5.1. Linear stability

To begin with the instability mechanism, first, we analyse the basic state using linear theory to identify instabilities. As discussed in § 3, the flow is comprised of an attached boundary over the cone and a long separated zone, characterised by negative velocity gradient and mean flow distortion, starting over the cylinder and extending over most of the flare. Since linear theory neglects streamwise gradients, these calculations are only performed in regions where its assumptions are reasonably valid, i.e. in the attached boundary layer over the cone. Figure 11(a) shows the growth rate ( $-\alpha _i$ ) contour obtained by solving the eigenvalue problem (2.9) over the cone, with clear shaded unstable regions in the $f{-}X$ plane, bounded by the outer contour line, which is the neutral curve. First-mode ( ${\sim} 35{-}185\,\rm kHz$ ) and second-mode ( ${{\gt}} 185\,\rm kHz$ ) unstable regions are distinctly evident.

Figure 11.

(a) Growth rate contour plot showing neutral curve. Streamwise distribution of (b) growth rate and (c) N-factor at several distinct frequencies over the cone.

Figures 11(b) and 11(c) plot the growth rate ( $-\alpha _i$ ) and N-factor distributions in the streamwise direction for various individual frequencies. The higher second-mode frequencies ( ${{\geqslant}} 250\,\rm kHz$ ) amplify and quickly damp over relatively short distances in the upstream part of the cone; these results are similar to those obtained for an isothermal wall (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020) with STABL-LST. Initially, the higher frequencies on the cone experience rapid detuning within a narrow amplification band. However, as these disturbance waves propagate downstream, they become more closely aligned with the local boundary-layer thickness, resulting in more sustained growth over extended distances. The peak amplification gradually shifts towards lower frequencies in the downstream direction, with mid- to low-frequency components ( $37$ – $212\,\rm kHz$ ) showing significant amplification by the end of the cone. The maximum growth rate is obtained for frequencies around $62.5\,\rm kHz$ (first mode), followed by those around $200\,\rm kHz$ (second mode); the peak for both lies in the vicinity of $X=10$ . The STABL-LST prediction over an isothermal wall (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020) indicated only second-mode amplification, suggesting an increased significance of the first mode in the case of an adiabatic wall. The N-factor distribution reported in figure 11(c) is computed using growth rate as

(5.1) \begin{align} {\rm e}^N = {\rm e}^{\int _{x_0}^x (-\alpha _i) {\rm d}x}, \end{align}

where $x_0$ is the location where the disturbance first becomes unstable. The higher frequencies, indicating early growth, attain their maximum N value early and then decay. The frequency $212.5\,\rm kHz$ has a maximum N value exceeding four by the end of the cone. Frequencies such as $62.5$ and $200\,\rm kHz$ , which showed the highest growth rates, reach N values of approximately 3.5. A higher growth rate thus does not necessarily guarantee a higher N-factor; rather, the spatial extent over which a particular frequency remains unstable also plays a critical role. For example, $212.5\,\rm kHz$ displays a lower growth rate than $62.5$ and $200\,\rm kHz$ , but has a higher N value due to its longer instability spatial extent.

5.2. Receptivity over cone

We now attempt to understand how the inflow forcing used in DNS produces instability waves that align with theoretical predictions, i.e. receptivity. The response of the system to each free-stream disturbance environment may be represented through a receptivity coefficient. Fedorov (Reference Fedorov2003) and Tumin et al. (Reference Tumin, Wang and Zhong2007) presented mathematical details on estimating the receptivity coefficient, which relates the amplitude of an instability wave excited in the boundary layer to the amplitude of free-stream disturbance (Saric, Reed & Kerschen Reference Saric, Reed and Kerschen2002; Balakumar Reference Balakumar2015):

(5.2) \begin{align} C_{\textit{rec}}(f) = \frac {A^{\textit{DNS}}_{\textit{inst}}(x_n,f)}{A^{\textit{DNS}}_{\textit{fst}} (f)}. \end{align}

Here $A^{\textit{DNS}}_{\textit{fst}}(f)$ is the amplitude of the free-stream disturbance imposed at frequency $f$ , while $A^{\textit{DNS}}_{\textit{inst}}(x_n,f)$ is the wall pressure amplitude for frequency $f$ from the DNS at the branch I neutral location $x_n$ . Neutral points for individual frequencies are identified from the neutral curve and growth rate plot shown in figure 11. The variation of pressure amplitude at the wall for individual frequencies from the DNS (details of which are discussed in the next section) is shown in figure 14. Some literature (Ma & Zhong Reference Ma and Zhong2001, Reference Ma and Zhong2003) reports the receptivity coefficient based on the instability wave amplitude at the branch II neutral point, denoted as the response coefficient.

To assess the efficiency of the receptivity process, figure 12 shows the variation of the receptivity coefficient based on linear stability analysis of the attached boundary layer over the cone, for the instability wave amplitude at both the branch I ( $A_{{N}_{\mathrm{I}}}$ ) and branch II ( $A_{{N}_{\mathrm{II}}}$ ) neutral points. Here $C_{\textit{rec}}=1$ represents the generation of instability waves with the same amplitude as the external disturbance, indicating moderate receptivity, while $C_{\textit{rep}}\lt 1$ represents low receptivity, indicating weak response. In contrast, $C_{\textit{rep}}\gt 1$ is indicative of a highly receptive response, where the boundary layer acts as an amplifier to the given free-stream disturbance. Based on $A_{{N}_{\mathrm{I}}}$ , lower frequencies ( ${\leqslant} 80\,\rm kHz$ ) show moderate to low receptivity, which starts to grow in a nearly linear manner with the increase in frequency. A local maximum is observed at $125\,\rm kHz$ with $C_{\textit{rec}} = 4.41$ that reduces to a local minimum at $175\,\rm kHz$ and then grows again to an absolute maximum at $187.5\,\rm kHz$ with $C_{\textit{rec}} = 7.68$ . This indicates $187.5\,\rm kHz$ as being highly receptive, generating instability waves of amplitude $7.68$ times the free-stream amplitude. Further, higher frequencies show decreasing coefficient, with weakest receptivity for $300\,\rm kHz$ . The green line indicates the receptivity coefficient based on the branch II neutral point, where the unstable waves become stable. The line for this plot is discontinuous as some of the frequencies do not become stable over the streamwise extent of the cone. It is evident that the receptivity for the frequency range $225{-}275\,\rm kHz$ at the branch II neutral point is higher than at the branch I neutral point. For frequencies outside this range, at branch II, the same or lower receptivity coefficient is observed as that of branch I. The observation that the receptivity coefficient at certain frequencies is lower at the second neutral point than at the first is a result of the surface amplitudes used to compute the receptivity coefficient being extracted from DNS signals. Unlike theoretical predictions, the numerically recorded wall amplitudes are more sensitive to local fluctuations (see figure 14) and to the precise spatial positioning of the neutral points. Overall, higher receptivity ( $C_{\textit{rec}} \geqslant 1$ ) is observed for the current configuration, indicating that the free-stream disturbances are relatively efficient at generating instability waves that tune with the boundary-layer modes.

Figure 12.

Variation of receptivity coefficient with frequency.

