3.1. Flow conditions

A compression ramp consisting of a flat plate with a sharp leading edge and a ramp with a deflection angle of 15 $^\circ$ is investigated. We previously simulated such a transitional compression ramp flow with a small Reynolds number shown in table 1 (Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025), in which the Reynolds number based on the flat plate length is ${\textit{Re}_{L}}$ = $3.9\times {10^5}$ . Due to the small Reynolds number, the flow is purely induced by convective instability. We refer to this low-Reynolds-number case as Case A to distinguish it from Case B in the current study. The flow condition of Case A was identical to the experiment of Chuvakhov & Radchenko (Reference Chuvakhov and Radchenko2020): free stream Mach number $\textit{Ma}_\infty =8$ , static temperature ${T_\infty }=53.3$ K, ${\textit{Re}_{L}}$ = $3.9\times {10^5}$ , static pressure ${P_\infty }=353$ Pa and wall temperature ${{T}_{w}}=293$ K, corresponding to a wall-to-total temperature ratio ( ${{T}_{w}}/{{T}_{0}}$ ) of 0.4. The DNS result has been comprehensively compared with the experimental one to demonstrate the reliability and high fidelity of our DNS.

Table 1.

Flow conditions of Case A in our previous work (Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025).

Table 1. Long description

A table summarizing flow conditions for Case A. The table has one row and eight columns. Column headers are Case, Ma∞, ReL, T∞, P∞, Tw, Tw/T0, and Ramp angle. Row A: Case, 8; Ma∞, 3.9 x 10^5; ReL, 53.3 K; T∞, 353 Pa; P∞, 293 K; Tw, 0.4; Ramp angle, 15°.

The case currently under consideration, referred to as Case B in table 2, is an extension of the previous study. The detailed flow conditions of Case B are: $\textit{Ma}_\infty =8$ , ${T_\infty }=53.3$ K, ${\textit{Re}_{L}}$ = $9.2\times {10^5}$ , ${P_\infty }=832$ Pa and ${{T}_{w}}=293$ K. The flat-plate and ramp lengths adopted in Case B are $L=50$ mm and ${{L}_{r}}=75$ mm, respectively. Clearly, the Reynolds number is now increased, which serves to create an absolutely unstable state. The highly unstable nature of Case B with self-excited instabilities will be elaborated via GSA in § 4.1.

Table 2.

Flow conditions of all the present simulations.

Table 2. Long description

A table with one row and eight columns summarizing the flow conditions for Case B. The columns are labeled Case, Ma∞, ReL, T∞, P∞, Tw, Tw/T0, and Ramp angle. The row provides specific values for each condition: Case B, Ma∞ of 8, ReL of 9.2 x 10^5, T∞ of 53.3 K, P∞ of 832 Pa, Tw of 293 K, Tw/T0 of 0.4, and a Ramp angle of 15 degrees.

Before moving forward, it is beneficial to establish the conceptual relationship between the current high-Reynolds-number flow (Case B) and the low-Reynolds-number flow (Case A) that was previously investigated (Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025). The flow is globally stable and functions as a pure noise amplifier in Case A, as the separation bubble is insufficiently robust to support self-excited global oscillations. Accordingly, the linear stability framework employed in that low-Reynolds-number case, which includes linear stability theory for Mack second modes as well as resolvent analysis for convective amplification, is directly applicable for describing the long-time response to external disturbances. In comparison, the separation bubble is completely unstable in Case B due to the increased Reynolds number, as demonstrated by the unstable stationary and oscillatory global modes identified by GSA in § 4.1. In this regime, the linearised dynamics of the clean base flow are intrinsically unstable.

The resolvent analysis framework, which is applied to the converged steady laminar base flow, is a carry-over from the Case A framework. This operator is well defined and can describe the early-time convective amplification of external white-noise before the absolute instability has grown to a non-negligible amplitude (Regime I in § 3.4). What does not persist is the assumption that the base flow will remain unaltered. In the long term, the absolute instability will saturate nonlinearly and fundamentally alter the flow. The decoupling strategy (see table 3) is fundamentally based on this conceptual distinction, which explains why resolvent analysis is exclusively employed for Cases B2/B4 (simulated within Regime I) and not as a predictive tool for the saturated coexistence regimes (B1/B3). The reader is referred to § 4.2 for a detailed discussion of the scope and limitations of resolvent analysis in the present globally unstable setting.

Table 3.

Case details for numerical simulations. The classification of Cases B2 and B4 is supported by the projection analysis presented below.

Table 3. Long description

A table comparing different cases of numerical simulations with various parameters. The table has six rows and six columns. Column headers are: Cases, 3-D DNS start time, 3-D DNS end time, The timing of introducing external disturbances, Disturbance amplitude, Contributed by which type of instability. Row 1: Cases, B; 3-D DNS start time, t U infinity / L = 0; 3-D DNS end time, t U infinity / L = 28.8; The timing of introducing external disturbances, No external disturbances; Disturbance amplitude, 0.0 percent; Contributed by which type of instability, Entirely from absolute instability. Row 2: Cases, B 1; 3-D DNS start time, t U infinity / L = 28.8; 3-D DNS end time, t U infinity / L = 34.8; The timing of introducing external disturbances, t U infinity / L = 28.8; Disturbance amplitude, 2.5 percent; Contributed by which type of instability, Absolute and convective instabilities coexist. Row 3: Cases, B 2; 3-D DNS start time, t U infinity / L = 0; 3-D DNS end time, t U infinity / L = 6.0; The timing of introducing external disturbances, t U infinity / L = 0; Disturbance amplitude, 2.5 percent; Contributed by which type of instability, Predominantly convective instability. Row 4: Cases, B 3; 3-D DNS start time, t U infinity / L = 28.8; 3-D DNS end time, t U infinity / L = 34.8; The timing of introducing external disturbances, t U infinity / L = 28.8; Disturbance amplitude, 6.0 percent; Contributed by which type of instability, Absolute and convective instabilities coexist. Row 5: Cases, B 4; 3-D DNS start time, t U infinity / L = 0; 3-D DNS end time, t U infinity / L = 6.0; The timing of introducing external disturbances, t U infinity / L = 0; Disturbance amplitude, 6.0 percent; Contributed by which type of instability, Predominantly convective instability.

3.2. Two-dimensional base flow

Figure 2 illustrates the Mach number distribution of the 2-D base flow as well as the enlarged view within the separation bubble, superimposed with streamlines. A large separation bubble is formed near the corner due to the adverse pressure gradient. The separation and reattachment points are located at $x/L\approx$ 0.35 and 1.44, respectively. As shown in figure 2( $b$ ), there is a strong secondary separation, with the primary separation bubble fragmenting into multiple secondary bubbles. The preliminary observation may imply a globally unstable state under the current condition. In §§ 4.1 and 4.2, stability analysis will be performed to examine the stability of the 2-D base flow within the separation bubble, as well as near and downstream of reattachment.

Figure 2.

Visualisation of 2-D base flow: ( $a$ ) Mach number contour and ( $b$ ) enlarged view superimposed with streamlines within the separation bubble.

3.3. Three-dimensional benchmark flow

Before clarifying the roles of absolute and convective instabilities, it is useful to illustrate the specifics of the 3-D benchmark flow (i.e. Case B). For this case, the destabilisation mechanism is entirely dominated by absolute instability, as no external disturbances are introduced.

Figure 3 shows the DNS results of instantaneous wall Stanton number distributions of Case B at different time instants. Here, the Stanton number is defined by

(3.1) \begin{equation} St=\frac {{{q}_{w}}}{{{\rho }_{\infty }}{{U}_{\infty }}{{C}_{p}}({{T}_{aw}}-{{T}_{w}})}, \end{equation}

where ${q}_{w}$ denotes the heat flux, ${C}_{p}$ is the specific heat capacity and ${T}_{aw}$ is the adiabatic wall temperature for turbulent flows. As seen in figure 3( $a$ ), at ${t{U}_{\infty }/L}= 3.2$ , St exhibits a clearly 2-D pattern. At this stage, the influence of absolute instability on the entire flow field is negligible, with no evidence of three-dimensionality (i.e. Regime I in figure 4 $a$ ). In figure 3( $b$ ), weak three-dimensionality emerges near the reattachment line at ${t{U}_{\infty }/L}= 14.4$ , marking the approach to the end of the exponential stage (Regime II) in figure 4( $a$ ). As time progresses, the streaks on the ramp become increasingly pronounced, reaching near full development in figure 3( $d$ ) at ${t{U}_{\infty }/L}= 20.8$ (i.e. Regime III in figure 4 a). The exponential and nonlinear growth dynamics of the flow field will be analysed through the temporal growth of the spanwise velocity shortly downstream of the ramp within the separation bubble.

Figure 3.

Instantaneous wall Stanton number of Case B at different time instants: ( $a$ ) ${t{U}_{\infty }/L}= 3.2$ ; ( $b$ ) ${t{U}_{\infty }/L}= 14.4$ ; ( $c$ ) ${t{U}_{\infty }/L}= 17.6$ ; ( $d$ ) ${t{U}_{\infty }/L}= 20.8$ and ( $e$ ) ${t{U}_{\infty }/L}= 25.6$ . Separation and reattachment positions are highlighted by iso-lines of zero skin-friction coefficient ( ${{C}_{f}}=0$ ).

To quantify the growth of three-dimensionality, the temporal evolution of the quadratic mean of spanwise velocity at a streamwise position is evaluated by

(3.2) \begin{equation} {{A}_{w}}=\sqrt {\frac {1}{{{N}_{y}}{{N}_{z}}}\sum \limits _{j=1}^{{{N}_{y}}}{\sum \limits _{k=1}^{{{N}_{z}}}{(w/{{U}_{\infty }})_{j,k}^{2}}}}. \end{equation}

In (3.2), ${N}_{y}$ and ${N}_{z}$ represent the total number of grid points in the normal and spanwise directions, respectively. Within the separation bubble (at $x/L=1.07$ ), the temporal evolution of ${A}_{w}/{A}_{s}$ is illustrated in figure 4( $a$ ). Here, ${A}_{s}$ denotes the average value of $A_w$ in the nonlinear saturation stage of Case B. This parameter ${A}_{w}/{A}_{s}$ can effectively characterise to what degree the absolute instability affects the entire flow field during time marching. Figure 4( $a$ ) demonstrates various instability regimes, which will be elaborated in § 3.4. In the range ${t{U}_{\infty }/L}= 0$ to 7.0, there exists a transient growth stage of ${A}_{w}/{A}_{s}$ . After that, ${A}_{w}/{A}_{s}$ begins to grow exponentially until ${t{U}_{\infty }/L}= 17.6$ and then reaches the nonlinear saturation stage. Figure 4( $b$ ) shows the temporal evolution of ${\textit{St}}$ downstream of reattachment at $x/L=1.8$ and $z/L=0.126$ . Obviously, ${\textit{St}}$ is nearly constant until it reaches the nonlinear saturation stage at the dimensionless instant ${\textit{tU}_{\infty }}/L=17.6$ . The observation suggests that the evolution of ${\textit{St}}$ downstream of reattachment is closely coupled with the absolute instability, which is intrinsic to the separation bubble (Cao et al. Reference Cao, Hao, Klioutchnikov, Wen, Olivier and Heufer2022).

