymbol{\nabla} \bar{\rho} \times \boldsymbol{\nabla} (\bar{p} + 2/3 \bar{\rho}\tilde{k})}{\overline{\rho}^3} + \frac{1}{\bar{\rho}} \boldsymbol{\nabla} \times \left( \frac{\boldsymbol{\nabla} \boldsymbol{\cdot} \tau^{t,d}}{\bar{\rho}} \right), \end{equation}

where the first term at the right-hand side is the compressible vortex stretching and tilting, the second term is the baroclinic term and the third term is the diffusion of vorticity by the action of viscous and turbulent stresses. We indicated with $\mathrm {D}/\mathrm {D}t$ the material derivative, and with $\tau _{ij}^t = \overline {\tau _{ij}} + \widetilde {\tau _{ij}}^R$ the total stress tensor including both the Reynolds-averaged molecular stress tensor $\overline {\tau _{ij}}$ and the Favre-averaged Reynolds stress tensor $\widetilde {\tau _{ij}}^R = - \overline {\rho u_i'' u_j''}$, while the Reynolds-averaged pressure is indicated as $\bar {p}$, the Favre-averaged turbulent kinetic energy as $\tilde {k}$ and the deviatoric part of the total stress tensor as $\tau _{ij}^{t,d}$. Compressibility effects in the conservation of mass for a generic fluid element are thus correctly accounted for in the Favre-averaged vorticity equation only if we introduce density-weighting. Indeed, by dividing the Favre-averaged vorticity by the Reynolds-averaged density, the classical equations describing vorticity dynamics are recovered. Therefore, if we observe the contours of the specific vorticity $\widetilde {\zeta _z}/\bar {\rho }$ (figure 17), we can see that the sudden jump observed in correspondence with the incident shock is mainly an effect of the conservation of mass associated with the sudden density rise, which is also observable in the USBLI case. Figure 19 further confirms this conclusion, suggesting that only a limited generation of mean vorticity in the top shear layer of the CSBLI case can be related to baroclinic or turbulent effects across the shock, as the vortex stretching and tilting term is null for the spanwise vorticity component because of the flow symmetry at $z^* = 0$. Looking at the specific vorticity, the decay of the shear layer intensity is now almost undisturbed by the presence of the shock, which confirms the qualitative belief that the arch-like vortices are robust enough not to be affected by SBLI. Figure 18, however, shows that there is no relevant difference between the trajectories of the wall-normal coordinates of the $\widetilde {\zeta _z}$ and $\widetilde {\zeta _z}/\bar {\rho }$ peaks. Considering the CSBLI case, the bottom shear layer surrounding the separation region follows the behaviour of the one in the USBLI case up to reattachment ($x^* \approx 0$). Conversely, the upper shear layer defined by the arch-like vortices and delimiting the edge of the boundary layer, first rises slowly until the interaction region, because of the known lift-up at the symmetry plane in the microramp wake, and then follows the triangular shape of the separation region with a slow recover after reattachment. Indeed, the wall-normal position is affected by the separation up to $x^* \approx 1$. After the reattachment point, the bottom shear layer converges to the upper one, which thus delimits the new boundary layer downstream of the interaction, becoming significantly thicker than in the USBLI case.

4.5. Flow unsteadiness

Except for very few recent studies (Dong et al. Reference Dong, Yan, Yang, Dong and Liu2018; Grébert et al. Reference Grébert, Bodart, Jamme and Joly2018; Sun et al. Reference Sun, Chen, Li, Liu, Guo and Yuan2020; Grébert et al. Reference Grébert, Jamme, Joly and Bodart2023), the features of the flow unsteadiness associated with MVG-controlled SBLI have been scarcely considered in the literature, despite the shock low-frequency unsteadiness being one of the most critical aspects for SBLI. Indeed, long integration periods to capture this spectral property are computationally demanding for high-fidelity methods, and RANS methods are notoriously unable to properly describe flow unsteadiness. To shed light on the spectral characteristics of the flow under consideration, in the following, we analyse the time evolution of the pressure at the wall. We first present a Fourier analysis of the pressure along the $xz$ and $xy$ planes and then a wavelet analysis of $p_w(t)/p_\infty$ for selected probes.