6.1. Spectra comparison

For instability quantification, the unstable spectra from the forced DNS are first compared with prior estimates of the CCF model with an isothermal wall. Experimental PCB data (Benitez et al. Reference Benitez, Jewell and Schneider2021) and linear PSE predictions are obtained from Paredes et al. (Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022) for comparison at four distinct locations marked in figure 13(a). The DNS spectra are presented for both isothermal and adiabatic walls to understand the effects of wall thermal conditions. Given the attached boundary layer over the cone, spectra from linear stability analysis are also compared, details of which are discussed earlier in § 5. The LST results in regions further downstream are not discussed because the extensive separated zone, shown in figure 7(f), characterised by negative velocity gradients and mean flow distortion, violates the underlying assumptions.

Figure 13.

(a) Streamwise locations for comparison. Power spectra at (b) $X= 11.6$ – $0.387\,\rm m$ , (c) $X= 14.61 $ – $0.51\,\rm m$ , (d) $X= 17.31 $ – $0.606\,\rm m$ and (e) $X= 18.37 $ – $0.643\,\rm m$ . Experimental PCB (Benitez, Jewell & Schneider Reference Benitez, Jewell and Schneider2021) and PSE data are obtained from Paredes et al. (Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022). LST plots $A^2$ , as $A^2 = A_i^2 \exp(2N)$ with $A_i=7.44 \times 10^{-9}$ .

Focusing on the cone region, to compare LST predictions with the experimental and DNS power spectra, the approach used by Paredes et al. (Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022) for PSE analysis is adopted as follows:

(6.1) \begin{align} A/A_i = \exp (N )\Rightarrow A^2 = A_i^2 \exp(2N). \end{align}

In the above, $A_i$ is the initial disturbance amplitude and N is the amplification factor (N-factor). Amplitude $A_i$ is fixed as $7.44\times 10^{-9}$ for all disturbances so as to match the second-mode peak of the DNS predicted spectra. A similar approach was used in the linear PSE analysis (Paredes et al. Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022) to match the experimental peak (Benitez et al. Reference Benitez, Jewell and Schneider2021). It should be noted that the comparison of outcomes of stability theories with DNS or experiments is performed in terms of unstable spectra and their locations, rather than absolute magnitudes of amplification. Power spectra over the cone at $X=11.6$ ( $0.387\,\rm m$ ) in figure 13(b) show good agreement between the current DNS and previous studies. The DNS with the isothermal wall shows second-mode amplification between $180\text{ and }240\,\rm kHz$ , while experiment and PSE indicate $195{-}260$ and $205{-}250\,\rm kHz$ , respectively. Comparing isothermal and adiabatic DNS cases, it is evident that the second-mode range of $180{-}240\,\rm kHz$ for the isothermal wall is reduced to $183{-}215\,\rm kHz$ in the adiabatic case. For the isothermal case, the maximum second-mode amplification is observed at $220\,\rm kHz$ , while it is $195\,\rm kHz$ for the adiabatic wall case. The overall second-mode amplification is suppressed for the adiabatic wall compared with the isothermal wall. At lower frequencies, it is clear that frequencies in the range $55{-}145\,\rm kHz$ show higher amplification over the adiabatic wall than the isothermal wall. However, no distinct frequency emerges as the dominant destabilised component. Linear stability amplification based on N-factor (using (6.1)) shows the first-mode amplification between $35\text{ and }175\,\rm kHz$ , while the second mode amplification occurs between $180\text{ and }235\,\rm kHz$ . Maximum amplification of the first and the second modes is found to be comparable, with the second mode slightly surpassing the first-mode peak. The spatial support of these frequencies over the cone from DNS (to be discussed later) connects the frequencies with the spatial structures of each mode.

Turning to the separation bubble over the cylinder at $X=14.6$ ( $0.51\,\rm m$ ) in figure 13(c), the higher frequencies do not exhibit any distinct peak, contrary to observations in the cone boundary layer. Consistent with experimental observations that did not report any distinct amplification band, DNS indicates no distinct frequency with gradual decay in power spectra with increasing frequency. Lower frequencies ( ${\lt} 130\,\rm kHz$ ) show higher amplification for the adiabatic wall compared with the isothermal wall, with distinct peaks emerging at $63,\ 94$ and $130\,\rm kHz$ , with a maximum at $63\,\rm kHz$ . Such distinct peaks are also observed for the isothermal case at the same frequencies, but their amplification is lower than in the adiabatic case. Spatial support of these low frequencies ( ${\lt} 130\,\rm kHz$ ), discussed later, displays coherent structures within the separation bubble that diminishes towards the shear-layer edge.

Proceeding further downstream, over the flare at $X=17.31$ (figure 13d ), the DNS prediction shows second-mode amplification in the range of $190{-}265\,\rm kHz$ , which aligns well with the experimental observation of amplification between $200\text{ and }270\,\rm kHz$ , and with PSE predictions in the narrower range of $200{-}235\,\rm kHz$ . Additionally, lower frequencies in the range $90{-}135\,\rm kHz$ amplify in the DNS, again consistent with the experimentally observed amplification between $62\text{ and }150\,\rm kHz$ . In contrast, the adiabatic wall case does not exhibit a distinct second-mode amplification band at higher frequencies, and the second-mode amplification remains suppressed. However, lower frequencies ( $56{-}77\,\rm kHz$ ) amplify with peak amplitude at $66\,\rm kHz$ . Over the rear part of the flare, at $X=18.37$ ( $0.643\,\rm m$ ), shown in figure 13(f), isothermal DNS is in good agreement with experimental and PSE spectra, where the boundary-layer modes re-amplify following reattachment, indicating second-mode amplification for $155{-}250\,\rm kHz$ . Experiment and PSE data showed the same trends at $170{-}265\,\rm kHz$ and $175{-}250\,\rm kHz$ , respectively. For the adiabatic case, it should be noted that the boundary layer still has not reattached at this location ( $X=18.37$ ) as indicated in figure 7(f), hence suppressing the second mode. However, the low frequencies show higher amplification with a peak at $95\,\rm kHz$ .

The above discussion clearly highlights a key impact of the wall thermal boundary condition: the adiabatic wall promotes amplification of lower frequencies over the higher frequencies, in particular, those belonging to the first-mode family, as compared with the isothermal case. This observation is consistent across the attached boundary layer over the cone and the separated zone on the cylinder and flare; however, the amplification differences between the isothermal and adiabatic cases for lower frequencies are more prominent in the separated zone than the attached boundary layer. Moreover, in the adiabatic case, the suppression of second-mode frequencies is most effective in the reattachment zone over the flare. This is because the adiabatic case represents a hotter wall condition than the isothermal case, thus increasing the boundary-layer thickness and amplifying low-frequency disturbances as observed in past stability investigations (Ma & Zhong Reference Ma and Zhong2003; Polivanov et al. Reference Polivanov, Gromyko, Sidorenko, Maslov, Keller, Groskopf and Kloker2011). Further, an adiabatic wall condition reduces the wall-normal temperature gradient, weakening the damping of low-frequency disturbances. In contrast, second-mode (high-frequency) instabilities (like Mack modes) are more sensitive to temperature gradients and viscosity stratification, which are stronger in isothermal walls, favouring their amplification. Moreover, in the isothermal case, strong temperature gradients near the wall lead to stronger acoustic coupling that effectively amplifies second-mode instabilities. This is also consistent with a prior normal mode investigation (Eissler & Bestek Reference Eissler and Bestek1996) showing lower frequencies being more unstable under adiabatic conditions compared with the isothermal case.