Figure 4.

( $a$ ) Temporal evolution of ${A}_{w}/{A}_{s}$ within the separation bubble (at $x/L=1.07$ ) of Case B. The slope of the red and green dashed lines denotes the respective growth rate of Mode 1 and Mode 3 obtained from GSA under the most unstable spanwise wavelength (see figure 5 $c$ ). The grey dashed line represents ${t{U}_{\infty }/L}= 6.0$ . Pink dashed lines represent ${t{U}_{\infty }/L}= 7.0$ and ${t{U}_{\infty }/L}= 17.6$ . ( $b$ ) Temporal evolution of ${\textit{St}}$ downstream of reattachment at $x/L=1.8$ ( $z/L=0.126$ ).

3.4. Physical interpretation of instability regimes and disturbance injection strategies for the examined flows

In this subsection, the physical interpretation of instability regimes in Case B and perturbation injection strategies for the examined flows will be elaborated.

To decouple the absolute and convective instabilities, it is imperative to meticulously design the timing and amplitude when adding external disturbances in the Case B series. Cases B2 and B4 are designed to isolate the convective instability by introducing the external forcing at ${t{U}_{\infty }/L}= 0$ and terminating the simulation at ${t{U}_{\infty }/L}= 6.0$ . This timing ensures that the simulation is confined to the initial stage, during which the absolute instability has not yet developed to a noticeable level. Contrastingly, Cases B1 and B3 are initiated from the saturated state of Case B at ${t{U}_{\infty }/L}= 28.8$ , where the absolute instability has already fully developed. The flow state is established by external forcing that is introduced at this instant, resulting in the coexistence of absolute and convective instabilities. As illustrated in figure 4( $a$ ), the degree of absolute instability can be quantitatively represented by the amplitude of ${A}_{w}/{A}_{s}$ . The degree of absolute instability increases as the amplitude of ${A}_{w}/{A}_{s}$ gets closer to unity. Nonlinear saturation is achieved when the amplitude reaches unity, which is the stage of fully developed absolute instability. In Regime I (from ${t{U}_{\infty }/L}= 0$ to ${t{U}_{\infty }/L}= 6.0$ ), the amplitude of ${A}_{w}/{A}_{s}$ is exceedingly small, measured at less than $2 \times 10^{-5}$ . In Regime III (from ${t{U}_{\infty }/L}= 17.6$ to ${t{U}_{\infty }/L}= 28.8$ ), i.e. the nonlinear saturation stage, the amplitude of ${A}_{w}/{A}_{s}$ is of the order of unity, approximately 50 000 times that of Regime I. Such a small level of ${A}_{w}/{A}_{s}$ in Regime I indicates that the contribution of absolute instability to the three-dimensionality is particularly weak (see figure 3 $a$ ). Therefore, Regime I is referred to as the weak absolute instability stage. Note that the regime separation primarily comes from the empirical comparison between saturated absolute instability cases (B1 and B3) and suppressed absolute instability cases (B2 and B4). The current ${A}_{w}/{A}_{s}$ serves as a supporting indicator. The flow parameters of Cases B1, B2, B3 and B4 are identical to those of Case B (see table 2). The differences among these five cases lie in whether external disturbances are applied or not and the injection timing of external disturbances, as shown in table 3.

For Case B, no external disturbance is introduced into the flow field during the DNS. Therefore, the destabilisation mechanism is entirely from the absolute instability, i.e. self-excited. For Cases B1, B2, B3 and B4, random forcing is introduced upstream of the separation point, taking the form of white noise $w_{j,k}^{\prime }/{{U}_{\infty }}={{\phi }_{n}}(2r-1)$ . Here, $w_{j,k}^{\prime }$ denotes the spanwise velocity perturbation imposed on all the grid nodes in the $y$ – $z$ plane at $x/L=0.2$ , and the indices $j$ and $k$ are along the normal and spanwise directions, respectively. Here, ${\phi }_{n}$ represents the amplitude of perturbations and $r$ denotes a pseudo-random number that varies from 0 to 1 (Cao et al. Reference Cao, Hao, Guo, Wen and Klioutchnikov2023; Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025). For Cases B1 and B2, the amplitude of external disturbances ${{\phi }_{n}}=2.5\, \%$ . For Cases B3 and B4, ${{\phi }_{n}}=6.0\, \%$ . Conventional hypersonic wind tunnels generally exhibit free stream noise levels ranging from 1 % to 5 % (Bounitch et al. Reference Bounitch, Lewis and Lafferty2011; Casper et al. Reference Casper, Beresh and Schneider2014; Munoz et al. Reference Munoz, Wu, Radespiel, Semper, Cummings, Duan and Schilden2019; Bearden et al. Reference Bearden, Padilla, Taubert and Craig2022; Hoffman et al. Reference Hoffman, LaLonde, Andrade, Chen, Bilbo and Combs2023; Zhao et al. Reference Zhao, Ma, Chen, Zhang, Hao and Wen2024), e.g. AEDC Tunnel 9 at Mach 10 reports 4.94 % (Bounitch et al. Reference Bounitch, Lewis and Lafferty2011) and the HHK-6 tunnel shows up to 3.4 % (Munoz et al. Reference Munoz, Wu, Radespiel, Semper, Cummings, Duan and Schilden2019). The disturbance amplitude of 2.5 % in our DNS is evidently within this range and the disturbance amplitude of 6.0 % is close to the wind tunnel noise level of AEDC Tunnel 9 at Mach 10. The above-mentioned data confirm that the selected perturbation amplitudes of 2.5 % and 6 % in our DNS can physically reflect the noise levels in conventional wind tunnels. The amplitude of 2.5 % is indicative of a conventional wind tunnel at a moderate noise level, while a disturbance amplitude of 6.0 % is at a relatively high level. In short, the currently selected noise levels are sufficient to cover most tunnel conditions.

For Case B1, the external disturbance is introduced at the end of the simulation of Case B (i.e. ${\textit{tU}_{\infty }}/L=28.8$ ). At that time, absolute instability has already evolved to a saturated state and excited strong unsteadiness, evidenced by figures 3( $e$ ) and 4(a,b). This signifies that in the 3-D flow field where absolute instability has been fully developed, convective external disturbance is superimposed onto the flow field. In this case, the upstream disturbance begins to be at play. Within the simulation range of Case B1, i.e. from ${\textit{tU}_{\infty }}/L=28.8$ to ${\textit{tU}_{\infty }}/L=34.8$ , absolute and convective instabilities coexist. The external disturbance is in the form of white noise (spanwise velocity perturbations $w'/{{U}_{\infty }}$ , where ${U}_{\infty }$ is the streamwise velocity of the free stream) on the whole $y$ – $z$ plane and is placed upstream of the separation point and downstream of the leading-edge shock (specifically, at $x/L=0.2$ ). In Case B1, the amplitude of the white noise is 2.5 %. Hence, the breakdown to turbulence in Case B1 is triggered by the conjunction of absolute and convective instabilities.

For Case B2, the simulated time range is from ${\textit{tU}_{\infty }}/L=0$ to ${\textit{tU}_{\infty }}/L=6$ , which corresponds to Regime I in figure 4( $a$ ). The disturbance is introduced at the start of the simulation of Case B2, i.e. ${\textit{tU}_{\infty }}/L=0$ . In this simulated time interval, the disturbances due to absolute instabilities have not had time to grow to a non-negligible level (see Regime I in figure 4 $a$ , the level of ${A}_{w}/{A}_{s}$ is less than one in 50 000 of the fully developed flow dominated by absolute instability). The minor value of the parameter ${A}_{w}/{A}_{s}$ indicates that the base flow is clean and must not be affected by the self-excited instability visibly. The only underlying contribution comes from the upstream disturbance. Therefore, the impact of the absolute instability is intended to be minimised and the convective instability is considered isolated from the absolute instability. In other words, the destabilisation mechanism for this case is predominantly convective. The other setup, including the disturbance, and its form and amplitude remain identical to those in Case B1.

Cases B3 and B4 possess the same computational setup as Cases B1 and B2, respectively, except that the amplitude of the external disturbance is increased to 6.0 %. They are designed to reveal the effect of the disturbance amplitude on the transition. The other setup is kept unchanged to conduct comparative research.

In addition to the qualitative characterisation based on $A_w/A_s$ , we further quantify the signature of absolute instability by projecting the DNS perturbation field onto the real part of the most unstable GSA mode (Mode 1), consistent with the real-valued nature of the DNS perturbation field. The projection is performed over the separation bubble region $\varOmega$ ( $x/L \in [0.35, 1.44]$ ) using the spanwise velocity component.

To obtain a dimensionless and case-independent measure of the absolute instability signature, we introduce a modal projection coefficient $\mathcal{P}$ , defined as the normalised inner product between the DNS perturbation field and the GSA mode:

(3.3) \begin{equation} \mathcal{P} = \frac {|\langle w'_{\textit{DNS}},\ \mathrm{Re}(\hat {w}_{\textit{GSA}}) \rangle |}{\|w'_{\textit{DNS}}\| \boldsymbol{\cdot }\|\mathrm{Re}(\hat {w}_{\textit{GSA}})\|}, \quad \langle f,g \rangle = \sum _{\varOmega } f \boldsymbol{\cdot }g,\quad \|f\| = \sqrt {\langle f,f \rangle }. \end{equation}

This coefficient is the direct analogue of the cosine of the angle between two vectors, generalised to functions defined over the spatial domain. This normalisation eliminates the dependence on the arbitrary scaling of the GSA eigenfunction, making $\mathcal{P}$ a robust metric for comparing the absolute instability signature across different cases.

