3.1. The low-frequency unsteadiness

Hao (Reference Hao2023) demonstrated that 2-D shock motion is primarily driven by a 2-D global mode (referred to as the shock mode), characterised by $\beta =0$ and ${\omega }_{r} =0$ , through a modal resonance mechanism. The interpretation, however, relies on Reynolds-averaged Navier–Stokes (RANS) simulations and RANS-based GSA. As a result, direct evidence for the physical existence of the mode was not established. In the present work, this issue is revisited by comparing 2-D coherent structures extracted from LES with the 2-D global mode obtained from LES-based GSA, thereby providing higher-fidelity support for the underlying mechanism. Spectral proper orthogonal decomposition (SPOD) analysis is performed to extract coherent structures from the unsteady spanwise-averaged data. The input data consist of the spanwise-averaged state vector $ {q}^{\prime }= [ \rho ^{\prime },u^{\prime },v^{\prime },w^{\prime },T^{\prime } ]$ . Welch’s method is used with a Hamming window to minimise spectral leakage, segmenting the dataset into eight blocks with 75 $ \,\%$ overlap. The Chu energy norm (Chu Reference Chu1965) is used as the inner product. The use of spanwise-averaged SPOD is appropriate owing to the quasi-2-D character of the shock motion in ramp flows (Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014). Figure 1( $a{,}b$ ) compares the 2-D shock mode and the leading 2-D SPOD mode at the normalised frequency $ St=fL_{\textit{sep}}/u_{\infty }=0.026$ , where $L_{\textit{sep}}$ is the separation length and $u_{\infty }$ is the free-stream velocity. Note that $L_{\textit{sep}}$ from the uncontrolled case is used in the definition of $ \textit{St}$ throughout this study to ensure consistent comparison. This frequency corresponds the energy peak of the SPOD modes in the low-frequency regime, as shown in figure 1( $c$ ), and aligns well with the characteristic frequency of the shock motion presented later in figure 2( $ c$ ). The shock mode maintains high values of $ u^{\prime }$ near the shock foot and along the separation shock. Notably, the SPOD mode closely resembles the shock mode, with elevated $ u^{\prime }$ values similarly concentrated around the shock foot and along the shock. Convergence of the SPOD modes is discussed in Appendix C.

A quantitative comparison between the two modes can be conducted using two methods. First, the integrated Chu energy density distributions along the streamwise direction are compared, as shown in figure 1( $ d$ ). Both distributions are normalised by their respective peak values located just downstream of the mean separation point $x_{\textit{sep}}$ . Slightly upstream of $x_{\textit{sep}}$ , both profiles increase rapidly towards their peaks. Their growth rates and positions closely match. Downstream of these peaks, both distributions exhibit a gradual decline, followed by an increase beyond the corner. Despite minor differences in the decay rates, the overall trends remain highly similar. This consistent behaviour provides compelling evidence that the two modes share fundamental similarities. The second perspective is to make a projection between the two modes, given by

(3.1) \begin{align} \gamma =\frac {\big | {\parallel \boldsymbol{{\hat {\varPsi }}_{\textit{SPOD}}}(x,y), \boldsymbol{{\hat {U}}_{\textit{GSA}}}(x,y)\parallel }_{E} \big |}{ {\parallel \boldsymbol{{\hat {\varPsi }}_{\textit{SPOD}}}(x,y) \parallel }_{E} \times {\parallel \boldsymbol{{\hat {U}}_{\textit{GSA}}}(x,y)\parallel }_{E} }, \end{align}

where $ \boldsymbol{{\hat {\varPsi }}_{\textit{SPOD}}}(x,y)$ represents the 2-D leading SPOD mode at $ \textit{St}=0.026$ , while $ \boldsymbol{{\hat {U}}_{\textit{GSA}}}(x,y)$ denotes the shock mode. The notation $ \left \| \boldsymbol{\cdot }\right \|_{E}$ means the Chu norm (Chu Reference Chu1965). A value of $ \gamma =1$ represents a perfect match of the two modes. The projection coefficient between the two modes shown in figure 1( $ a{,}b$ ) is 0.93, indicating a strong correlation between them. This further supports the conclusion that the low-frequency unsteadiness of the turbulent separation bubble is predominantly driven by the shock mode. Moreover, the low-frequency shock motion can be effectively reconstructed by superimposing the leading SPOD mode at $ \textit{St}=0.026$ onto the mean flow (see supplementary movie available at https://doi.org/10.1017/jfm.2026.11679).