4.5.1. Spectra

Streamwise spectra. First, we consider the overall picture given by the spanwise-averaged premultiplied spectra of the wall pressure along the streamwise coordinate. Spectra have been evaluated with the Welch method, using 16 (USBLI) and 8 (CSBLI) segments with 50 % overlap and a rectangular window, and have then been smoothed using a Konno–Ohmachi filter (Konno & Ohmachi Reference Konno and Ohmachi1998). A summary of the main results is reported in table 4, where $x^*_{low}$ is the location of the low-frequency peak, $x^*_{sep}$ is the separation point, $x^*_{peak}$ is the foremost wall-pressure standard deviation peak, $x^*_{reat}$ is the reattachment point, $L_{sep}/\delta _{shk}$ is the separation length with respect to the boundary layer thickness at inviscid shock impingement, $f\, L_{sep}^u/U_\infty$ is the non-dimensional value of the peak low frequency with respect to the free-stream velocity and the USBLI separation length and $f\, L_{sep}/U_\infty$ is the non-dimensional value of the peak low frequency with respect to the free-stream velocity and the local separation length.

Table 4. Main streamwise locations of interest, separation length and value of the peak low frequency.

The contours in figure 20 show the typical behaviour of the wall-pressure spectra. A broadband low-frequency peak in correspondence with the onset of the separation is associated with the oscillations of the separation shock foot. The low-frequency peak is followed by an intermediate region with mid-to-high frequencies associated with the development of the separation bubble. Here, the dominant frequencies drop in the first half of the bubble and then rise in correspondence with the reattachment. After this point, high-frequency pressure fluctuations indicate the increased turbulent activity in the boundary layer following the interaction. In the CSBLI case, we can also notice a high-frequency content before the interaction onset due to the microramp wake flow. The energetic region, sharp in space at $x^* \approx -2.4$ and broadband in frequency, corresponds to the peak of the wall-pressure standard deviation observed just after the ramp trailing edge (see figure 8) and is associated with the trace on the wall of the conical shock around the microramp wake (Della Posta et al. Reference Della Posta, Blandino, Modesti, Salvadore and Bernardini2023). Comparing the USBLI and CSBLI cases, we can notice that the entire frequency content is slightly shifted towards higher frequencies. As a result, the low-frequency broadband peak, whose absolute maximum is located at $f\, L_{sep}^u/U_\infty \approx 0.052$ for the USBLI case, increases slightly up to $f\, L_{sep}^u/U_\infty \approx 0.079$ for the CSBLI case. Similar values were obtained from LES data in Grébert et al. (Reference Grébert, Jamme, Joly and Bodart2023), with the maximum low-frequency activity concentrated around $f\, L_{sep}^u/U_\infty \approx [0.03, 0.05]$ and not significantly affected by the MVGs. In addition, another low-frequency energetic content is visible at $x^* \in [-0.5,-0.25]$, where the second peak of wall-pressure standard deviation is present in figure 8. This interval corresponds to the first half of the interaction, where the height of the separation bubble is increasing and the impinging shock interacts with the shear layer surrounding the recirculating flow. Finally, although an increased energy content is observable at $f\, L_{sep}^u/U_\infty \in [3,4]$, corresponding to the shedding frequency of the arch-like vortices (Bo et al. Reference Bo, Weidong, Yuxin, Xiaoqiang and Chao2012; Della Posta et al. Reference Della Posta, Fratini, Salvadore and Bernardini2023a), we cannot conclude that this is a trace on the wall of these structures, as an analogous increase at the same frequencies is also observable in the uncontrolled case.

Figure 20. Streamwise distribution of the spanwise-averaged premultiplied wall-pressure spectra: (a) USBLI and (b) CSBLI. The dashed white line indicates the frequency of the low-frequency peak in the shock region.

Given the relevant spanwise modulation of the flow induced by the microramp, it is worth considering the premultiplied spectra along the streamwise direction at single spanwise sections. As in the previous sections, we examine the notable stations at $z^* \approx -0.3$, $-0.05$ and $0$. Figure 21 shows that there is no qualitative distinction between the spectra in the span and also the low-frequency peak takes place at approximately the same Strouhal number. Slight quantitative differences however are present in the reattachment region and beyond ($x^* \approx [-0.25, 0.5]$) for high frequencies ($\,f\, L_{sep}^u/U_\infty \in [2,5]$). At $z^* \approx -0.05$, the intensity of the spectra is higher, suggesting an increased activity of the eddies in the boundary layer. In this section, the local separation length is minimum and the height of the recirculation bubble is smaller (see figure 12). Given this observation, we could speculate that the action of the arch-like vortices, whose characteristic shedding frequency is precisely in this range, is here felt stronger close to the wall and thus contributes to the surge in the spectra. Moreover, looking at the distribution of the skin friction lines in figure 11, the spanwise section with minimum separation length corresponds approximately to the location of a focus on the reattachment line, where the convergence of the flow could further contribute to the observed increase in high-frequency wall-pressure fluctuations. Unsteady data in this spanwise section have not been considered in this work and will be the subject of future research. An increased intensity is also visible close to the ramp trailing edge at $z^* \approx -0.05$. Here, at the sides of the primary vortex pair, the conical shock wave around the microramp wake can penetrate the flow close to the wall, thus leaving a stronger imprint on the spectra.