The projection analysis was performed on the final time instant of each case. For the pure absolute instability case (Case B), $\mathcal{P} = 0.79$ . For the coexistence regimes, the values are $\mathcal{P} = 0.69$ (Case B1, $2.5$ % forcing) and $\mathcal{P} = 0.61$ (Case B3, $6.0$ % forcing). In contrast, for the convective-dominated regimes (Cases B2 and B4, both confined to Regime I), the coefficients drop to $\mathcal{P} = 0.07$ and $0.05$ , respectively. These results indicate a clear quantitative separation between the coexistence regimes ( $\mathcal{P} \gt 0.6$ ) and the convective-dominated regimes ( $\mathcal{P} \lt 0.1$ ), providing quantitative corroboration that the absolute instability is indeed negligible in Cases B2 and B4, while it remains strong whenever the flow is initialised from the saturated state of Case B. This analysis provides supporting evidence for the regime classification in table 3 and supports the conclusion that although the absolute instability does not significantly affect the transition onset location, it remains active in coexistence regimes and can therefore modulate the detailed transition features.

Figure 5.

( $a$ ) Growth rates of the most unstable stationary Mode 1 and ( $b$ ) the frequency of the most unstable modes as a function of the spanwise wavelength, provided by GSA. ( $c$ ) Eigenvalue spectrum at ${{\lambda }_{z}/L} = 0.084$ ( ${\lambda }_{z}$ = 4.2 mm) corresponding to the largest growth rate in panel (a). The blue highlighted symbols in panels (a) and (b) denote ${{\lambda }_{z}/L} = 0.084$ and ${{\lambda }_{z}/L} = 0.04833$ ( ${\lambda }_{z}$ = 2.42 mm, the switching wavelength), respectively. Stationary modes: Modes 1, 2 and 5. Oscillatory modes: Modes 3 and 4.

Figure 5. Long description

Panel A: A scatter plot shows the growth rates of the most unstable stationary Mode 1 as a function of the spanwise wavelength. The x-axis is labeled λz/L and ranges from 0 to 0.4. The y-axis is labeled ωiL/U∞ and ranges from −1.0 to 1.5. A vertical dashed line at approximately 0.1 on the x-axis and a blue highlighted symbol at the same x-value are visible. Panel B: A scatter plot shows the frequency of the most unstable modes as a function of the spanwise wavelength. The x-axis is labeled λz/L and ranges from 0 to 0.4. The y-axis is labeled fL/U∞ and ranges from −0.25 to 0.5. A vertical dashed line at approximately 0.1 on the x-axis and a blue highlighted symbol at the same x-value are visible. Panel C: A scatter plot shows the eigenvalue spectrum at a specific spanwise wavelength corresponding to the largest growth rate in Panel A. The x-axis is labeled fL/U∞ and ranges from −0.4 to 0.4. The y-axis is labeled ωiL/U∞ and ranges from −2 to 1. Modes 1, 2, and 5 are stationary, while Modes 3 and 4 are oscillatory.

4.1. Global stability analysis

Figures 5( $a$ ) and 5( $b$ ) depict the non-dimensional growth rates and frequencies of the most unstable stationary mode (Mode 1) in figure 5( $c$ ) as a function of the spanwise wavelength, respectively. As shown in figure 5( $a$ ), the largest growth rate of Mode 1 occurs at ${{\lambda }_{z}/L}\approx 0.084$ ( ${\lambda }_{z}\approx$ 4.2 mm), denoted by the blue highlighted symbol. This growth rate is used to compare with the slope of the exponential growth stage of DNS in figure 4( $a$ ). Figure 5( $b$ ) shows the frequency of the most unstable mode against the spanwise wavelength. In figure 5( $b$ ), the most unstable global modes are stationary when ${{\lambda }_{z}/L}\gt 0.04833$ , whereas unstable oscillatory modes are in the range ${{\lambda }_{z}/L} \approx 0.025{-} 0.04833$ . The value ${{\lambda }_{z}/L}= 0.04833$ is called the switching wavelength here and the global mode is stable below ${{\lambda }_{z}/L}\approx 0.025$ . The eigenvalue spectrum at the wavelength ${{\lambda }_{z}/L}=0.084$ is shown in figure 5( $c$ ). In addition to Mode 1, two other stationary modes (Mode 2 and Mode 5) and two oscillating modes (Mode 3 and Mode 4) are identified. At the most unstable spanwise wavelength, the growth rate of Mode 1 is approximately 2.3 times that of Mode 3. As illustrated in figure 4( $a$ ), the slope of the red and green dashed lines denotes the respective growth rate of Mode 1 and Mode 3 obtained from GSA under the most unstable spanwise wavelength. The slope of the red dashed line matches well with the evolution of ${A}_{w}/{A}_{s}$ in the exponential growth stage. It signifies that Mode 1 dominates the evolution of perturbations until the separation bubble reaches its nonlinear saturation stage. For Mode 3 and Mode 4, the corresponding Strouhal numbers are $fL/{{U}_{\infty }}\approx 0.25599$ ( $f\approx$ 6 kHz) and $fL/{{U}_{\infty }}\approx 0.43607$ ( $f\approx$ 10.2 kHz), respectively. Mode 3 possesses a slightly higher growth rate than Mode 4. The oscillation frequency of Mode 3 will then be compared with DNS later.

Figure 6.

Iso-surfaces of $w/{{U}_{\infty }}= \pm 0.0014$ from DNS at ${\textit{tU}_{\infty }/L}=14.4$ of Case B. The numerical schlieren is superimposed at $z/L=0$ to demonstrate positions of separation and reattachment points.

Figure 7.

( $a$ ) Instantaneous distribution of the spanwise velocity $w/{{U}_{\infty }}$ obtained from DNS in the exponential growth stage ( ${\textit{tU}_{\infty }/L}=14.4$ ) in the $x$ – $y$ plane at $z/L=0.126$ of Case B. ( $b$ ) Real part of the spanwise velocity perturbation $\hat {w}/{{U}_{\infty }}$ for Mode 1 at ${{\lambda }_{z}/L}=0.084$ obtained from GSA. ( $c$ ) Enlarged view of $w/{{U}_{\infty }}$ in the $y$ – $z$ plane at $x/L=1.07$ at the same time instant with panel ( $a$ ).

Figure 6 displays the iso-surface of spanwise velocity $w/{{U}_{\infty }}= \pm 0.0014$ obtained from DNS at ${\textit{tU}_{\infty }/L}=14.4$ , which is in the exponential growth stage of Case B. As shown, the spanwise velocity obtained from DNS is primarily constrained within the separation bubble with an alternating positive–negative distribution pattern. As a comparison between GSA and DNS, figure 7 shows the instantaneous distribution of $w/{{U}_{\infty }}$ in the $x$ – $y$ plane (at $z/L=0.126$ ) and $y$ – $z$ plane (at $x/L=1.07$ ) obtained from DNS at ${\textit{tU}_{\infty }/L}=14.4$ of Case B, and the real part of the spanwise velocity perturbation $\hat {w}/{{U}_{\infty }}$ for Mode 1 at ${{\lambda }_{z}/L}=0.084$ obtained from GSA. As illustrated again, the spanwise velocity obtained from DNS and GSA exhibits a similar alternating positive–negative distribution pattern. An average spanwise wavelength ${\lambda }_{z}/L\approx$ 0.084 ( ${\lambda }_{z}\approx$ 4.2 mm) is reported from DNS, which agrees well with the GSA. The good agreement implies that the exponential growth of the separation bubble is dominated by the GSA mode, provided that no external disturbance is imposed.

Estimated from figures 3( $d$ ) and 3( $e$ ), the mean spanwise wavelength is about 2.1 mm, which is about one half of the GSA result (4.2 mm). To clarify the discrepancy between DNS and GSA results, the quantified spatial spectra will be analysed later to reveal the space scales in the exponential growth and nonlinear saturation stages. Figure 8 presents the spanwise Fourier transforms of the spanwise velocity on the $x$ – $z$ plane at ${{y}_{n}}/L=0.005$ within the boundary layer. The contour is plotted versus the streamwise coordinate $x/L$ and the spanwise wavelength ${{\lambda }_{z}}/L$ in the exponential growth and nonlinear saturation stages. The ensemble average of the Fourier-transformed spanwise velocity with hundreds of snapshots is taken to minimise the randomness effect. In the exponential growth stage, the preferential spanwise wavelength spans from ${{\lambda }_{z}}/L\approx 0.08$ to ${{\lambda }_{z}}/L\approx 0.145$ , and the most amplified spanwise wavelength 0.084 obtained from GSA falls within this range. This confirms that DNS results clearly capture the spatial characteristics in the exponential growth stage. In figure 8( $b$ ), two preferential spanwise wavelengths occur in the nonlinear saturation stage: one corresponds to the most amplified GSA mode, while the other pertains to the spanwise wavelength ${{\lambda }_{z}}/L\approx 0.042$ ( ${{\lambda }_{z}}\approx$ 2.1 mm). This new peak is the second harmonic mode of the most amplified GSA mode.

Figure 8.

Spanwise Fourier transforms of the spanwise velocity on the $x$ – $z$ plane at ${{y}_{n}}/L=0.005$ in the ( $a$ ) exponential growth stage and ( $b$ ) nonlinear saturation stage. Ensemble average of the result is taken from ${t{U}_{\infty }/L}= 8.0$ to 16.0 in the exponential growth stage and from ${t{U}_{\infty }/L}= 20.8$ to 28.8 in the nonlinear saturation stage. ${y}_{n}$ denotes the wall-normal coordinate that starts from the wall.

From the perspective of mode-mode interactions in the Fourier space, the emergence of the second harmonic mode can be described by

(4.1) \begin{equation} (0,{{\beta }_{0}})\text{ }+\text{ }(0,{{\beta }_{0}})\to (0,2{{\beta }_{0}}), \end{equation}

where ${\beta }_{0}$ represents the fundamental spanwise wavenumber (i.e. ${{\beta }_{0}}={2\pi }/{{{\lambda }_{z0}}}$ , ${{{\lambda }_{z0}}}=$ 4.2 mm). Mode $(0,{{\beta }_{0}})$ is the most amplified stationary GSA mode (Mode 1) and Mode $(0,2{{\beta }_{0}})$ is the generated second harmonic through nonlinear interactions. This signifies that during the switch from the linear growth stage to the nonlinear saturation stage, each streamwise streak bifurcates into two branches. Note that the small-wavelength range can also belong to the primary unstable oscillatory mode in figure 5( $b$ ). Nonetheless, the wavelength of the oscillatory GSA mode is not constant and growth is dominated by the stationary mode in figure 4( $a$ ). Therefore, the 2.1 mm component is the second harmonic of the stationary Mode 1.