The laminar solution of the separation zone also induces differences in the amplification of linear and nonlinear perturbations over the cylinder–flare junction. The separation bubble over cylinder and flare has been observed to amplify low frequencies primarily due to shear-layer dynamics as reported in prior work on separated flows (Rist & Augustin Reference Rist and Augustin2006; Robinet Reference Robinet2013). The presence of a free shear layer separating the outer high-speed flow and the reversed/recirculating flow inside the bubble is highly susceptible to K–H instabilities, which naturally amplify low-frequency, long-wavelength disturbances. In contrast, the second-mode waves (high frequency), which are trapped acoustic waves (Kuehl Reference Kuehl2018), amplify when their phase speed closely matches the local flow velocity at a particular wall-normal location. In the separated zone, this synchronisation is disrupted due to strong velocity gradients and reverse flows, leading to suppression of higher frequencies. A lack of acoustic trapping further subdues high-frequency second-mode instabilities in the separated zone. Additionally, in the adiabatic case, a zero temperature gradient at the wall further stabilises the second mode.

6.2. Instabilities over cone

The streamwise variation of surface pressure amplitude for individual frequencies is analysed to identify the instability characteristics and dominant frequency ranges over the cone region. Figure 14 presents the pressure amplitude distribution along the streamwise direction, divided into four subfigures for clarity: figure 14(a–d), covering frequency ranges of $30{-}137$ , $152{-}256$ , $271{-}376$ and $391{-}495\,\rm kHz$ , respectively. The selected frequencies within each band are spaced at approximately $15\,\rm kHz$ intervals. Comparing the amplitude across all frequency bands, it is evident that the higher amplification occurs for the ranges $30{-}137$ and $152{-}256\,\rm kHz$ shown in figures 14(a) and 14(b). Peak pressure amplitude is observed for $152.4\,\rm kHz$ at a very upstream region of the cone. For most frequencies, the highest amplitudes are observed in the vicinity of the nose of the cone, suggesting a dominant influence of the leading-edge shock, which induces high amplitude immediately downstream. An exception to this is for $195.6\,\rm kHz$ , which exhibits growth in the aft section of the cone, downstream of $X=9$ as indicated in figure 14(b). Higher frequencies, mostly above $256.6\,\rm kHz$ , display lower amplitude, gradually decaying in the streamwise direction. This is consistent with the power spectra shown earlier in figure 13(b) over the cone at $X=11.6$ where the higher frequencies ( ${\gt} 250\,\rm kHz$ ) remain unamplified.

Figure 14.

Surface pressure amplitude over the cone for various ranges of frequencies with an interval of ${\sim} 15.0\,\rm kHz$ : (a) $30.5{-}137.2\,\rm kHz$ , (b) $152.4{-}256.6\,\rm kHz$ , (c) $271.9{-}376.0\,\rm kHz$ and (b) $391.3{-}495.5\,\rm kHz$ .

The spatial distribution of the Fourier components of flow disturbances provides insights into the dominant regions of instability. Figure 15 illustrates the distribution of the real component of pressure perturbation over the cone for frequencies ranging from $106.7$ to $210.9\,\rm kHz$ . Enlarged views of flow structures over the surface near the aft portion of the cone, prior to the cylinder junction, are highlighted within black boxes on the right-hand side of the figure. At $106.7\,\rm kHz$ , prominent alternating positive–negative structures (APNS) are observed, extending from the surface to the oblique shock originating at the cone tip and persisting downstream. The emergence of these dominant vertical APNS in the region between the boundary layer and the shock is attributed to transient dynamics associated with the shock as it interacts with the forcing disturbances, and has been observed in past receptivity studies (Egorov, Fedorov & Soudakov Reference Egorov, Fedorov and Soudakov2008; Parsons et al. Reference Parsons, Zhong, Kim and Eldredge2010; Balakumar Reference Balakumar2015). It is noteworthy that acoustic waves at this frequency amplify strongly through the shock, with amplification diminishing with increasing frequency. A similar pattern is observed at $137.2\,\rm kHz$ , although the dominant APNS between the shock and the boundary layer is reduced and primarily confined to the upstream region. The strong shock characteristics have a significant impact on the flow stability mechanisms, altering the receptivity process (Balakumar et al. Reference Balakumar, Zhao and Atkins2005; Egorov et al. Reference Egorov, Fedorov and Soudakov2008). The shock wave acts as a nonlinear interface where a part of the incident wave (incoming disturbances) is reflected, depending on the shock strength and wave angle, and a part of it is transmitted, with modified spectral content such as amplitude, direction or wavelength. For some frequencies, the transmitted energy is focused and reinforced, showing amplification immediately downstream, while for others, the wave may be damped, scattered or deflected.

Figure 15.

Spatial distribution of the real component of pressure perturbation for the selected amplified frequencies over the cone. A magnified view of the rear part of the cone relevant to the boundary layer is shown in the insets on the right. Dashed black lines highlight the boundary-layer edge.

The flow near the surface within the developed boundary layer on the aft section of the cone is shown in the boxed regions on the right-hand side of figure 15, where the boundary-layer edge is marked by a black dashed line. At $106.7\,\rm kHz$ , distinct wall-normal elongated structures are observed that amplify over the surface and extend beyond the boundary-layer height. As the frequency increases to $137.2\,\rm kHz$ , the wall-normal extent of these structures diminishes, becoming predominantly confined within the boundary layer. At $152.4\,\rm kHz$ , the spatial distribution within the boundary layer exhibits first-mode rollers (Mack Reference Mack1984; Smith & Stewart Reference Smith and Stewart1987; Fedorov Reference Fedorov2003); this frequency is initially amplified near the cone tip due to transient shock dynamics observed in figure 14 and later evolves as a boundary-layer first mode in the aft portion of the cone. Under adiabatic wall conditions, the elevated surface temperature shifts the unstable frequency band towards lower frequencies, thereby reducing the stability of the first mode (Polivanov et al. Reference Polivanov, Gromyko, Sidorenko, Maslov, Keller, Groskopf and Kloker2011). Higher frequencies, specifically $195.6$ and $210.9\,\rm kHz$ , clearly exhibit the second-mode (Mack Reference Mack1984; Malik Reference Malik1990) features, characterised by forward-stretched two-lobe structures. For $195.6\,\rm kHz$ , the second-mode instability emerges downstream of $X=9$ and continues to grow along the length of the cone. Conversely, at $210.9\,\rm kHz$ , the onset of the second-mode behaviour occurs slightly upstream ( $X=8$ ), and stabilises towards the downstream end of the cone.