3.2. Controlled breathing motion with a micro-bump

Having established the central role of the 2-D global mode in driving the low-frequency unsteadiness, we now turn to its targeted suppression by exploiting its sensitivity distribution. Following Paladini et al. (Reference Paladini, Marquet, Sipp, Robinet and Dandois2019), the core of the 2-D instability can be quantified by the complex density $ d= \tilde {U} ^{\dagger } \hat {U}$ , where $ \tilde {U}$ is the 2-D adjoint mode and the superscript $ \dagger$ refers to the transconjugate. The adjoint mode is computed by solving the adjoint eigenvalue problem. The real part of $ d$ represents the local contribution to the growth rate of the zero-frequency shock mode. Note that the integration of $ d$ over the computational domain equals 1. Figure 2( $ a$ ) shows the growth rate density map for the shock mode. The highest density values are concentrated near the shock foot, consistent with the mode structure shown in figure 1( $a$ ). In contrast, the density remains nearly zero elsewhere, including across the separation shock. This localised distribution indicates that the shock-foot region dominates the sensitivity to global instabilities, a finding consistent with Paladini et al. (Reference Paladini, Marquet, Sipp, Robinet and Dandois2019) in the research of transonic shock buffet, which is governed by a globally unstable, oscillatory mode.

Table 1.

Parameters in (3.2): $h$ , bump height; $x_{l}$ and $x_{r}$ , left and right boundaries; $x_{p}$ , bump peak location; $A$ , $B$ , $C$ and $D$ , shape-control constants.

Figure 2.

( $a$ ) Real part of the shock mode’s growth rate density map, with the bump edges $x_{l}$ and $x_{r}$ marked by green lines. ( $b$ ) Comparison of skin-friction ( $C_f$ ) and pressure ( $C_p$ ) coefficients. ( $c$ ) Pre-multiplied wall-pressure spectra at station $x_{\textit{sep}}$ (case $C_{0}$ ) and station $x_{p}$ (case $C_{35}$ ).

To mitigate the instability contribution from the shock-foot region, we introduce a local surface modification. Inspired by Martinez-Cava et al. (Reference Martinez-Cava, Chávez-Modena, Valero, de Vicente and Ferrer2020), who showed that the optimal surface deformation for weakening shock-buffet instability resembles a shock-control bump, we adopt a similar control bump around the shock-foot region, as highlighted by the purple line in the inset of figure 2( $a$ ). This small-scale bump is designed to control the shock mode and suppress the low-frequency shock motion. The geometric configuration of the bump is defined by the following shape function:

(3.2) \begin{align} y=\begin{cases}h_{1} \times (1-\mathrm{exp}(-(x-x_{p}+A )^{B})) & \text{ if } x_{l}\le x\lt x_{p}, \\h_{2} \times (1-\mathrm{tanh}(C\times (x-x_{p}+D))) & \text{ if } x_{p}\le x\le x_{r}. \end{cases} \end{align}

Here, $ h_{1} =h/(1-\mathrm{exp}(-A^{B} ))$ and $ h_{2} =h/(1-\mathrm{tanh}(C\times D))$ are scaling factors, where $h$ is the bump height. The streamwise extent and positioning of the bump are defined by $x_{l}$ and $x_{r}$ marking the left and right boundaries, respectively, and $x_{p}$ indicating the peak location. Constants $ A$ , $ B$ , $ C$ and $D$ govern the bump profile. Values of these parameters are listed in table 1. The bump is designed primarily to envelop the high-sensitivity region and centre around the separation point $x_{\textit{sep}}$ , featuring a gradual ascent followed by a steep descent. The smooth forward rise prevents additional separation, while the steeper rear section serves to modify the separation bubble. The bump size is relatively small, with a height 15 $\,\%$ of the boundary-layer thickness and a streamwise length 20 $\,\%$ of the separation length. Hereafter, the uncontrolled case is denoted as $C_{0}$ and the controlled case as $C_{35}$ , where the subscript denotes the bump height.

The flow response to the introduction of the micro-bump is examined next. Figure 2( $b$ ) compares the skin-friction coefficient $ C_{f}$ and pressure coefficient $ C_{p}$ . For $C_{35}$ , the bump induces a mild adverse pressure gradient upstream, leading to a minor localised separation ahead of the bump. Near the bump peak, the $ C_{f}$ distribution shows a pronounced increase, while that of $ C_{p}$ exhibits slight fluctuations followed by a recovery. Away from the bump region, both $ C_{f}$ and $ C_{p}$ distributions closely match those of $ C_{0}$ , confirming that the influence of the bump remains highly localised. Over the range $(-10\delta , 8\delta )$ , the integrated skin-friction drag of the controlled case $C_{35}$ is reduced by $14\,\%$ compared with that of case $C_{0}$ , while the pressure drag decreases by 0.8 $\,\%$ .