Figure 21. Streamwise distribution of the premultiplied wall-pressure spectra of the CSBLI case at: (a) $z^* \approx -0.3$, (b) $z^* \approx -0.05$ and (c) $z^* \approx 0$. The dashed white line indicates the frequency of the local low-frequency peak in the shock region.

Finally, it is worth noting that, despite its increase in absolute terms, if the local peak frequency is scaled by the local separation length (table 4), the resulting non-dimensional frequency goes back approximately to the value of the 2-D USBLI. The slight differences that are still present may be due to 2-D issues that make the 2-D scaling not completely effective.

Spanwise spectra. To better investigate the relative distribution in frequency in the $z^*$ direction, figure 22 reports the premultiplied spectra along notable curves in the span. The previous analysis confirmed that the microramp wake modulates the flow in the spanwise direction, with a mild variation also in the spectral features of the SBLI. For this reason, to compare analogous conditions along $z^*$, we sampled the wall-pressure signals following the separation line, the foremost wall-pressure standard deviation peak location, and the reattachment line (see figure 11c). However, the integral in frequency of the spectra is not constant along these curvilinear coordinates, as it corresponds to the variance of the local wall-pressure signals (Parseval's theorem), which changes along the $xz$ plane in the CSBLI case. Since we want to compare only the relative distribution in frequency at different sections, we normalised the spectra with the local wall-pressure variance. The results allow us to examine how the relative magnitude of the spectral content associated with the shock oscillation, turbulent fluctuations and other features varies along the span.

Figure 22. Spanwise distribution of locally normalised premultiplied wall-pressure spectra: USBLI (a–c) and CSBLI (d–f), along the separation (a,d), along the wall-pressure standard deviation peak (b,e), and along the reattachment (c,f).

From the contours in the first row, as expected, the USBLI results show that the relative distribution is approximately constant along the span. Along the (straight) separation line, the energy of the wall-pressure signal is distributed evenly between the low- and high-frequency ranges. Moving downstream, towards the standard deviation peak and then the reattachment (straight) lines, the dominant spectral content shifts to higher frequencies, as the influence of the reflected shock unsteadiness vanishes progressively. At the reattachment, the peak frequency related to the eddies in the reattaching boundary layer is $f\, L_{sep}^u /U_\infty \approx 2$.

The contours of the second rows report, instead, the results of the CSBLI case. Along the separation, compared with the USBLI case, the scenario at the side and central regions is different. Close to the lateral boundaries, the energy of the signal at the low-frequency peak is larger than the energy at high frequencies. The opposite takes place at the centre of the domain, behind the microramp, where the contribution from the low frequencies is dampened and the high-frequency contributions are stronger. According to the previous results, the spectra in the span confirm that although microramps do not cancel the global unsteadiness of the shock, they provide a local attenuation of its relative impact in the region of their wake. Moreover, as the frequency associated with the shock unsteadiness stays almost constant, we can deduce that, despite its geometry being altered by the impingement of the ramp wake, the front of the reflected shock remains coherent in the span during its low-frequency oscillation. The distribution along the peak standard deviation is similar to that along the separation, although we note that the contribution from the low-frequency range in the CSBLI case is still relevant compared with the USBLI case. In both figures 22(d) and 22(e), we also note a mild shift towards higher frequencies close to the symmetry plane. Although there may be effects associated with the local reduction of the separation extent, we can speculate that the shift in value of the high-frequency content is primarily due to the different streamwise positions of the loci considered along the spanwise direction. Indeed, the inner portion of the separation line and of the wall-pressure standard deviation peak in the CSBLI case are generally more downstream, where high-frequency turbulent fluctuations count more. Interesting differences are present instead along the reattachment line. Similarly to the USBLI case, the energy is here concentrated in the high-frequency range, and especially at the lateral boundaries the contour is almost the same as without microramp. However, close to the symmetry plane, the dominant contribution increases slightly in magnitude and frequency, reaching exactly the shedding frequency of the arch-like vortices in a peaked fashion.