The profiles of perturbations associated with velocity and pressure elucidate the relevant instability characteristics linked to the boundary-layer modes. Figure 16 plots the streamwise velocity and pressure perturbation amplitudes in the wall-normal direction for several amplified frequencies at $X=10$ on the cone (location indicated by a green circle in figure 16a ). Here, $\eta$ denotes the distance measured in the wall-normal direction, with $\eta =0$ representing the model surface. Figures 16(b) and 16(c) depict the velocity $u$ and pressure amplitude, respectively, for frequencies ranging from $106.7$ to $226.1\,\rm kHz$ . The dotted lines show the perturbation eigenfunction calculated using local linear stability at $X=10$ for the respective frequency. It is observed that the linear stability prediction matches well with the DNS, except for the highest frequency, $226.1\,\rm kHz$ , where the second peaks in velocity and pressure profiles predicted by LST are much lower than those of the DNS calculation. Notably, the velocity fluctuations within the boundary layer are observed to exceed the amplitude of the incoming acoustic field, as the maximum amplitude within the boundary layer for $106.7{-}152\,\rm kHz$ surpasses disturbance group amplitude ( $A_0 = 1\times 10^{-3}$ ) at the inflow. However, the higher frequencies ( $165.2{-}226.1\,\rm kHz$ ) exhibit smaller velocity amplitude than the inflow disturbance amplitude (Balakumar & Kegerise Reference Balakumar and Kegerise2011). Higher amplitude indicates that the initial perturbation resonates well within the boundary layer, where the boundary layer acts as an amplifier to that mode. Such a disturbance extracts energy from the mean flow and amplifies, leading to higher amplitudes than at the inflow. Conversely, smaller instability amplitudes than at the inflow indicate poor coupling within the boundary layer, which attenuates the disturbances, leading to a weaker response.

Figure 16.

Region over the cone at $X=10$ indicated with a green circle in (a): (b) streamwise velocity and (c) pressure perturbation amplitude varying in the wall-normal direction for frequencies $106.7{-}226.1\,\rm kHz$ . Dotted lines indicate perturbations from the linear stability calculation.

The velocity perturbation is zero at the surface and reaches its peak value within the boundary layer. As the frequency increases, the wall-normal position of the maximum velocity amplitude shifts closer to the surface, a sign of increasing boundary-layer instability (Malik & Spall Reference Malik and Spall1991; Esfahanian Reference Esfahanian1991). The maximum of the pressure amplitude distribution is consistently located at the surface across all frequencies. Frequencies below $195.6\,\rm kHz$ display characteristics of the first mode, exhibiting a smooth decay profile in the wall-normal direction (Malik Reference Malik1987; Zhong & Ma Reference Zhong and Ma2006). In contrast, frequencies equal to or above $195.6\,\rm kHz$ initially demonstrate a gradual decay in the wall-normal direction, followed by a notable increase in amplitude prior to decay in the free stream. This amplitude jump is particularly pronounced at $226.1\,\rm kHz$ , which serves as a key indicator of the second mode (Mack Reference Mack1984). The observed variation in profile behaviour with increasing frequency aligns with the spatial distribution depicted in figure 15.

6.3. Instability over cylinder and flare

To elucidate the dominant acoustic behaviour within the separation bubble and reattachment zone, surface pressure amplitudes are analysed across the cylinder and flare, covering a frequency spectrum from low to high frequencies ( $30.5 {-} 256.6\,\rm kHz$ ). Figures 17(a) and 17(b) illustrate the amplitude variation along the streamwise direction for frequencies spanning $30.5{-}137.2$ and $152.4{-}256.6\,\rm kHz$ , respectively. The X-axis spans from the cone–cylinder junction to the flare end. Low frequencies demonstrate significant amplification within the separation bubble, with peak amplitude at $30.5\,\rm kHz$ . The amplification of low frequencies in the separation bubble aligns with experimental observations (Benitez et al. Reference Benitez, Borg, Scholten, Paredes, McDaniel and Jewell2023b ), and is attributed to shear-layer instabilities. Shear-generated travelling waves, originating from the shear layer, are characterised by these lower frequencies (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020). In the reattachment zone over the flare, $61.0\,\rm kHz$ emerges as the most dominant frequency, exhibiting notable growth. Additionally, higher frequencies within the range of $76.2{-}106.7\,\rm kHz$ , which are suppressed within the separation bubble, are amplified over the flare in the reattachment zone. The stabilising results are consistent with the flow expansion at the cone–cylinder juncture and attendant favourable pressure gradients. In contrast, the adverse pressure gradients experienced over the flare serve to amplify boundary-layer modes (Esquieu et al. Reference Esquieu, Benitez, Schneider and Brazier2019). High second-mode frequencies ( ${\geqslant} 195.6\,\rm kHz$ ) observed over the cone do not exhibit significant amplification within the separation bubble, as illustrated in figure 17(b). Within the separation bubble, the second mode becomes neutrally stable, with its fluctuations primarily confined to the outer regions of the shear layer, failing to substantially penetrate the bubble (Balakumar et al. Reference Balakumar, Zhao and Atkins2005).

Figure 17.

Pressure amplitude distribution along the surface of the cylinder and flare for various ranges of frequencies with an interval of ${\sim} 15.0\,\rm kHz$ : (a) $30.5{-}137.2\,\rm kHz$ and (b) $152.4{-}256.6\,\rm kHz$ . Dashed vertical line indicates the cylinder–flare corner.

Figure 18 illustrates the spatial distribution of the real component of the pressure perturbation for a range of amplified frequencies from $30.5$ to $106\,\rm kHz$ over the cylinder and flare. The dashed black line highlights the separated shear- and boundary-layer edge line. The pink- and black-bordered boxes on the right provide an enlarged view of the flow structures near the corner within the separation bubble and in the reattachment zone over the flare, respectively. At $30.5\,\rm kHz$ , the separation bubble exhibits larger flow structures associated with smaller streamwise wavenumbers, which progressively decrease in size as the frequency increases. Low-frequency perturbations dominate within the separation bubble (as seen in the pink boxes), but their intensity diminishes with increasing frequency. Based on the spatial distribution of these perturbations, it is difficult to distinctly separate mutually exclusive modes (e.g. shear-layer or boundary-layer modes), as they appear to coexist and influence each other. The irregular spatial distribution of disturbances suggests a strong sensitivity of the disturbance field to the rapidly varying mean flow near the corner. This behaviour is likely governed by local scattering and amplification mechanisms tied to the distorted basic state.

Figure 18.

Spatial distribution of real components of pressure perturbation over cylinder and flare encompassing separation bubble and reattachment region. Dashed black lines mark the location of the separated shear that reattaches over the flare, forming a boundary layer (BL). Pink and black boxes on the right show magnified views of the flow at the corner and in the reattachment zone.