Figure 9 presents the PSD results of the DNS spanwise velocity, temperature and pressure within the separation bubble and shortly downstream of reattachment. It is evident that the PSD results exhibit a broadband low-frequency feature. The dominant 6 kHz component of the GSA oscillatory mode (Mode 3) also falls well in the pronounced low-frequency band. Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021b ) have confirmed that such a low-frequency feature originates from the separation bubble. After examining the PSD results in the transitional region ( $x/L=2.1$ ), such a low-frequency feature still exists (not shown here). It is inferred that such a low-frequency feature can be seeded to propagate until the transitional region. In Appendix C, transitional images of Case B triggered by absolute instability are clearly identified, which resembles the results of Fu et al. (Reference Fu, Karp, Bose, Moin and Urzay2021) and Cao et al. (Reference Cao, Hao, Klioutchnikov, Wen, Olivier and Heufer2022) without introducing any external disturbances.

Figure 9.

PSD results of ( $a$ ) spanwise velocity ( $b$ ) temperature and ( $c$ ) pressure within the separation bubble ( $x/L=1.1$ ), and shortly downstream of reattachment ( $x/L=1.5$ ) at $z/L=0.126$ and ${{y}_{n}}/L=0.005$ for Case B.

Figure 9. Long description

Panel A: A line graph shows the power spectral density (PSD) of spanwise velocity within the separation bubble. The x-axis represents frequency (f) in kilohertz (kHz) on a logarithmic scale, and the y-axis represents PSD on a logarithmic scale. Two lines are plotted: one for x/L = 1.1 (black) and another for x/L = 1.5 (red). The black line shows a steeper decline in PSD with increasing frequency compared to the red line. Panel B: A line graph displays the PSD of temperature within the separation bubble. The axes are the same as in Panel A. The black line (x/L = 1.1) shows a more pronounced drop in PSD with increasing frequency than the red line (x/L = 1.5). Panel C: A line graph illustrates the PSD of pressure within the separation bubble. The axes are identical to those in Panels A and B. The black line (x/L = 1.1) again shows a steeper decline in PSD with increasing frequency compared to the red line (x/L = 1.5).

4.2. Resolvent analysis

In the present study, the base flow used for resolvent analysis is a converged 2-D steady solution of the Navier–Stokes equations. The relation $\hat {\boldsymbol{U}}={\unicode{x1D64D}} \boldsymbol{\mathcal{B}} \hat {\kern-2.5pt\boldsymbol{f}}$ conveys information about the input–output linear mapping, irrespective of the global instability status. In this context, the resolvent operator $\unicode{x1D64D}$ manifests the linear dynamics about the clean base flow $\boldsymbol{\overline {U}}$ , where $\boldsymbol{\overline {U}}$ exactly satisfies the steady governing equations. The clean flow is the state at the very beginning when the absolute instability has no visible impact on the entire flow field. Thus, the eigen analysis of the resolvent operator constructed from this $\boldsymbol{\overline {U}}$ characterises how external white noise is amplified before the absolute instability increases to a certain amplitude and can then alter the base state. The resolvent analysis is strictly constrained to the discussion in a finite time interval, ensuring that the base flow remains valid. For instance, the resolvent analysis is only compared with DNS results in Cases B2 and B4, where the simulation time ( ${\textit{tU}_{\infty }/L}=6.0$ ) is earlier than the instant when the absolute instability begins to increase exponentially. In this stage, the normalised amplitude ${A}_{w}/{A}_{s}$ , characterising the baseline global instability, is less than $2 \times 10^{-5}$ . Under these conditions, the resolvent operator provides a reasonable model for the forced response, essentially capturing the convective-instability mechanism in the initial stage.

Figure 10.

( $a$ ) Contour of the optimal gain in the space of spanwise wavenumber and angular frequency. The gain is maximal when $\beta L\approx 260.8$ (marked with a pink circle), as the frequency ${{\omega }_{r}}L/{{U}_{\infty }}$ is approaching zero. An additional local maximum of the optimal gain can be identified at $\beta L=0$ (blue circle) when ${{\omega }_{r}}L/{{U}_{\infty }}\approx 135$ . ( $b$ ) Optimal gain as a function of the angular frequency at the most amplified spanwise wavenumber $\beta L=260.8$ .

Note that this approach has a limited application scope: resolvent analysis here is not employed to predict a long-term instability or the fully saturated nonlinear state. Rather, it functions as a supplementary tool to identify the most responsive convective modes to external forcing. The resolvent analysis establishes a baseline to understand the amplification mechanism before absolute instabilities have a notable impact on the entire flow field. It provides a mechanistic interpretation of the early transition stages. More importantly, resolvent analysis is not used in isolation. The spatial and frequency scales predicted by the resolvent analysis will be compared with DNS results of Cases B2/B4 with caution. The mutual verification between DNS and resolvent analysis shown in § 5.2 will further justify this issue.

2-D base flow solutions are utilized to conduct resolvent analysis that spans a wide range of spanwise wavelengths and angular frequencies. The spanwise wavenumber ranges from $\beta L=0$ to 600 and ${{\omega }_{r}}L/{{U}_{\infty }}$ from 0.1 ( $f$ = 372 Hz) to 199.5 ( $f$ = 742 kHz). It is worth noting that the forcing is located at $x/L=0.2$ , which is consistent with the 3-D DNS.

The resulting optimal gain is depicted in the $\beta$ – ${\omega }_{r}$ space in figure 10( $a$ ). The gain is maximal when $\beta L\approx 260.8$ ( ${{\lambda }_{z}}\approx$ 1.2 mm) and the frequency ${{\omega }_{r}}L/{{U}_{\infty }}$ is approaching zero. Figure 10( $b$ ) gives the optimal gain as a function of the angular frequency at the most amplified spanwise wavenumber $\beta L\approx 260.8$ . Clearly, the optimal gain almost remains constant at low frequencies and decays dramatically when ${{\omega }_{r}}L/{{U}_{\infty }}\gt 2$ . Such a low-frequency feature has also been observed in other studies of SWBLIs (Dwivedi et al. Reference Dwivedi, Sidharth, Nichols, Candler and Jovanović2019; Bugeat et al. Reference Bugeat, Robinet, Chassaing and Sagaut2022; Cao et al. Reference Cao, Hao, Guo, Wen and Klioutchnikov2023; Hao et al. Reference Hao, Cao, Guo and Wen2023). It has been confirmed that Görtler instability is the dominant amplification mechanism for streamwise streaks (Cao et al. Reference Cao, Hao, Guo, Wen and Klioutchnikov2023; Hao et al. Reference Hao, Cao, Guo and Wen2023). In figure 10( $a$ ), an additional local maximum of the optimal gain can be identified at $\beta L=0$ when ${{\omega }_{r}}L/{{U}_{\infty }}\approx 135$ ( $f\approx$ 502 kHz), corresponding to the most amplified planar Mack second mode.

5.1. Transition onset and breakdown to turbulence

Figure 11 illustrates the distribution of the spanwise- and time-averaged Stanton number in the streamwise direction. The theoretical correlations are plotted based on the following equation (Neumann et al. Reference Neumann and Hayes1986; Simeonides et al. Reference Simeonides1992):

(5.1) \begin{equation} {\textit{St}_{\textit {theory}}}=\frac {A}{s}{{\left[\frac {{{u}_{e}}{{P}_{e}}}{{{u}_{\infty }}{{P}_{\infty }}}\right]}^{1-n}}{{\left[\frac {{{T}_{\infty }}}{{{T}^{*}}}\right]}^{1-2n}}{{\left[\frac {{{C}^{*}}}{{{\textit{Re}}_{\infty ,x}}}\right]}^{n}}. \end{equation}

For a Prandtl number 0.72, the parameters ( $A$ , $s$ , $n$ , $r$ ) in (5.1) are (0.332, ${\textit{Pr}}^{2/3}$ , 0.5, ${\textit{Pr}}^{1/2}$ ) and (0.0296, 1.0, 0.2, ${\textit{Pr}}^{1/3}$ ) for laminar and turbulent flows, respectively. Here, ${T}^{*}$ is the reference temperature, ${C}^{*}$ denotes the Chapman–Rubesin linear viscosity factor (Chapman & Rubesin Reference Chapman and Rubesin1949) and ${u}_{e}$ , ${P}_{e}$ are local values at the edge of boundary layers. For more details, please refer to Roghelia et al. (Reference Roghelia, Olivier, Egorov and Chuvakhov2017).

Figure 11.

Distributions of spanwise- and time-averaged wall Stanton number in the streamwise direction. Black and red solid lines represent laminar and turbulent theoretical correlations, respectively. Green squares, 2-D DNS results.

As shown in figure 11, downstream of reattachment, the 2-D DNS results coincide well with the laminar theoretical formula. For Case B, the flow does not develop to a fully developed turbulence flow (see also figure 33 in Appendix C), which yields deviations from the turbulent correlation. From Case B1 to Case B4, near the outflow boundary, the ${\textit{St}}$ numbers obtained from DNS collapse well onto the turbulent correlation, which indicates a fully developed turbulent state.

The elevated Stanton number observed in Cases B1 and B2 is a classic heat flux overshoot phenomenon, well documented in SWBLI in the transitional state. The overshoot occurs when the flow undergoes transition, but has not yet reached a fully developed turbulent state within the computational or experimental domain. This overshoot phenomenon has been extensively reported in both experimental (Benay et al. Reference Benay, Chanetz, Mangin, Vandomme and Perraud2006; Bur & Chanetz Reference Bur and Chanetz2009; Knight & Mortazavi Reference Knight and Mortazavi2018; Currao et al. Reference Currao, Choudhury, Gai, Neely and Buttsworth2020) and DNS studies (Cao et al. Reference Cao, Hao and Guo2024; Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025) of hypersonic SWBLI flows. Specifically, Currao et al. (Reference Currao, Choudhury, Gai, Neely and Buttsworth2020) observed overshoots of approximately 15 % in hypersonic transitional SWBLI, while Lugrin et al. (Reference Lugrin, Nicolas, Severac, Tobeli, Beneddine, Garnier, Esquieu and Bur2022b ) reported overshoots up to 25 %. The causes of heat-flux overshoot are closely linked to the Görtler instability pertaining to the streamline curvature and the rapid transition at reattachment (Benay et al. Reference Benay, Chanetz, Mangin, Vandomme and Perraud2006; Currao et al. Reference Currao, Choudhury, Gai, Neely and Buttsworth2020).