A trace at the wall of the K–H vortices around the wake is noticeable only at the reattachment, while the typical tonal signature at $f\, L_{sep}^u/U_\infty \in [3,4]$ is absent in the first part of the interaction. Considering also their reduced spanwise vorticity compared with that of the bottom shear layer (see figure 19), it is possible to believe that the action of the arch-like vortices is first shielded by the 3-D shear layer developing around the separation region, and thus that they may play a limited role in the delay of the separation. Their role may instead be more relevant in the mechanism to close the recirculation bubble.

Wall-normal spectra. We complete the picture of the Fourier analysis with figure 23, which shows the premultiplied spectra of the pressure along the wall-normal coordinate, at the symmetry plane. Time signals are sampled in correspondence with a streamwise location close to separation (a,b) and reattachment (c,d) for the USBLI (a,c) and CSBLI (b,d) cases. These spectra allow us to locate the characteristic frequencies of the flow in the wall-normal direction, and thus relate the behaviour of the fluctuations at the wall with those far from the wall.

Figure 23. Wall-normal distribution of the premultiplied wall-pressure spectra at the symmetry plane in the (a,b) separation and (c,d) reattachment regions. USBLI on (a,c) and CSBLI on (b,d). The dashed white line indicates the local low-frequency peak associated with the oscillation of the separation shock.

The USBLI results at the separation document the energetic fluctuations associated with the low-frequency motion of the reflected shock wave, which initiates the separation penetrating the boundary layer up to a short wall-normal distance. The high-frequency content close to the wall is instead the trace of the vortical structures living at the edge of the recirculation bubble. Indeed, their intensity is small at separation, as the separation shear layer is here still at its onset, while being diffused across the vertical direction at reattachment, where pressure fluctuations are associated with the eddies of the new boundary layer downstream of the interaction. Finally, the contour at reattachment shows also an interesting feature related to the reattachment shock taking place downstream of the separation, observable at $y^* \approx 0.38$. This shock is characterised by a broadband shape, as for the separation shock, despite its peak non-dimensional frequency being considerably higher (at least a decade more than the traditional low-frequency unsteadiness).

The CSBLI contours show analogous results for what concerns the trace of the low-frequency peak of the separation shock, although the trace now covers a larger wall-normal range, suggesting an increased shock smearing.

The most interesting difference, however, is the strong peak at $y^* \in [0.2,0.3]$ and $f\, L_{sep}^u / U_\infty \in [2,5]$, which marks the spectral content of the arch-like vortices. This contribution remains generally separate from the high-frequency one located close to the wall, especially in the separation region, and its intensity is comparable with that of the most energetic features in the spectra. At reattachment, the rise in intensity of the fluctuations in the high-frequency range induced by the shocks all along the wall-normal direction causes a larger peak in correspondence with the arch-like vortices, whose influence is able to reach the wall, as we observed in the spanwise spectra reported in figure 22( f). The spectral analysis thus confirms that the arch-like vortices’ shedding frequency is in the same range as the turbulent fluctuations taking place in traditional SBLI and that their influence directly reaches the wall only at reattachment. However, despite providing interesting information about the spatial and spectral features of the flow, spectra do not provide information about the actual reattachment mechanism in microramp-controlled SBLIs and about the role arch-like vortices play in it, which thus remains an open question for future work.

4.5.2. Wavelet analysis

In this section, we want to characterise the degree of intermittency of the wall-pressure signals in the interaction region, in particular at the onset of the separation and at reattachment. Similarly to the approach adopted in Bernardini et al. (Reference Bernardini, Della Posta, Salvadore and Martelli2023a), wavelet analysis is applied to extract energetic intermittent events. Indeed, this approach provides a more direct measure of the degree of intermittency of the wall-pressure field and allows for the extraction of local (in time) features that may be partially lost using Fourier analysis.