In the reattachment region over the flare (black boxes), the interaction between shear-generated travelling waves and the boundary-layer modes, a phenomenon referred to as mode switching (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020), is clearly evident across the boundary-layer edge line, as indicated by distinct spatial patterns. The shear-generated waves, which originate in the separated shear layer upstream of reattachment, extend towards the wall and penetrate into the reattaching boundary layer. As the boundary layer reforms along the flare surface, these incoming shear-layer disturbances interact with the intrinsic boundary-layer modes, depending on the local amplification and frequency. This interaction results in an exchange of energy between the outer (shear-driven) and inner (wall-bounded) instability modes, leading to a reorganisation of the disturbance field. The outcome is a spatial modulation in the structure of the waveforms, characterised by a shift in dominance from one mode to another, which manifests as mode switching. This is evident, for example, at $45.7\,\rm kHz$ , where the smooth positive–negative patterns within the boundary layer transition to more oscillatory structures above it, particularly across the boundary-layer edge line. This distinction is noticeable across all the frequencies shown. At lower frequencies ( ${\leqslant} 61.0\,\rm kHz$ ), the amplified interaction between shear-generated and boundary-layer waves results in an identical positive–negative distribution across the boundary-layer edge line, making it difficult to clearly and exclusively identify boundary-layer modes. However, as the frequency increases to $91.5\,\rm kHz$ and above, boundary-layer modes become more evident, characterised by well-defined roller-like structures, similar to those observed over a cone, while the shear-generated waves on top of it diminish.

Velocity and pressure perturbations are analysed across various frequencies to evaluate the instability characteristics within the separation bubble. Figure 19 illustrates the variation of streamwise velocity and pressure amplitudes in the wall-normal direction at a streamwise location of $X=14$ , situated within the separation bubble over the cylinder, as indicated by the green circle in figure 19(a). Each line corresponds to a specific frequency, indicated by the legend on the right-hand side of the figure. A comparison of the velocity and pressure perturbation amplitudes over the cylinder with those observed over the cone (as in figure 16) reveals that none of the amplified frequencies over the cylinder display characteristics typical of the boundary-layer modes where perturbations have decayed smoothly away from the wall. The peak velocity amplitude occurs at a distinct distance from the wall, which progressively shifts closer to the surface as frequency increases to $76.2\,\rm kHz$ . The overall amplitude of $u'$ is notably higher for lower frequencies, with a significant reduction beyond $106.7\,\rm kHz$ , accompanied by more peaks. The pressure perturbation also reveals multiple peaks with increasing frequency, which may be attributed to the emergence of finer structures within the separation bubble, as observed earlier in figure 18. These flow structures possess an inherent capacity to amplify even minor disturbances into large-scale, three-dimensional structures, ultimately leading to global instability (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021; Paredes et al. Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022), thereby influencing the reattachment process. Notably, in contrast to the maximum pressure amplitude observed at the surface for boundary-layer modes over the cone (refer to figure 16c ), the maximum pressure amplitude over the cylinder occurs at locations away from the wall, particularly for frequencies exceeding $61\,\rm kHz$ . In the separation bubble, neither of these perturbation profiles exhibits purely boundary-layer-mode (Malik Reference Malik1990) or purely shear-layer-mode (Macaraeg & Streett Reference Macaraeg and Streett1991) characteristics, indicating more complex behaviour across different scales that led to the appearance of sporadic structures.

Figure 19.

In the separation bubble over the cylinder at $X=14$ indicated with a green circle in (a): (b) streamwise velocity and (c) pressure perturbation amplitude for frequencies $30.5{-}152.4\,\rm kHz$ .

The velocity and pressure amplitude distributions in the wall-normal direction within the reattachment zone over the flare at $X=18$ are depicted in figure 20. With increasing frequency, the amplitude profiles resemble characteristics of smooth boundary-layer modes, notably displaying first-mode behaviour at $91.5\,\rm kHz$ , which is visually analogous to that for $137.2\,\rm kHz$ observed over the cone, as illustrated in figures 16(b) and 16(c). The smooth perturbation profile adjacent to the wall, combined with oscillatory behaviour further from the wall, in particular for $30.5{-}45.7\,\rm kHz$ and ${\geqslant} 122\,\rm kHz$ , indicates interactions between the boundary-layer modes and shear-layer instability as observed in experiment (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020). This is evident in figure 18 discussed earlier, where boundary-layer rollers are clearly visible and remain confined below the reattachment line, while a distinct pattern associated with the shear layer is observed over the reattachment line. The maximum pressure amplitude is recorded at the surface for most frequencies; however, at $122.0$ and $152.0\,\rm kHz$ , notable peaks occur around $\eta = 0.07$ , indicating a dominant influence of shear-layer modes on the pressure fluctuations at these frequencies.

Figure 20.

Results over the flare at $X=18$ indicated with a green circle in (a): (b) streamwise velocity and (c) pressure perturbation amplitude for frequencies $30.5{-}152.4\,\rm kHz$ in the reattachment zone.

7.1. Phase-locked triadic interaction

The Fourier representation of discrete-time signals, as given by (4.6), yields a complex number output that holds amplitude and phase details. The signatures of physical processes are contained in the transient signal spectrum, which may be processed in different ways to highlight mechanisms of interest (Bendat & Piersol Reference Bendat and Piersol2011). The power spectrum used earlier is the Fourier transform of the autocorrelation ( $E[\boldsymbol{\cdot }]$ ) with the unit of fluctuation quantity squared per hertz (Welch Reference Welch1967), given as

(7.1) \begin{align} P(f_1) = \lim _{T\to \infty }\frac {1}{T}E[\boldsymbol{R}'(f_1)\boldsymbol{R}^{\prime *}(f_1)]= \int _{-\infty }^{+\infty } S_{qq}{\rm e}^{-2{\rm i}\pi f_1 \tau } \rm d \tau , \end{align}

where $S_{qq} = E[\boldsymbol{q}'(t)\boldsymbol{q}'(t-\tau )]$ , $\boldsymbol{q}'(t)$ is the fluctuating time signal, $\tau$ is the autocorrelation time delay, $\boldsymbol{R}'(f_1)$ is the Fourier transform of signal $\boldsymbol{q}'(t)$ at frequency $f_1$ and the asterisk represents the complex conjugate. The two-dimensional Fourier transform of the second-order autocorrelation is the bispectrum with the unit of fluctuation quantity cubed per hertz squared, given as

Here

(7.2) \begin{align} b(f_1,f_2) &= \lim _{T\to \infty }\frac {1}{T}E[\boldsymbol{R}'(f_1)\boldsymbol{R}'(f_2)\boldsymbol{R}^{\prime *}(f_1+f_2)] \end{align}

(7.3) \begin{align} & =\int _{-\infty }^{+\infty } \int _{-\infty }^{+\infty }S_{qqq}{\rm e}^{-2{\rm i}\pi (f_1 \tau _1 +f_2 \tau _2)}\, {\rm d} \tau _1 \,{\rm d} \tau _2. \end{align}

$S_{qqq}$ is the second-order autocorrelation as $ S_{qqq} = E[\boldsymbol{q}(t)\boldsymbol{q}'(t-\tau _1) \boldsymbol{q}'(t-\tau _2)]$ , $\tau _1$ and $\tau _2$ are time delays, while $f_1$ and $f_2$ are their associated frequencies. In contrast to the power spectrum, which does not retain phase information, the bispectrum accounts for phase details and facilitates the identification of the phase-coupled waves (Kim & Powers Reference Kim and Powers1979; Nikias & Raghuveer Reference Nikias and Raghuveer1987). The value of the bispectrum depends on wave amplitude and hence is usually presented as a normalised quantity, the bicoherence, given as