The observed discrepancy in the ${\textit{St}}$ distribution is a result of the distinction in the disturbance amplitude and its subsequent impact on the transition physics. Cases B1 and B2 represent moderate disturbance levels (2.5 %), while Cases B3 and B4 correspond to large disturbance levels (6.0 %). This amplitude disparity fundamentally alters the transition dynamics. For large disturbance amplitudes, i.e. Cases B3 and B4, the strong convective disturbance accelerates the breakdown process, leading to rapid transition completion (see figures 15 and 16). Consequently, the spanwise- and time-averaged ${\textit{St}}$ profiles collapse onto the turbulent theoretical correlation just shortly downstream of reattachment. Under the moderate disturbance levels, i.e. Cases B1 and B2, the weak external disturbance triggers a slower and more gradual transition. The transition requires a longer distance to reach its fully turbulent state compared with the noisy cases. This developing state is characterised by elevated heat flux known as the overshoot phenomenon. This disturbance-amplitude dependent behaviour is physically consistent, since a large noise level case injects more energy into the instability modes, accelerates the nonlinear breakdown and shortens the transitional-region length. This interpretation is consistent with the findings of Benay et al. (Reference Benay, Chanetz, Mangin, Vandomme and Perraud2006) and Bur & Chanetz (Reference Bur and Chanetz2009), who showed that the heat flux overshoot can be mitigated by triggering an earlier transition to a fully turbulent SWBLI.

Figure 12.

( $a$ ) The r.m.s. distribution of the Stanton number fluctuation ${\textit{St}}{{'}_{\textit {rms}}}$ along the streamwise direction. The black dash dotted line denotes the threshold criterion ${\textit{St}}{{'}_{\textit {rms}}}=4\times {{10}^{-3}}$ . ( $b$ ) Transition onset locations based on different thresholds of ${\textit{St}}{{'}_{\textit {rms}}}$ for Cases B1–B4.

Figure 12. Long description

Panel A: A line graph shows the root mean square (r.m.s.) distribution of the Stanton number fluctuation along the streamwise direction. The x-axis is labeled x/L and the y-axis is labeled St_rms. The graph includes data for five cases: Case B, Case B1, Case B2, Case B3, and Case B4, each represented by different symbols and colors. A black dash-dotted line denotes the threshold criterion. Panel B: Another line graph displays transition onset locations based on different thresholds of St_rms for Cases B1 to B4. The x-axis is labeled Threshold and the y-axis is labeled Transition onset location. Each case is represented by different symbols and colors.

To reveal the impact of absolute and convective instabilities, the transition onset and end locations are imperative to be identified. Figure 12( $a$ ) shows the root mean squared (r.m.s.) distribution of the Stanton number fluctuation ${\textit{St}}{{'}_{\textit {rms}}}$ averaged in the spanwise direction along the streamwise direction for all the five cases. The definition of ${\textit{St}}{{'}_{\textit {rms}}}$ is given by

(5.2) \begin{equation} \textit{St}{{'}_{\textit {rms}}}=\sqrt {\overline {{\textit{St}}^{\prime 2}}}, \end{equation}

where the long overline ‘—’ represents the arithmetic mean. Meanwhile, ${\textit{St}}'=St-{{\left \langle St \right \rangle }_{t}}$ , where ${\left \langle St \right \rangle }_{t}$ is the time-averaged value. As shown in figure 12( $a$ ), the peak value of ${\textit{St}}{{'}_{\textit {rms}}}$ of Case B remains below $1.5\times {{10}^{-3}}$ . This value is considerably lower than the values of Cases B1–B4, implying that the transitional process of Case B is not as quick as the others. Consequently, the transition onset criterion proposed based on Cases B1–B4, which relies on a fixed threshold, was not applied to Case B. As a special case, the transition onset of Case B is determined based on the streamwise position where the spanwise- and time-averaged ${\textit{St}}$ number begins to exhibit a secondary increase. In the study of transitions induced by pure absolute instability by Cao et al. (Reference Cao, Hao, Klioutchnikov, Wen, Olivier and Heufer2022), they adopted the same criteria to determine the transition onset location. With the presence of convective instabilities, the transition is accelerated for Cases B1–B4 and there is no secondary increase in ${\textit{St}}$ . The different criteria indicate that the transition onset locations of Case B cannot be directly compared with the others. Nonetheless, it is clear that Case B transitions slower according to the mean ${\textit{St}}$ curves.

As shown in figure 12( $b$ ), the selection of the threshold does not affect the relative magnitude of transition onset locations for different cases. The central conclusion is robust with respect to the choice, since a varied threshold within a reasonable range yields the same conclusion regarding the relative magnitude of transition onset locations. In this paper, the transition onset for Cases B1–B4 takes the streamwise position at which ${\textit{St}}{{'}_{\textit {rms}}}$ reaches a threshold ${\textit{St}}{{'}_{\textit {rms}}}\approx 4\times {{10}^{-3}}$ , which is nearly the peak value. This maximum-amplitude location for instability waves is usually in the vicinity of the breakdown point for the 3-D hypersonic transitional boundary layer (Guo et al. Reference Guo, Hao and Wen2023). Thus, the threshold value of $4\times {{10}^{-3}}$ serves as a representative criterion. The instantaneous visualisation of vortex structures of different cases is depicted in figures 13–16 and 33 to reveal the transitional and breakdown events of the flow field. For the moderate noise level cases (Cases B1 and B2), coherent vortices emerge near the reattachment line, followed by the formation of hairpin-like vortex packets that undergo gradual breakdown further downstream. For the large noise level cases (Cases B3 and B4), the vortex structures are more vigorous: streamwise vortices appear immediately downstream of reattachment and rapidly disintegrate into small-scale coherent structures. This amplitude-dependent behaviour confirms that a stronger convective forcing accelerates the nonlinear breakdown process, while the presence of absolute instability provides a secondary modulation to the overall breakdown scenarios.

Figure 13.

Instantaneous visualisation of vortex structures of Case B1. The $Q$ -criterion coloured by $u/{{U}_{\infty }}$ is utilized to visualise the transitional and fully developed turbulent flow. The Stanton number contour is shown superimposed with isolines of zero skin friction coefficient. Numerical schlieren at $z/L=0$ is additionally added to highlight the separation and reattachment points. The upper left panel represents the zoomed-in view of the shock–shear layer patterns and coherent structures near the transitional boundary layer.

Based on the above-mentioned criterion, the transition onset locations are listed in table 4. For Case B, the onset of transition is at $x/L\approx 2.10$ and Appendix C provides a comprehensive overview of its transitional scenario. For Cases B1 and B2 where the external disturbance amplitude is 2.5 %, the transition onset locations are approximately $x/L\approx 1.68$ and $x/L\approx 1.77$ , respectively. This signifies that when absolute and convective instabilities coexist (Case B1), the position of transition onset is $\Delta x/L\approx 0.1$ earlier than where only convective instability is evident (Case B2). This minor advance indicates that the absolute instability does not affect the transition onset remarkably. As shown in figures 12 and 15–16, the locations of transition onset of Cases B3 and B4 are both $x/L\approx 1.55$ , given that the amplitude of external disturbances is increased to 6 %. In such situations, the role of the absolute instability is even weaker than that of the convective instability, thereby showing no visible difference in the transition onset. In other words, Cases B3 and B4 are dominated by the convective-type instability. By comparing Cases B1 and B3 (or B2 and B4), one finds that as the strength of the external disturbance increases, the transition onset position is moved forward more evidently than by the effect of the absolute instability.

Table 4.

Transition onset locations of the five cases.

Table 4. Long description

A table with two columns and five rows. The first column is labeled ‘Cases’ and the second column is labeled ‘Transition onset locations’. The rows are as follows: Row 1: Case B, x / L ≈ 2.10; Row 2: Case B1, x / L ≈ 1.68; Row 3: Case B2, x / L ≈ 1.77; Row 4: Case B3, x / L ≈ 1.55; Row 5: Case B4, x / L ≈ 1.55.

Figure 14.

Instantaneous visualisation of vortex structures of Case B2. The $Q$ -criterion coloured by $u/{{U}_{\infty }}$ is utilized to visualise the transitional and fully developed turbulent flow. The Stanton number contour is shown superimposed with isolines of zero skin friction coefficient. Numerical schlieren at $z/L=0$ is additionally added to highlight the separation and reattachment points. The upper left panel represents the zoomed-in view of the shock–shear layer patterns and coherent structures near the transitional boundary layer.

Figure 14. Long description

Panel A: A heat map shows the Stanton number contour with isolines of zero skin friction coefficient. The x-axis represents the normalized streamwise distance (x/L) ranging from 0 to 2.50, and the y-axis represents the normalized spanwise distance (z/L) ranging from 0 to 1.50. The color scale indicates the Stanton number, with lighter colors representing higher values and darker colors representing lower values. Panel B: A heat map displays the numerical schlieren at a specific value, highlighting separation and reattachment points. The x-axis represents the normalized streamwise distance (x/L) ranging from 0 to 2.50, and the y-axis represents the normalized spanwise distance (z/L) ranging from 0 to 1.50. The color scale indicates the intensity of the numerical schlieren, with a gradient from blue to red. The upper left panel provides a zoomed-in view of the shock-shear layer patterns and coherent structures near the transitional boundary layer, with the -criterion colored by is.

Figure 15.

Instantaneous visualisation of vortex structures of Case B3. The $Q$ -criterion coloured by $u/{{U}_{\infty }}$ is utilized to visualise the transitional and fully developed turbulent flow. The Stanton number contour is shown superimposed with isolines of zero skin friction coefficient. Numerical schlieren at $z/L=0$ is additionally added to highlight the separation and reattachment points. The upper left panel represents the zoomed-in view of the shock–shear layer patterns and coherent structures near the transitional boundary layer.

Figure 16.