The wavelet transform is computed by the convolution of the wall-pressure signal $p_{w}(t)$ with the dilated (by the factor $k$) and translated (by the factor $t$) complex conjugate counterpart of a so-called mother wavelet, according to the following formalism:

(4.3)\begin{equation} G_{\varPsi}(k,t) = \frac{1}{\sqrt{k}} \int_{-\infty}^{+\infty} p_w(\tau)\varPsi^{*}\left(\frac{\tau-t}{k}\right)\,\mathrm{d}\tau, \end{equation}

where $\varPsi$ is the wavelet mother function, $k$ is a dilatation parameter indicating the time scale of the event under consideration, $t$ is the time-translation parameter and $*$ denotes the complex conjugate. A detailed theoretical framework can be found in Farge (Reference Farge1992) and Mallat (Reference Mallat1999). In this study, the Morlet wavelet has been chosen, as a higher resolution in frequency can be achieved compared with other mother functions. Owing to the wavelet admissibility condition (Torrence & Compo Reference Torrence and Compo1998), the frequency associated with the dilatation parameter for the Morler wavelet is defined as $f = 0.97/k$. Scalograms of the wall pressure at separation and reattachment on the symmetry plane for the two cases are reported in Appendix C.

In statistics, intermittency denotes the rare occurrence of exceptionally spiky events which are patchy and bursty. These instances cause higher-order moments (skewness and flatness) to converge with greater difficulty, suggesting a significant departure from Gaussian statistics and, hence, non-homogeneous distribution of energy in time (Camussi & Bogey Reference Camussi and Bogey2021).

Once computed the wavelet transform coefficients $G_{\varPsi }(k,t)$, it is possible to obtain the scale-time distribution of the energy density $|G_{\varPsi }(k,t)|^2$ of the wall-pressure signal. Thanks to this property, Meneveau (Reference Meneveau1991) and Camussi & Bogey (Reference Camussi and Bogey2021) suggested that an effective indicator of the intermittency is the squared local intermittency measure, denoted as $LIM2$:

(4.4)\begin{equation} LIM2(k,\tau)=\frac{|G_{\varPsi}(k,t)|^4}{\langle|G_{\varPsi}(k,t)|^2\rangle_{t}^2}, \end{equation}

where $\langle \bullet \rangle _t$ indicates the time average of the considered quantity. $LIM2$ can be interpreted as a time-scale dependent measure of the flatness factor or kurtosis of wall-pressure signals. Therefore, the $LIM2$ parameter will be equal to 3 when the probability distribution is Gaussian, while the condition $LIM2>3$ identifies only those rare bursts of energy contributing to the deviation of the wavelet coefficients from a normal, Gaussian distribution.

In the following, we apply the wavelet analysis to the wall-pressure signal at two relevant stations for both the USBLI and CSBLI cases: the separation point and the reattachment point at the symmetry plane. Figure 24(a) shows the LIM2 maps (only the levels greater than 3) at the separation point for USBLI. The occurrence of intermittent events in the frequency range characterising the shock unsteadiness, whose Fourier frequency peak is indicated with a dashed horizontal line, is immediately apparent. The most relevant characteristic feature of these bursts of energy is that they are rather scattered in time and clustered around the Fourier peak frequency, but with some very energetic events at lower frequencies. The long simulation time allows our DNS to capture these rare events. This picture again confirms that the low-frequency unsteadiness of SBLIs is composed of a collection of energetic events, whose rarity causes the difficulty of convergence of higher-order statistics. Nevertheless, it is important to underline that the computational time is largely sufficient to evaluate the means and the variances of the wall-pressure signals, as well as the Fourier spectra, as amply demonstrated in the literature (Touber & Sandham Reference Touber and Sandham2009; Bernardini et al. Reference Bernardini, Della Posta, Salvadore and Martelli2023a). Indeed, we verified that relative standard errors, evaluated using the bootstrap method (Efron & Tibshirani Reference Efron and Tibshirani1994), for the mean and variance at separation and reattachment on the symmetry plane are smaller than 1 %. At higher frequencies, it is possible to see the intermittency given by turbulence which is quite separate in the frequency space from the events caused by the shock/boundary layer interaction. If we now look at the CSBLI case in figure 24(b), it is evident that the intermittent events are grouped around a higher frequency, as indicated by the Fourier analysis, with the additional occurrence of a single extremely energetic low-frequency event at $t L^{u}_{sep}/U_{\infty } \approx 750$. The analysis of the LIM2 maps at the reattachment points for both cases, figures 24(c) and 24(d), shows a very similar behaviour. We can just note that the energetic bursts at lower frequencies survive for all the interaction length. It may be concluded therefore that, at least for these test case conditions, microramps do not alter significantly the highly intermittent nature of SBLI.

Figure 24. Contours of $LIM2$ of $p_w/p_{\infty }$ at the separation (a,b) and reattachment points (c,d) at the symmetry plane. USBLI (a,c) and CSBLI (b,d). Only $LIM2$ values above 3 are shown. Dashed black lines indicate the dominant low-frequency peak of each case.