(7.4) \begin{align} B^2(f_1,f_2)= \frac {|b(f_1,f_2)|^2}{E[|\boldsymbol{R}'(f_1)\boldsymbol{R}'(f_2)|^2] E[|\boldsymbol{R}'(f_1+f_2)|^2]}. \end{align}

The degree of quadratic phase coupling of waves with frequencies $f_1$ , $f_2$ and $f_1+f_2$ is identified through the bicoherence amplitude; with the above normalisation, its value is bounded between 0 and 1. Bicoherence $|B^2|=0$ represents completely independent linear waves while $|B^2|=1$ indicates completely coupled nonlinear waves (Kim & Powers Reference Kim and Powers1979). Hence, when $|B^2|\gt 0$ for the frequency components $f_1$ and $f_2$ , the triad of the waves with frequencies $f_1$ , $f_2$ and $f_1+f_2$ are phase-coupled. If waves are independent and randomly excited, their phase will also be random, independent and uniformly distributed over $[0, 2\pi ]$ . If any of the three waves is excited by a nonlinear interaction, its phase will be dependent and non-uniformly distributed with a non-zero bicoherence value (Kimmel & Kendall Reference Kimmel and Kendall1991).

Bicoherence values are calculated using (7.4) for the pressure signal, where the range is bounded between $0$ and $1$ , representing independently excited waves and a high degree of phase-locked triadic interaction, respectively. Figure 21 illustrates the bicoherence distribution over the cone, centred around a second-mode frequency of $195.6\,\rm kHz$ , as identified in figure 15, along with its subharmonic and higher harmonics. The primary disturbance frequency ( $195.6\,\rm kHz$ ) is referred to as the fundamental ( $f_0$ ), while $f_0/2$ , $2f_0$ and $3f_0$ indicate the subharmonic, first harmonic and second harmonic, respectively (Chokani Reference Chokani1999). Four triadic interaction scenarios are observed involving the fundamental second-mode wave ( $f_0 = 195.6\,\rm kHz$ ), namely subharmonic–fundamental ( $f_0/2, f_0$ ), fundamental–fundamental ( $f_0, f_0$ ), first harmonic–fundamental ( $2f_0, f_0$ ) and second harmonic–fundamental ( $3f_0, f_0$ ). The spatial oscillations inherent in the computed pressure signal, caused by unsteady disturbances, are reflected in the bicoherence values in the form of streamwise oscillations. As depicted in figure 21(a), the subharmonic–fundamental ( $f_0/2, f_0$ ) interaction shows weak phase coupling, with minimal bicoherence amplitude among the four interactions considered. Although a slight amplification is observed near the nosetip, likely influenced by shock-related dynamics, the bicoherence remains near zero through most of the downstream region. This indicates that the components of the triad ( $f_0/2$ , $f_0$ and $f_0/2 + f_0$ ) behave largely as independent waves, with negligible contributions towards nonlinear amplification. Consequently, the results suggest that subharmonic resonance plays a limited role in driving the fundamental nonlinear processes in the cone region, similar to that observed in a past study over a sharp cone by Bountin et al. (Reference Bountin, Shiplyuk and Maslov2008).

Figure 21.

Bicoherence amplitude over the cone pertinent to fundamental second-mode wave $f_0 = 195.6\,\rm kHz$ . (a) Subharmonic–fundamental ( $f_0/2,f_0$ ), (b) fundamental–fundamental ( $f_0,f_0$ ), (c) first harmonic–fundamental ( $2f_0,f_0$ ) and (d) second harmonic–fundamental ( $3f_0,f_0$ ).

Wave resonance plays a key role in amplifying disturbances and facilitating destabilisation, serving as a catalyst for phase-locked interactions that can initiate secondary instabilities (Wu & Stewart Reference Wu and Stewart1996). In figure 21(b), significant phase-locked nonlinear interactions are observed for self-interacting fundamental second-mode waves ( $f_0, f_0$ ) near $X=3$ , with the coupling intensity diminishing downstream. The early onset of these interactions can broaden the spectral sidebands of the involved Mack mode, enabling energy transfer among different wave components. In a past study (Chokani Reference Chokani2005), such mechanisms were associated with the growth of secondary disturbances, thereby accelerating the transition process. However, in the current case, the early interactions dissipate by the rear end of the cone, suggesting that the generation of secondary waves is unlikely. For higher-harmonic interactions, such as ( $2f_0, f_0$ ) and ( $3f_0, f_0$ ), dominant amplitude modulation is evident around $X=4$ and towards the rear of the cone, as shown in figure 21(c,d). The high bicoherence and unsteadiness associated with higher harmonics near the rear portion of the cone can lead to spanwise variations in the flow prior to separation over the cylinder, as observed in the case of a compression ramp (Cao et al. Reference Cao, Hao, Klioutchnikov, Wen, Olivier and Heufer2022).

The variation in bicoherence over the cylinder and flare, representing interactions between the fundamental frequency and its harmonics, is illustrated in figure 22. The green circle on the horizontal axis denotes the location of the cylinder–flare junction. As indicated by the surface pressure amplitude distribution in figure 17(a), a range of amplified frequencies ( $30.5{-}137.2\,\rm kHz$ ) is observed over the cylinder and flare. However, these frequencies do not exhibit characteristics typical of boundary-layer modes, except for a few frequencies ( $91.5$ and $106.7\,\rm kHz$ ) showing resemblance to the first mode in reattached flow over the aft section of the flare (in figure 18), albeit with lower pressure amplitude than that for $61.0\,\rm kHz$ , as seen in figure 17. Hence, to assess the nonlinear contributions, the most amplified frequency ( $61.0\,\rm kHz$ ) is chosen as the fundamental frequency (Miksad Reference Miksad1973; Chokani Reference Chokani1999), $f_0$ , and its interactions with harmonics are presented.

Figure 22.

Bicoherence amplitude over cylinder and flare concerning the most amplified frequency $f_0=61.0\,\rm kHz$ in figure 17. (a) Subharmonic–fundamental ( $f_0/2,f_0$ ), (b) fundamental–fundamental ( $f_0,f_0$ ), (c) first harmonic–fundamental ( $2f_0,f_0$ ) and (d) second harmonic–fundamental ( $3f_0,f_0$ ). Green circles along the X-axis indicate the cylinder–flare junction.

In figure 22, subharmonic–fundamental ( $f_0/2, f_0$ ) and fundamental–fundamental ( $f_0, f_0$ ) interactions exhibit prominent phase-locked nonlinear interactions in the rear part of the separation bubble, downstream of the corner, over the flare. Physically, such triadic interactions indicate that energy from the dominant fundamental disturbance ( $f_0 = 61.0\,\rm kHz$ ), which gains strength over the flare as observed earlier in figure 17, is redistributed to other frequencies through nonlinear processes. In figure 22(c,d), interactions involving higher harmonics ( $2f_0, f_0$ ; $3f_0, f_0$ ) exhibit elevated bicoherence near the mid-separation region, particularly between $X = 15$ and $16.5$ , close to the cylinder–flare junction. This region marks a zone of high shear and inflectional instability, where separated shear layers are prone to nonlinear amplification. The strong coupling between the fundamental and its harmonics indicates that nonlinear mechanisms are actively transferring energy to shorter wavelengths, intensifying flow gradients that can promote localised breakdown. Separated flows are highly susceptible to these interactions, and the amplification of higher harmonics can accelerate mixing, facilitating energy transfer across different flow scales and intensifying chaotic motion (Goldstein & Lee Reference Goldstein and Lee1992; Wu Reference Wu1995).