Instantaneous visualisation of vortex structures of Case B4. The $Q$ -criterion coloured by $u/{{U}_{\infty }}$ is utilized to visualise the transitional and fully developed turbulent flow. The Stanton number contour is shown superimposed with isolines of zero skin friction coefficient. Numerical schlieren at $z/L=0$ is additionally added to highlight the separation and reattachment points. The upper left panel represents the zoomed-in view of the shock–shear layer patterns and coherent structures near the transitional boundary layer.

Figure 16. Long description

Panel A: A heat map showing the Stanton number contour with isolines of zero skin friction coefficient. The color scale ranges from low to high values, indicating different levels of heat transfer. The numerical schlieren highlights separation and reattachment points. Panel B: A zoomed-in view of the shock-shear layer patterns and coherent structures near the transitional boundary layer. The -criterion colored by is used to visualize transitional and fully developed turbulent flow.

Apart from the differences in the transition onset location, the breakdown scenarios of Cases B1 to B4 are visibly similar. In proximity and downstream of reattachment lines, streamwise vortices appear, followed by the roll-up and lift of hairpin-like vortex structures. Taking Case B2 as an example, mesh sensitivity study of 3-D DNS is conducted in Appendix D, and one can observe a good agreement on the boundary layer profiles between DNS and theoretical predictions in Appendix E regarding the law of the wall and the mean temperature-velocity relation. As a result, a fully developed turbulent state is reached for most cases.

5.2. Spectral characteristics of transition scenarios

To further reveal the roles of absolute and convective instabilities, spatial and temporal spectral analysis is performed in this subsection. The analysis is categorised into two subsections. The first subsection (§ 5.2.1) analyses the fully evolved 3-D state, encompassing Cases B, B1 and B3, where the absolute instability either evolves purely from the outset (Case B) or has already been modulated by external white noise forcings after saturation (Cases B1 and B3). The second subsection (§ 5.2.2) examines the early-stage 2-D base state, encompassing Cases B2 and B4, where the absolute instability is suppressed by early termination and the flow is predominantly convective.

5.2.1. Role of convective instabilities on fully evolved 3-D base state: Cases B, B1 and B3

The long-duration evolved 3-D base state cases are first examined, which comprises the baseline absolutely unstable case (Case B) and the coexistence regimes at two disturbance amplitudes (Case B1, 2.5 % and Case B3, 6 %).

Figure 17.

Common logarithm of modulus of spanwise Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the streamwise coordinate $x/L$ for ( $a$ ) Case B1 (coexistence, 2.5 %) and ( $b$ ) Case B3 (coexistence, 6 %). The ensemble average is taken using several hundreds of snapshots. The key spanwise wavelengths are ${{\lambda }_{z}/L}\approx 0.084$ ( ${{\lambda }_{z}}\approx 4.2$ mm, GSA fundamental) and ${{\lambda }_{z}/L}\approx 0.042$ ( ${{\lambda }_{z}}\approx 2.1$ mm, GSA second harmonic).

Figure 18.

Amplitude of spanwise Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the spanwise wavelength ${\lambda }_{z}$ extracted at $x/L=1.5$ in figure 17 for ( $a$ ) Case B1 (coexistence, 2.5 %) and ( $b$ ) Case B3 (coexistence, 6 %).

Figure 17 depicts the common logarithm of modulus of spanwise Fourier transforms of ${\textit{St}}$ for Cases B1 and B3. The data are retrieved on the wall ( $x$ – $z$ plane) versus the streamwise coordinate $x/L$ . Figure 18 shows the amplitude versus the spanwise wavelength ${\lambda }_{z}$ extracted at $x/L=1.5$ in figure 17 for Cases B1 and B3. For Case B1 in which absolute and convective instabilities coexist, the GSA mode wavelength ${{\lambda }_{z}}\approx 4.2$ mm becomes weaker, while ${{\lambda }_{z}}\approx 2.1$ mm of the second harmonic tends to be more pronounced. The 2.1 mm component arises from nonlinear mode-mode interactions (see (4.1)), where the fundamental GSA mode ( ${{\lambda }_{z}}\approx 4.2$ mm) generates its second harmonic as the flow transitions from the linear growth stage to nonlinear saturation stage (Cao et al. Reference Cao, Hao, Klioutchnikov, Wen, Olivier and Heufer2022). Notably, by comparing figures 8, 17( $a$ ) and 18( $a$ ), the introduction of external noise serves to add broadband components to the wavenumber spectrum. Nonetheless, the signature of the absolute instability is not wiped by the external forcing in Case B1. As a result, the impact of the absolute instability remains in the transition process, albeit the transition onset location is not affected remarkably.

For Case B3, where the external forcing is added after absolute instability saturates, the GSA mode wavelengths 4.2 and 2.1 mm are less clear owing to a larger-amplitude forcing. The side band is filled by external noise to a larger extent. Additionally, we note that the resolvent optimal gain peak at ${{\lambda }_{z}}\approx 1.2$ mm appears in the purely convective regimes (Cases B2 and B4, see figures 21 and 22), not in Case B3.

Figures 19 and 20 characterise the spanwise-averaged modulus of the temporal Fourier transforms of the observed signals for Cases B, B1 and B3. The signals include the Stanton number in figure 19 and wall pressure in figure 20. In figure 20, dashed and solid lines represent the oscillatory GSA mode frequency $f\approx 6$ kHz and the frequency of the most unstable Mack second mode, respectively. The second-mode frequency is predicted by LSA.

Figure 19.

Common logarithm of spanwise-averaged modulus of temporal Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the streamwise coordinate $x/L$ for ( $a$ ) Case B (pure absolute instability), ( $b$ ) Case B1 (coexistence, 2.5 %) and ( $c$ ) Case B3 (coexistence, 6.0 %). The left vertical axis denotes the physical frequency, and the right vertical axis represents the dimensionless frequency, where ${{\omega }_{r}}=2\pi f$ .

Figure 20.

Common logarithm of spanwise-averaged modulus of temporal Fourier transforms of the wall pressure on the $x$ – $z$ plane versus the streamwise coordinate $x/L$ for ( $a$ ) Case B (pure absolute instability), ( $b$ ) Case B1 (coexistence, 2.5 %) and ( $c$ ) Case B3 (coexistence, 6.0 %). Dashed and solid lines represent $f\approx 6$ kHz and the frequency of the most unstable planar Mack second mode, respectively. The left vertical axis denotes the physical frequency, and the right vertical axis represents the dimensionless frequency, where ${{\omega }_{r}}=2\pi f$ .

Figure 20. Long description

Panel A: A heatmap shows the common logarithm of spanwise-averaged modulus of temporal Fourier transforms of the wall pressure on the plane versus the streamwise coordinate for Case B (pure absolute instability). The x-axis is labeled x/L, and the y-axis is labeled f(kHz). The color bar indicates values ranging from −2.5 to −1.0. A dashed line represents a specific frequency. Panel B: A heatmap shows the same data for Case B1 (coexistence, 2.5 %). The x-axis is labeled x/L, and the y-axis is labeled f(kHz). A black line indicates a trend. Panel C: A heatmap shows the same data for Case B3 (coexistence, 6.0 %). The x-axis is labeled x/L, and the y-axis is labeled f(kHz). A black line indicates a trend.

For Case B, as shown in figures 19( $a$ ) and 20( $a$ ), the low-frequency signature below the frequency 6 kHz can propagate downstream into the transitional region ( $x/L\gt 2.1$ ). Ahead of the transition onset, no obvious high-frequency components are identified. This observation confirms that the transition of Case B is triggered by the low-frequency feature intrinsic to the separation bubble.

For Case B1, as shown in figure 19( $b$ ), in addition to low-frequency components, the band with hundreds of kilohertz emerges. In figure 20( $b$ ), the most unstable second mode frequency by LSA is consistent with DNS in the pre-transitional region. Such evidence signifies that the transition in Case B1 is triggered by the combination effects of the low-frequency component and the Mack second mode. The low-frequency component can be contributed by the GSA mode and/or the resolvent low-frequency streaks, which is difficult to clarify by figure 19( $b$ ) only.

As shown in figure 20( $c$ ), with the external disturbance amplitude increased to 6 % in Case B3, the strength of the Mack second mode is notably stronger than that of Case B1. In Case B3, the signature of absolute instability is even weaker compared with the second mode. In this case, the convective instability counts more than Case B1.

5.2.2. Role of convective instabilities on the early-stage 2-D base state: Cases B2 and B4

We now proceed to the early-stage 2-D base state modulated by the white noise, which includes Cases B2 and B4. In this state, the simulation is limited to Regime I ( ${\textit{tU}_{\infty }}/L \leqslant 6.0$ ) and the absolute instability has a negligible effect. These cases isolate the purely convective response to upstream forcing at moderate (Case B2, 2.5 %) and high (Case B4, 6 %) amplitudes.

Regarding Case B2, figures 21( $a$ ) and 22( $a$ ) indicate that the GSA second harmonic wavelength ${{\lambda }_{z}}\approx 2.1$ mm is not obvious. This is because the forcing is imposed at ${t{U}_{\infty }/L}= 0$ , at which the absolute instability has not yet developed. Another wavelength peak emerges at ${{\lambda }_{z}}\approx 1.26$ mm is observed. As demonstrated in figure 10( $a$ ), the most amplified optimal gain peaks at ${{\lambda }_{z}}\approx$ 1.2 mm, which is close to the spanwise wavelength obtained from figures 21( $a$ ) and 22( $a$ ). The absence of the GSA mode wavelength and the presence of the resolvent like wavelength indicate that the instability is dominated by the convective type. The mutual confirmation between DNS and resolvent analysis results confirms that the destabilising mechanism in Case B2 is dominated by a convective-type instability. Regarding Case B4 under the large-amplitude external forcing, the optimal wavelength of 1.26 mm is consistent with the results provided by resolvent analysis, further confirming the convective-dominated mechanisms.

Figure 21.

Common logarithm of modulus of spanwise Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the streamwise coordinate $x/L$ for ( $a$ ) Case B2 (convective-dominated, 2.5 %) and ( $b$ ) Case B4 (convective-dominated, 6.0 %). The ensemble average is taken using several hundreds of snapshots. The resolvent optimal gain wavelength is ${{\lambda }_{z}/L}\approx 0.024$ ( ${{\lambda }_{z}}\approx 1.2$ mm).

Figure 22.