The primary differences emerge in the power spectral density (PSD) spectra near the bump region, as illustrated in figure 2( $c$ ). Here, we choose the station $ x_{\textit{sep}}$ from case $C_{0}$ and $x_{p}$ from case $C_{35}$ for comparison, where the low-frequency motion is evident. In the low-frequency regime ( $ \textit{St}\lt 0.05$ ), a predominant peak is observed at $ \textit{St}=0.031$ , which corresponds to the low-frequency breathing motion. However, this peak diminishes in the $C_{35}$ case. With increasing $ \textit{St}$ , high-frequency energy is gradually amplified. A similar trend was also reported by Ceci (Reference Ceci2025), where a large shock-control bump was implemented.

Figure 3.

Instantaneous vortices from ( $ a$ ) $C_{0}$ and ( $ b$ ) $C_{35}$ , extracted using the $ Q$ -criterion, 5 $ \,\%$ of the maximum of $C_{0}$ , coloured by the Favre-fluctuating velocity $ u^{\prime\prime}/u_{\infty }$ . Transparent iso-surfaces of density-gradient magnitude are used to highlight the shock position.

A comparison of the instantaneous vortices, coloured by the fluctuating velocity $ u^{\prime\prime}/u_{\infty }$ , is presented in figure 3 for the two cases. Upstream of the separation shock, both cases exhibit negligible vortex activity. In the uncontrolled case $C_{0}$ , vortices primarily originate near the shock foot and subsequently propagate downstream. While for the controlled $C_{35}$ case, the micro-bump acts as an effective vortex generator, shedding numerous small-scale vortices in its immediate vicinity. Downstream of the bump, the large-scale vortices are significantly reduced compared with case $C_{0}$ . This distinct alteration in the flow pattern demonstrates that the bump fundamentally modifies the vortex dynamics of the interaction.

Figure 4.

( $a$ ) Comparison of the eigenvalues for the 2-D modes from cases $C_{0}$ and $C_{35}$ . Real parts of ( $ b$ ) the 2-D mode and ( $ c$ ) the leading SPOD mode at $ \textit{St}=0.03$ for case $C_{35}$ . ( $d$ ) Low-pass-filtered planar TKE distributions ( $ \textit{St}_{cut}=0.1$ ), normalised by the square of the friction velocity $u_{\tau }^{2}$ at the reference station. Streamwise distributions of ( $e$ ) the wall-normal integrated TKE ( $ \textit{St}_{cut}=0.1$ ) and ( $f$ ) the normalised wall-pressure root mean square ( $ \textit{St}_{cut}=0.1$ ) for both cases. Black lines in ( $d$ ) indicate dividing streamlines. Green dash-dot and solid lines in ( $c$ , $d$ ) mark the edges and peak positions of the bump, respectively. Both TKE and $p^{\prime}_{\textit{rms}} /\bar {p}_{w}$ in ( $e$ , $f$ ) are normalised by the peak values of the $C_{0}$ case.

The GSA reveals a fundamental change in the 2-D dynamics. As shown in figure 4( $a$ ), the most unstable 2-D global mode in case $C_{0}$ is strongly damped in case $C_{35}$ , transitioning to a stable mode. Its spatial structure also shifts, with high $u^{\prime}$ amplitudes now concentrated along the shear layer from the bump and the separation shock, rather than at the shock foot, as shown in figure 4( $b$ ). Although stabilised, this mode is still excited by disturbances through modal resonance. The leading SPOD mode at $ \textit{St}=0.03$ (figure 4 $c$ ) confirms this, showing strong similarity to the global mode (projection coefficient of 0.89 with the global mode). Crucially, the stable mode is associated with weak breathing motion, as evidenced by the reconstructed unsteady flow fields obtained using the specific SPOD mode (see supplementary movie). Furthermore, the long-time filtered unsteady density gradient magnitude with $ \textit{St}_{cut}=0.05$ for the two cases is also provided in the supplementary movies, which clearly show weakened shock motion. Convergence of the SPOD modes is verified in Appendix C.