7.2. Multidimensional phase-locked interaction

Bispectral mode decomposition proposed by Schmidt (Reference Schmidt2020) extends the above typical bispectral approach, using higher-order spectra, to multidimensional data. The classical bispectral technique measures the extent of quadratic phase coupling locally for given transient point data (Nikias & Raghuveer Reference Nikias and Raghuveer1987; Mendel Reference Mendel1991). In contrast, the bispectral decomposition provides spatially coherent structures associated with triadic interaction among $f_1$ , $f_2$ and $f_1+f_2$ . For the sake of completeness and consistency with the above discussion, mathematical details of the BMD are presented here briefly. Considering spatio-temporal fluctuating flow field $ \boldsymbol{q}'_i = \boldsymbol{q}'(\boldsymbol{s}, t_i)$ , where $\boldsymbol{s}$ is space and $i$ represents the index of time instances, the bispectral mode is defined as an integral measure of bispectral density over spatial domain $\varOmega$ , mathematically

(7.5) \begin{align} b(f_1,f_2) = E\!\left [\int _{\varOmega }\left (\boldsymbol{R}'(\boldsymbol{s},f_1)\boldsymbol{\cdot }\boldsymbol{R}'(\boldsymbol{s},f_2)\boldsymbol{\cdot }\boldsymbol{R}^{\prime *}(\boldsymbol{s},f_1+f_2)\right ){\rm d}\boldsymbol{s}\right ]\!. \end{align}

Here $\boldsymbol{R}'(\boldsymbol{s},f_1)$ is a temporal Fourier transform of $\boldsymbol{q}'(\boldsymbol{s},t)$ at frequency $f_1$ and ‘ $\boldsymbol{\cdot}$ ’ represents an element-wise dot product. One can contrast the BMD with the classical bispectrum formulation given by (7.2) that uses a single-point Fourier transform $\boldsymbol{R}'(f_1)$ . In the case of resonance among triad members, the sum-frequency component $\boldsymbol{R}'(\boldsymbol{s},f_1+f_2)$ represents the effect caused by the product of the other two components $\boldsymbol{R}'(\boldsymbol{s},f_1)$ and $\boldsymbol{R}'(\boldsymbol{s},f_2)$ . Arranging the spatio-temporal Fourier realisation in the form of matrices:

(7.6) \begin{align} \boldsymbol{Q}_1 = [\boldsymbol{R}^{\prime (1)}_{1}, \boldsymbol{R}^{\prime (2)}_{1},\ldots , \boldsymbol{R}^{\prime (n_{\textit{blk}})}_{1}], \end{align}

(7.7) \begin{align} \boldsymbol{Q}_2 = [\boldsymbol{R}^{\prime (1)}_{2}, \boldsymbol{R}^{\prime (2)}_{2},\ldots , \boldsymbol{R}^{\prime (n_{\textit{blk}})}_{2}], \end{align}

(7.8) \begin{align} \boldsymbol{Q}_{1+2} = [\boldsymbol{R}^{\prime (1)}_{1+2}, \boldsymbol{R}^{\prime (2)}_{1+2},\ldots , \boldsymbol{R}^{\prime (n_{\textit{blk}})}_{1+2}], \end{align}

where $\boldsymbol{R}'_{1} = \boldsymbol{R}'(\boldsymbol{s},f_1)$ , $\boldsymbol{R}'_{2} = \boldsymbol{R}'(\boldsymbol{s},f_2)$ and $\boldsymbol{R}'_{1+2} = \boldsymbol{R}'(\boldsymbol{s},f_1+f_2)$ , $n_{\textit{blk}}$ is the number of blocks computed by rounding off the expression $(n_t -n_{ov})/(n_{fft}-n_{ov})$ , $n_{t}$ is the number of transient instances, $n_{fft}$ is the number of output from Fourier transform and $n_{ov}$ is the number of overlap between two consecutive blocks. The sum-frequency component is associated with the bispectral mode, while the product of two interacting components provides a cross-frequency field given as

(7.9) \begin{align} \boldsymbol{\phi }_{1+2} &= \boldsymbol{Q}_{1+2}\boldsymbol{a}, \end{align}

(7.10) \begin{align} \boldsymbol{\phi }_{1\boldsymbol{\cdot }2} &= \boldsymbol{Q}_{1\boldsymbol{\cdot }2}\boldsymbol{a}, \end{align}

respectively, where $\boldsymbol{Q}_{1\boldsymbol{\cdot }2}= \boldsymbol{Q}^*_1 \boldsymbol{\cdot }\boldsymbol{Q}^*_2$ and $\boldsymbol{a}$ is an optimal expansion coefficient that ensure the maximisation of $b(f_1,f_2)$ . Analogous to the bispectrum for time series in the classical approach, the mode spectrum for spatio-temporal data in BMD is defined as

(7.11) \begin{align} \lambda (f_1,f_2) = \left\lvert \frac {\boldsymbol{a}^H\boldsymbol{B}\boldsymbol{a}}{\boldsymbol{a}^H \boldsymbol{a}}\right\rvert \!. \end{align}

Superscript ‘ $H$ ’ indicates complex transpose and $\boldsymbol{B}$ is the weighted bispectral density given as

(7.12) \begin{align} \boldsymbol{B} = \frac {1}{n_{\textit{blk}}}\boldsymbol{Q}_{1\boldsymbol{\cdot }2}^H \boldsymbol{W}\boldsymbol{Q}_{1+2}, \end{align}

where $\boldsymbol{W}$ is a diagonal weight matrix. The interaction map that quantifies the region of triadic interaction ( $f_1$ , $f_2$ and $f_1+f_2$ ) is given by the product of (7.9) and (7.10) as

(7.13) \begin{align} \boldsymbol{\psi }_{\boldsymbol{s},f_1,f_2} = \lvert { \boldsymbol{\phi }_{1\boldsymbol{\cdot }2}\boldsymbol{\cdot }\boldsymbol{\phi }_{1+2} \rvert }. \end{align}

7.2.1. Multidimensional phase-locked Iinteraction: cone

The mode bispectrum ( $\rm log|\lambda |$ ) for flow over the cone, given by expression (7.11), is shown in figure 23. The upper triangular region defined by the vertices ( $0,0$ ), ( $f_N,0$ ), ( $f_N/2,f_N/2$ ) indicates sum interactions while the lower triangular region defined by the vertices ( $0,0$ ), ( $f_N,0$ ), ( $f_N,{-}f_N$ ) indicates difference interactions (Kim & Powers Reference Kim and Powers1979; Nikias & Raghuveer Reference Nikias and Raghuveer1987; Schmidt Reference Schmidt2020). Here, $f_N=760\,\rm kHz$ is the Nyquist frequency, which is half the sampling rate of the transient dataset (Oppenheim & Schafer Reference Oppenheim and Schafer1975). Spectral peaks indicative of nonlinearity are predominantly observed at low frequencies ( ${{\leqslant}} 350\,\rm kHz$ ), and figure 23(b) presents the magnification of the low-frequency section. To extract the spatial bispectral mode structures and their interaction maps, five spectral combinations are identified corresponding to the peak bicoherence in figure 23(b), marked as ‘ $+$ ’ and named as: bispectral mode (BM) 1, BM2, BM3, BM4 and BM5. Mode BM1 ( $f_1$ , $f_2$ , $f_1+f_2$ $\rightarrow 156, 150, 306\,\rm kHz$ ) and BM2 ( $138, 18, 156\,\rm kHz$ ) are in the upper half of the figure representing the sum interaction while BM3 ( $156, -12, 144\,\rm kHz$ ), BM4 ( $156, -138, 18\,\rm kHz$ ) and BM5 ( $306, -150, 156\,\rm kHz$ ) correspond to the difference interaction.