Amplitude of spanwise Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the spanwise wavelength ${\lambda }_{z}$ extracted at $x/L=1.5$ in figure 21 for ( $a$ ) Case B2 (convective-dominated, 2.5 %) and ( $b$ ) Case B4 (convective-dominated, 6.0 %).

Figure 23.

Common logarithm of spanwise-averaged modulus of temporal Fourier transforms of ${\textit{St}}$ number on the wall ( $x$ – $z$ plane) versus the streamwise coordinate $x/L$ for ( $a$ ) Case B2 (convective-dominated, 2.5 %) and ( $b$ ) Case B4 (convective-dominated, 6.0 %). The left vertical axis denotes the physical frequency and the right vertical axis represents the dimensionless frequency, where ${{\omega }_{r}}=2\pi f$ .

Figure 24.

Common logarithm of spanwise-averaged modulus of temporal Fourier transforms of the wall pressure on the $x$ – $z$ plane versus the streamwise coordinate $x/L$ for ( $a$ ) Case B2 (convective-dominated, 2.5 %) and ( $b$ ) Case B4 (convective-dominated, 6.0 %). Solid lines represent the frequency of the most unstable planar Mack second mode. The left vertical axis denotes the physical frequency and the right vertical axis represents the dimensionless frequency, where ${{\omega }_{r}}=2\pi f$ .

Figure 24. Long description

Panel A: A heat map showing the common logarithm of spanwise-averaged modulus of temporal Fourier transforms of wall pressure versus streamwise coordinate for Case B2 (convective-dominated, 2.5 %). The x-axis represents the streamwise coordinate (x/L) and the y-axis represents the physical frequency (f in kHz) on the left and dimensionless frequency (ωrL/U∞) on the right. The color scale ranges from −2.5 to −1.0, with darker colors indicating lower values and lighter colors indicating higher values. A solid line represents the frequency of the most unstable planar Mack second mode. Panel B: A heat map showing the common logarithm of spanwise-averaged modulus of temporal Fourier transforms of wall pressure versus streamwise coordinate for Case B4 (convective-dominated, 6.0 %). The x-axis represents the streamwise coordinate (x/L) and the y-axis represents the physical frequency (f in kHz) on the left and dimensionless frequency (ωrL/U∞) on the right. The color scale ranges from −2.5 to −1.0, with darker colors indicating lower values and lighter colors indicating higher values. A solid line represents the frequency of the most unstable planar Mack second mode.

The corresponding temporal spectra for Cases B2 and B4 are shown in figures 23 and 24. In comparison, for Case B2, the strength of the low frequency feature becomes too weak compared with the second mode to be well identifiable in figure 23( $a$ ). Note that Case B2 eliminates the effect of a strong absolute instability in Case B1. As a result, the low-frequency component is now supposed to be mainly induced by the low-frequency resolvent mode rather than the absolute instability in Case B2. Moreover, due to the weak low-frequency band, the second mode should play a more active role in the transition of Case B2. Furthermore, analogous to Case B2 where absolute instability is minimised, transition of Case B4 is dominated by convective instabilities (see figures 23 $b$ and 24 $b$ ).

5.3. Spectral proper orthogonal decomposition

SPOD, a data-driven method to extract coherent structures or modes from flow fields, is further employed. At each frequency, the SPOD modes represent the dominant coherent structures, which are mathematically the eigenvectors of the cross-spectral density tensor. At first, a snapshot matrix is generated by combining ${N}_{t}$ snapshots of the fluctuating flow field:

(5.3) \begin{equation} \boldsymbol{Q} = \left [ \boldsymbol{q}_1, \boldsymbol{q}_2, \ldots , \boldsymbol{q}_{N_t} \right ]\!. \end{equation}

To estimate the cross-spectral density, the dataset is divided into ${N}_{\textit {blk}}$ overlapping segments, each containing ${N}_{\textit {FFT}}$ snapshots:

(5.4) \begin{equation} \boldsymbol{Q}^{(k)} = \big [ \boldsymbol{q}_{1}^{(k)}, \boldsymbol{q}_{2}^{(k)}, \ldots , \boldsymbol{q}_{N_{\textit {FFT}}}^{(k)} \big ], \end{equation}

where the $j$ th column in the $k$ th block is

(5.5) \begin{equation} \boldsymbol{q}_{\!j}^{(k)} = \boldsymbol{q}_{j + (k - 1)(N_{\textit {FFT}} - N_{\textit {ovlp}}) + 1}. \end{equation}

In this equation, $N_{\textit {ovlp}}$ denotes the number of overlapping snapshots for each block. The data segmentation procedure is employed to enhance statistical reliability by increasing the number of available ensemble members. For each block, a Hamming window is applied. A discrete Fourier transform is subsequently applied to each windowed block and the cross-spectral density matrix is generated for each discrete frequency. The SPOD modes are subsequently calculated as the eigenvectors of the cross-spectral density matrix for each frequency, which are arranged in a descending order of energy. Here, the SPOD mode energy is evaluated using the Chu’s energy norm. The SPOD analysis is conducted based on the open-source code of Schmidt & Colonius (Reference Schmidt and Colonius2020).

Regarding the implementation, a minimum of $N_{\textit {blk}} \geqslant 20$ flow realisations was deemed sufficient to ensure statistical convergence (Guo et al. Reference Guo, Hao and Wen2025a ). Here, $N_{\textit {blk}} = 20$ is applied, and a reasonable resolution of $N_{\textit {FFT}}=256$ with 75 % overlapping is adopted. The total number of snapshots should be at least 1472, according to the following relationship:

(5.6) \begin{equation} {{N}_{\textit {blk}}}=\left \lfloor \frac {{{N}_{t}}-{{N}_{\textit {ovlp}}}}{{{N}_{\textit {FFT}}}-{{N}_{\textit {ovlp}}}} \right \rfloor \!, \end{equation}

where $\lfloor \boldsymbol{\cdot }\rfloor$ represents the floor operator.

Figure 25 shows the real part of the spanwise velocity perturbation $\hat {w}/{{U}_{\infty }}$ for Mode 1 at ${{\lambda }_{z}/L}=0.084$ obtained from GSA (see figure 5 $c$ ). In comparison, the spanwise velocity for the first leading SPOD $x$ – $y$ mode at ${{f}}=0$ kHz of Case B is also displayed. Unsurprisingly, the spanwise velocity exhibits an alternating positive–negative distribution pattern, predominantly confined within the separation bubble. More notably, the SPOD results show a perfect agreement with GSA, confirming the reliability of the current SPOD setup.

Figure 25.

( $a$ ) Real part of the spanwise velocity perturbation $\hat {w}/{{U}_{\infty }}$ for Mode 1 at ${{\lambda }_{z}/L}=0.084$ obtained from GSA. ( $b$ ) Spanwise velocity for the first leading SPOD $x$ – $y$ mode at ${{f}}=0$ kHz of Case B in the exponential growth stage.

Figure 26.

Spanwise velocity for the first leading SPOD $x$ – $y$ mode at ${{f}}=23$ kHz of ( $a$ ) Case B, ( $b$ ) Case B1, ( $c$ ) Case B2, ( $d$ ) Case B3 and ( $e$ ) Case B4.

Figure 26. Long description

Five line graphs show spanwise velocity for the first leading SPOD mode at kHz for different cases. Each graph is labeled as Panel A, Panel B, Panel C, Panel D, and Panel E. The x-axis represents the normalized streamwise position (x/L) ranging from 0 to 2.4, and the y-axis represents the normalized spanwise position (y/L) ranging from 0 to 0.6. Panel A shows data for Case B, Panel B for Case B1, Panel C for Case B2, Panel D for Case B3, and Panel E for Case B4. Each graph includes multiple lines in different colors representing the spanwise velocity profiles. The lines exhibit various trends and patterns, indicating the behavior of spanwise velocity across different cases.

Figure 27.

Maximum of spanwise velocity ${(w/{{U}_{\infty }})}_{\textit {max} }$ versus $x/L$ at ${{f}}=23$ kHz extracted from figure 26 for ( $a$ ) Case B, ( $b$ ) Case B1, ( $c$ ) Case B2, ( $d$ ) Case B3 and ( $e$ ) Case B4.

Figure 28.

Spanwise velocity for the first leading SPOD $x$ – $y$ mode at ${{f}}=114$ kHz of ( $a$ ) Case B, ( $b$ ) Case B1, ( $c$ ) Case B2, ( $d$ ) Case B3 and ( $e$ ) Case B4.

Figure 28. Long description

Panel A: A line graph shows spanwise velocity for Case B. The x-axis is labeled x/L and ranges from 0 to 2.4. The y-axis is labeled y/L and ranges from 0 to 0.6. Panel B: A line graph shows spanwise velocity for Case B1. The x-axis is labeled x/L and ranges from 0 to 2.4. The y-axis is labeled y/L and ranges from 0 to 0.6. Panel C: A line graph shows spanwise velocity for Case B2. The x-axis is labeled x/L and ranges from 0 to 2.4. The y-axis is labeled y/L and ranges from 0 to 0.6. Panel D: A line graph shows spanwise velocity for Case B3. The x-axis is labeled x/L and ranges from 0 to 2.4. The y-axis is labeled y/L and ranges from 0 to 0.6. Panel E: A line graph shows spanwise velocity for Case B4. The x-axis is labeled x/L and ranges from 0 to 2.4. The y-axis is labeled y/L and ranges from 0 to 0.6.

Figures 26, 28 and 29 present the first leading SPOD modes of different cases in the $x$ – $y$ plane for selected frequencies: a low frequency ( ${{f}}=23$ kHz), a medium frequency ( ${{f}}=114$ kHz) and a high frequency ( ${{f}}=503$ kHz). Overall, the three frequencies exhibit remarkably different disturbance patterns and structures. For the low-frequency SPOD modes, figure 27 shows the maximum of spanwise velocity ${(w/{{U}_{\infty }})}_{\textit {max} }$ versus $x/L$ at ${{f}}=23$ kHz extracted from figure 26. Combined with figures 26 and 27, there are noticeable differences in the modal distribution among different cases. In cases where absolute instability is present (Cases B, B1 and B3), the pronounced spanwise velocity is primarily restricted to the separation bubble and the boundary layers downstream of reattachment. Conversely, the spanwise velocity is predominantly distributed in the boundary layer downstream of reattachment in the cases where only convective instability (Cases B2 and B4) is involved. Whether the absolute instability is at play can be well distinguishable from the low-frequency SPOD modal distribution, regardless of the disturbance strength.