The strength of the low-frequency breathing motion is then quantified and compared for cases $ C_{0}$ and $ C_{35}$ . Here, we compare their wall-normal integrated planar turbulent kinetic energy (TKE) distributions and wall-pressure fluctuations ( $ p^{\prime}_{\textit{rms}}$ ). Both quantities are low-pass filtered at $ \textit{St}_{cut}=0.1$ to focus on the low-frequency dynamics. The planar TKE is defined as follows:

(3.3) \begin{align} \mathrm{Planar\;TKE}=0.5\times \big [ \big \langle u^{\prime\prime}_{\textit{span}}u^{\prime\prime}_{\textit{span}} \big \rangle +\big\langle v^{\prime\prime}_{\textit{span}}v^{\prime\prime}_{\textit{span}} \big \rangle \big], \end{align}

where the spanwise averaging (denoted by the subscript ‘ $ span$ ’) and the omission of the small $\langle w^{\prime\prime}_{\textit{span}}w^{\prime\prime}_{\textit{span}} \rangle$ term are deliberate choices to isolate the dominant 2-D motion from 3-D turbulent fluctuations. This operation is reasonable since the total TKE is mainly contributed by the planar part, as compared in figure 4( $e$ ). The operator $\left \langle \boldsymbol{\cdot }\right \rangle$ signifies the Favre average and the double prime marks the Favre fluctuating variables.

In figure 4( $d$ ), the filtered TKE distributions of the two cases are presented, normalised by the square of the friction velocity at the reference station and superimposed with the dividing streamlines. In the uncontrolled $C_{0}$ case, regions of high TKE are concentrated around the shock foot and along the separation shock, exhibiting a striking resemblance to the structure of the 2-D global mode shown in figure 1( $a{,} b$ ). This spatial agreement strongly indicates that the low-frequency TKE is governed by this 2-D global mode. In the controlled $C_{35}$ case, the TKE distribution continues to broadly follow the shape of the stabilised global mode from figure 4( $b{,} c$ ), though with visibly attenuated amplitudes compared to the uncontrolled case.

Figure 4( $e$ ) compares the streamwise evolutions of the wall-normal integrated TKE for the two cases. For easy comparison, all quantities are normalised by the maximum value obtained in case $C_{0}$ , and the same normalisation is applied in figure 4( $f$ ). For case $C_{0}$ , the TKE begins to rise slightly upstream of $x_{\textit{sep}}$ , and reaches a peak just downstream of this point, corresponding directly to the high-intensity region observed in the spatial plot. In contrast, the distribution for case $C_{35}$ is fundamentally different. A minor peak with a value of 0.03 occurs upstream of the bump, attributable to the small separation discussed in figure 2( $b$ ). Within the bump region, the TKE drops to a minimum near the bump peak before increasing again downstream. The peak TKE value in the controlled interaction is approximately 0.15, which is considerably lower than that of the uncontrolled case. Details of this reduction in peak values are presented in table 2. Furthermore, the control effects are independent of the spanwise width, as confirmed by a wider case ( $8\delta$ ) presented in Appendix A.

Table 2.

Performance metrics. For easy comparison, both TKE and $p^{\prime}_{\textit{rms}} /\bar {p}_{w}$ are normalised by the peak values in case $C_{0}$ .

The control effect is also reflected in the wall-pressure fluctuations $ p^{\prime}_{\textit{rms}}/\bar {p}_{w}$ in figure 4( $f$ ). The $C_{0}$ case displays a single, prominent peak near $x_{\textit{sep}}$ , which decays rapidly downstream and plateaus at around 0.25. The $C_{35}$ case, however, exhibits a bifurcated response with two distinct peaks: one of magnitude 0.15 associated with the upstream separation, and a second peak of 0.3 near the bump peak. Critically, both peaks are substantially weaker than the dominant peak in the uncontrolled flow. Downstream of the bump, the $ p^{\prime}_{\textit{rms}}/\bar {p}_{w}$ values stabilise at a level comparable to the uncontrolled case.

Our results confirm that the micro-bump significantly attenuates the low-frequency breathing motion. The control mechanism operates by minimally modifying the base flow, which stabilises the dominant 2-D shock mode and thereby makes it more difficult to be excited by external disturbances. Another plausible explanation is the transformation of the separation mechanism from pressure-induced separation to geometry-induced separation, the latter being associated with comparatively weaker low-frequency dynamics due to the relatively fixed separation point (Eaton & Johnston Reference Eaton and Johnston1982; Borgmann et al. Reference Borgmann, Cura, Weiss and Little2024).

Figure 5.

Comparisons of ( $a$ ) pre-multiplied wall-pressure spectra, ( $b$ ) integrated planar TKE ( $ \textit{St}_{cut}=0.1$ ) and ( $c$ ) the maximum growth rate of the 2-D mode for cases with different bump heights. Streamwise stations in ( $a$ ): $x_{\textit{sep}}$ for uncontrolled case and $x_{p}$ for controlled cases.

Effects of bump height are then investigated. Figure 5( $a$ ) compares the pre-multiplied wall-pressure spectra for cases with different heights. The low-frequency peak diminishes at $h=0.15$ mm, indicating that the bump begins to effectively suppress the breathing motion at this height. The low-frequency TKE level drops sharply compared with the uncontrolled case $C_{0}$ , as seen in figure 5( $b$ ). Further increasing the bump height has little influence on the spectral shape, but the TKE level continues to decline with height up to $h = 0.35$ mm. At $h = 0.45$ mm, the TKE level is comparable to that at $h = 0.35$ mm, suggesting a local weakening of the suppressive effects. The weakening effects can be explained using the maximum growth rates of the 2-D mode, as illustrated in figure 5( $c$ ). The growth rate decreases monotonically with bump height, a trend that tends to saturate around $h = 0.45$ mm. In the presence of a bump, the mode shape resembles that shown in figure 4( $b$ ). As noted earlier, modal resonance can excite the 2-D mode, which is linked to the weakened breathing motion and can be triggered under modal resonance. Therefore, a sharp reduction in fluctuating energy results from a significantly damped growth rate. If the bump height is increased further, the flow may no longer remain only slightly disturbed and nonlinear effects could emerge, which might in turn reduce the effectiveness of the suppression mechanism.

Figure 6.

Comparisons of ( $a{,} b$ ) $C_{f}$ and $C_{p}$ , ( $c$ ) pre-multiplied wall-pressure spectra, ( $d$ ) integrated planar TKE ( $ \textit{St}_{cut}=0.1$ ) and ( $e$ ) the normalised wall-pressure root mean square ( $ \textit{St}_{cut}=0.1$ ) of cases at different positions.

The effects of bump position are examined by considering three additional cases: case $C_{35}$ –upstream–1.32 $\delta$ , with the bump shifted $1.32\delta$ upstream; $C_{35}$ –downstream–1.32 $\delta$ , with the bump shifted $1.32\delta$ downstream; and case $C_{35}$ –upstream–10 $\delta$ , with the bump shifted $10\delta$ upstream. The first two cases are designed to assess the impact of slight bump shifts, whereas the last case represents a fully off-design condition. Figure 6( $a{,} b$ ) compares the distributions of $C_{f}$ and $C_{p}$ for cases at different positions. Large separation bubbles are formed in the two upstream cases, whereas the downstream bump, concealed within the separation bubble, induces only minor changes in the two distributions relative to case $C_{0}$ . Figure 6( $c$ ) shows the pre-multiplied wall-pressure spectra for the original $C_{0}$ and $C_{35}$ cases, together with the three repositioned cases. The low-frequency peak is greatly amplified for the fully off-design case ( $C_{35}$ –upstream–10 $\delta$ ). For the two slightly shifted cases, the low-frequency components are enhanced compared with the well-controlled $C_{35}$ case, although both remain slightly lower than that in case $C_{0}$ . The peak TKE and $ p^{\prime}_{\textit{rms}}/\bar {p}_{w}$ values increase by 300 $\,\%$ and 150 $\,\%$ , respectively, in the $C_{35}$ –upstream–10 $\delta$ case. In contrast, the two slightly shifted cases show similar satisfactory reductions in the shock motion strength, with decreases of approximately $75\,\%$ in the peak TKE and $50\,\%$ in $ p^{\prime}_{\textit{rms}}/\bar {p}_{w}$ .

The conclusions drawn here regarding positioning the bump far upstream are consistent with Della Posta et al. (Reference Della Posta, Martelli, Salvadore and Bernardini2024), who reported that an upstream vortex generator amplified the low-frequency magnitude of shock motion. The two partially off-design conditions suggest that slight shifts of the bump can still mitigate the unsteadiness, although the effectiveness varies. Further efforts could employ techniques such as piezoelectrics and shape memory alloys to advance this control strategy towards practical implementation, ensuring the bump remains in the effective region across varying flow conditions.