Figure 29.

Spanwise velocity for the first leading SPOD $x$ – $y$ mode at ${{f}}=503$ kHz of ( $a$ ) Case B, ( $b$ ) Case B1, ( $c$ ) Case B2, ( $d$ ) Case B3 and ( $e$ ) Case B4.

Dwivedi et al. (Reference Dwivedi, Sidharth and Jovanović2022) investigated a Mach 5 flow over a slender double wedge that is globally (absolutely) stable. They found that upstream unsteady disturbances can activate the second leading stationary global mode (i.e. steady streaks) through nonlinear interactions, which corresponds to a resolvent mode. This confirms that even in the absence of absolute instability, convective disturbances could trigger specific stable global modes and lead to transition. Notably, Dwivedi et al. (Reference Dwivedi, Sidharth and Jovanović2022) employed resolvent analysis to identify the global (stable) mode instead of a GSA. Only the GSA can report the absolutely unstable mode.

In our study, Cases B1 and B3 are absolutely unstable flows with superimposed convective disturbances of different amplitudes (2.5 % and 6.0 %, respectively). Within these coexistence regimes, the low-frequency component in the flow arises from two distinct sources. The absolute instability modes identified by GSA, which are intrinsic to the separation bubble and the convective streak modes predicted by resolvent analysis (optimal gain at low frequencies and spanwise wavelengths around ${{\lambda }_{z}}\approx$ 1.2 mm), which are amplified from upstream disturbances. It is inevitable that the two families of low-frequency modes will coexist and be intermixed during the transition process of Cases B1 and B3. Distinguishing their individual contributions using available analysis tools (i.e. spectra analysis, SPOD) is challenging, because they occupy overlapping frequency bands (both are low-frequency). Their spatial structures can be similar in the reattachment region. Therefore, the spatial wavelengths in figures 17–18 and the low-frequency energy in figures 19–20 are a superposition of the GSA modes and the resolvent streaky modes.

We note that the total simulation time of Case B ( ${\textit{tU}_{\infty }}/L=28.8$ ) corresponds to approximately seven periods of the 6 kHz oscillatory global mode (Mode 3, see § 4.1). While this is sufficient to identify the low-frequency band, the spectral estimates at the very lowest frequencies should be interpreted with some caution due to the finite window length. The stationary modes (Mode 1 and its harmonic) are not affected by this limitation, as they are time-independent in the linear regime.

Nevertheless, the amplitude-dependent behaviour permits the establishment of certain conclusions. For Case B1, the GSA modes remain clearly identifiable, i.e. the stationary GSA wavelength ( ${{\lambda }_{z}}\approx$ 4.2 mm) and its harmonic (2.1 mm) are prominent in the spatial spectra (figures 17 $a$ and 18 $a$ ), and the low-frequency component is notable in the temporal spectra (figures 19 $b$ and 20 $b$ ). This confirms that the absolute-instability modes actively participate alongside the convective streaky mode. For Case B3, the strong convective disturbance amplifies the resolvent streak modes more strongly, making the GSA signatures less pronounced. Specifically, the 4.2 mm peak is weaker, suggesting a greater contribution from the convective streak modes. However, the structural footprint of the absolute instability persists in the flow field, implying that the GSA modes still provide a background modulation.

Regarding the SPOD modes at 23 kHz (figure 26), this frequency is above the low-frequency range of the GSA oscillatory modes. For Cases B1 and B3 where absolute instability is involved, the modal shape of the spanwise velocity can be identified in the separation bubble and downstream of reattachment. Thus, while the 23 kHz SPOD mode does not directly correspond to the GSA eigenmode, its modal shape provides indirect evidence that the GSA modes remain dynamically relevant in shaping the flow response in Cases B1 and B3.

At 23 kHz, the first leading SPOD mode exhibits different structural features between Cases B1 and B3. In Case B1 (figure 26 $b$ ), the mode is concentrated in the separation bubble and along the reattached boundary layer, reflecting the persistent influence of the absolute instability. For Case B3 (figure 26 $d$ ), the mode shows an alternating pattern in the upstream part of the separation bubble. However, the structural footprint of the absolute instability persists in the separation bubble, implying that the GSA modes still provide a background modulation. Thus, the two coexistence regimes do not activate different global modes; rather, the same absolute instability signature is modulated by the strength of convective forcing. This is distinct from Case B4 (figure 26 $e$ ), where the absolute instability is negligible and the flow is purely convective. Case B4 (figure 26 $e$ ) represents a purely convective regime where the absolute instability is suppressed by early termination ( ${\textit{tU}_{\infty }}/L \leqslant 6.0$ ). Its modal structure is therefore fundamentally different from the coexistence regimes (Cases B1 and B3), and it should not be used as a reference for comparing Cases B1 and B3.

We confirm that in our absolutely unstable flows with convective disturbances, the low-frequency dynamics are a mixture of GSA modes and convective streak modes. The intermixing of these two families precludes a clear separation of contributions, but the combined evidence from GSA, resolvent analysis, and DNS consistently supports the coexistence and interplay of both types of low-frequency mechanisms in the transition scenarios.

For the medium-frequency SPOD modes ( ${{f}}=114$ kHz) shown in figure 28, compared with low-frequency counterparts, the spanwise velocity within the separation bubble is significantly eliminated in Case B. This is associated with the low-frequency features of the absolute instability in the separation bubble. For Case B, wisp-like structures begin to appear in the vicinity of the transition onset $x/L \approx 2.10$ . The emergence of this structure may result from the breakdown of the streamwise streak downstream of reattachment. For Cases B1 and B2, their modal distributions exhibit remarkable similarity. A distinct rope-like coherent structure emerges at $x/L \approx 1.68$ in Case B1, while it appears slightly downstream at $x/L \approx 1.77$ in Case B2. Here, the positions at which the regular rope-like structure commences to emerge coincide with the onset of transition. For Cases B3 and B4, the amplitude of external disturbances increases to 6 %, and the modal structures are distinct from those of Case B, B1 and B2. Notably, the shear layer in the separation region exhibits prominent spanwise velocity structures, which are likely associated with the convective shear layer instability. The alternatingly distributed structure propagates downstream along the shear layer, intersects with the reattached boundary layer near the reattachment point and subsequently disintegrates into a multi-layered structure shortly downstream of reattachment at $x/L \approx 1.65$ .

The high-frequency SPOD modes at ${{f}}=503$ kHz are shown in figure 29. One noteworthy characteristic is that, in all cases, no discernible signal appears below the shear layer and within the separation bubble, which further reveals the low-frequency feature of the separation bubble. In Case B, upstream of the reattachment point, a clean flow field is presented, since no external forcing is applied. The high-frequency components downstream of the reattachment are more likely to originate from the secondary instability upon breakdown of streamwise streaks triggered by absolute instability. The modal structures of Cases B1 and B2, both with an amplitude of 2.5 %, exhibit similarity in their overall patterns. The difference lies in that the pronounced structure of Case B1 is located more upstream. For Cases B3 and B4 with an amplitude of 6.0 %, the shear layer instability also exists and resembles the medium frequency performance. In addition, there is a visible ‘contaminated region’ above the separation bubble. It suggests that the strong high-frequency component in the white noise, albeit decaying, is comparable to the secondary growing signals downstream of reattachment. The occurrence of the shear layer instability in figures 28( $d$ , $e$ ) and 29( $d$ , $e$ ) indicates that for compression ramp flows with large external-disturbance amplitudes, the shear layer instability of convective nature is predominant in the laminar–turbulent transition.

5.4. Synthesis

Based on the spectral analyses already presented, we now synthesise the relative roles of absolute and convective instabilities across the different base states.

For Cases B1 and B3, external disturbances were introduced after absolute instability had saturated ( ${\textit{tU}_{\infty }/L}=28.8$ ), representing the coexistence of absolute and convective instability types. Their transitional-mechanism difference lies in the relative dominance determined by the disturbance amplitude. For Case B1, the strengths of absolute and convective instabilities are comparable. The absolute instability provides the dominant low-frequency component intrinsic to the separation bubble, while convective forcing activates the Mack second mode and streaks mode. Thus, the transition is a product of both instability types. For Case B3, the stronger convective disturbance (6 %) amplifies the Mack second mode more strongly than in Case B1, making the low-frequency band less dominant relative to the high-frequency band. However, substantial low-frequency energy remains present due to the concurrent amplification of resolvent-predicted streak modes. Thus, the absolute instability is not ‘swamped’ in an absolute sense, rather, its signature becomes relatively weaker compared with the convectively driven high-frequency components. This is consistent with the observation that the low-frequency region in figure 20( $c$ ) (Case B3) still shows high energy. The transition onset is primarily governed by convective mechanisms, with absolute instability playing a secondary, structurally modulating role.

For Cases B2 and B4, external disturbances were introduced at the very beginning ( ${\textit{tU}_{\infty }/L}=0$ ). The simulation is conducted prior to the onset of the growth of absolute instability, representing a purely convective regime. The differences in the physical mechanism entirely depend on the amplitude of external white noise. For Case B2, convective instability manifests in a moderate amplification regime. The Mack second mode is present, but moderate in strength. Transition onset and vortex structures occur downstream of reattachment (see figure 14). For Case B4, convective instability approaches the strong amplification regime more rapidly. The Mack second mode is strongly amplified, and the transition onset moves upstream considerably. The flow reaches a fully turbulent state more rapidly, with vortex structures forming near the reattachment line with more vigorous breakdown (see figure 16).

The aforementioned spatial and temporal spectral analysis demonstrates that without external disturbances, the transition is induced by the low-frequency feature originated from the separation bubble. In scenarios where absolute and convective instability coexist, a small amplitude of external disturbances allows the absolute instability to directly participate in the transition process. Such a disturbance level of 2.5 % can be encountered in a moderate noise level wind tunnel (Munoz et al. Reference Munoz, Wu, Radespiel, Semper, Cummings, Duan and Schilden2019). The participation of the absolute instability generates a slightly earlier transition onset compared with the situation with pure convective instabilities. As the amplitude of external disturbances further increases (i.e. 6.0 %), the absolute instability will be further weaker than the convective instability, which corresponds to conventional noisy wind tunnel experiments (Bounitch et al. Reference Bounitch, Lewis and Lafferty2011; Casper et al. Reference Casper, Beresh and Schneider2014).