Figure 23.

(a) Mode bispectrum for triadic nonlinear interaction over the cone exhibiting sum and difference interactions in triangular regions bounded by [( $0,0$ ), ( $f_N,0$ ), ( $f_N/2,f_N/2$ )] and [( $0,0$ ), ( $f_N,0$ ), ( $f_N,-f_N$ )], respectively. Frequency $f_N$ is a Nyquist frequency. (b) An enlarged view of the low-frequency region indicates high nonlinear coupling. The amplified interactions are marked with green ellipses with ‘+’ and named sequentially as bispectral modes 1, 2, 3, 4 and 5, indicating sum and difference interactions. (c) Mode bispectrum for forcing amplitude $A_0 = 1\times 10^{-2}$ .

Notably, all identified bispectral modes linked to peak nonlinearity involve a frequency of $156\,\rm kHz$ in their triadic interactions, whether as a primary frequency ( $f_1,f_2$ ) or generated as a result of sum ( $f_1+f_2$ ) or difference ( $f_1-f_2$ ) interactions. The nearest frequency ( $152.4\,\rm kHz$ ) to $156\,\rm kHz$ displays first-mode characteristics in its spatial distribution and eigenfunction plot as shown earlier in figures 15 and 16. Consequently, the presence of $156\,\rm kHz$ across all peak triadic interactions (BM1–BM5) underscores the significant contribution of the first mode to nonlinearity. While the importance of the first mode, primarily as an oblique wave, in secondary mechanisms is well established (Smith & Stewart Reference Smith and Stewart1987; Hader & Fasel Reference Hader and Fasel2019), its role in nonlinear triadic interactions remains both unforeseen and intriguing, but nonetheless related to the conscious choice of adiabatic wall boundary condition, which enhances first-mode implications.

An additional case, where the inflow forcing amplitude ( $A_0$ ) is increased to $1\times 10^{-2}$ , is investigated to observe the amplitude effect on triadic interaction. Figure 23(c) presents the bicoherence distribution for a forcing amplitude of $1\times 10^{-2}$ . Amplification over broadband spectra is observed, where strong triadic interactions exist at frequencies lower than $120\,\rm kHz$ , highlighted using a green circle. Unlike low-amplitude forcing ( $1\times 10^{-3}$ ), where the distinct peaks are evident in figure 23(b), the increased amplitude leads to the disappearance of tonal peaks and exhibits broadband spectra. Triadic nonlinear interactions are selective and resonate only among compatible triads, i.e. satisfying $\omega _{1} \pm \omega _{2} = \omega _3$ . In the weakly nonlinear regime, only a few modes are excited significantly, which show up as sharp peaks in the bispectrum, since energy is being transferred through specific, efficient triadic pathways. As forcing increases, more modes across a wider range of frequencies get excited. This leads to several possible triadic interactions, spreading energy over a larger range of spectral combinations.

The spatial structures of the bispectral modes and corresponding interaction maps for the sum interactions BM1 and BM2 are presented in figures 24(a) and 24(b), respectively. For BM1, the bispectral mode exhibits predominant activity near the nosetip, with spatially evolving structures extending primarily upstream of $X=2$ , along the oblique shock and within the boundary layer, before dissipating downstream. The interaction map in figure 24(a) reveals that the associated triadic interaction ( $156, 150, 306\,\rm kHz$ ) is strongest near the nosetip and diminishes within the boundary layer. Strong nonlinearity at the nosetip results from the receptivity process where incoming disturbances interact with the shock, causing initial amplification and retaining its signature in the nearby developing boundary layer (Zhong & Ma Reference Zhong and Ma2006; Parsons et al. Reference Parsons, Zhong, Kim and Eldredge2010; Lei Reference Lei2013). In contrast, BM2, as shown in figure 24(b), displays spatially developing structures within the boundary layer that persist up to $X=8$ , continuously diminishing downstream, in addition to being present near the nosetip. The surface patterns for BM2 resemble those observed for the first mode at $152.4\,\rm kHz$ over the cone in figure 15, indicating that the structures arise from nonlinear sum interactions. Sum interactions facilitate energy transfer to higher frequencies, potentially corresponding to mechanisms such as harmonic generation (Kim & Powers Reference Kim and Powers1979; Hajj, Miksad & Powers Reference Hajj, Miksad and Powers1997). The interaction map for BM2 shows slightly enhanced intensity along the boundary layer compared with BM1, highlighting the dominance of nonlinear interactions associated with the sum triad ( $138, 18, 156\,\rm kHz$ ). This dominance is consistent with the contribution to the high amplitude of the $152.4\,\rm kHz$ component in figure 14 (closest to $156\,\rm kHz$ ) and suggests a significant role in the overall energy redistribution.

Figure 24.

Spatial distribution of bispectral mode and triadic interaction map indicating the region of nonlinearity for sum interactions. (a) BM1 ( $156, 150, 306\,\rm kHz$ ) and (b) BM2 ( $138, 18, 156\,\rm kHz$ ) are marked in figure 23.

The bispectral modes and interaction maps corresponding to the difference interactions BM3, BM4 and BM5 are presented in figures 25(a), 25(b) and 25(c), respectively. The bispectral mode distributions for BM3 and BM5 are similar to those of the sum interaction BM2, with surface lobe structures predominantly confined to the boundary layer. Likewise, the spatial distribution of the sum-generated $156\,\rm kHz$ (as in figure 24b ) closely resembles the difference-generated $156\,\rm kHz$ (shown in figure 25c ), underscoring the influence of both low ( $18\,\rm kHz$ ) and high ( $306\,\rm kHz$ ) frequencies in driving the amplification of the $152.4\,\rm kHz$ component (closest to $156\,\rm kHz$ ) over the cone, as illustrated in figure 14. Consistent with the sum interactions, the difference interaction maps indicate the leading shock and boundary-layer regions as the most susceptible to nonlinear interaction. Unlike sum interactions, however, difference interactions typically represent energy redistribution from higher to lower frequencies (Kim & Powers Reference Kim and Powers1979). A distinctive feature of BM4 is the presence of large-scale blob patterns that trace the path of the leading oblique shock; this behaviour is not observed in other sum or difference interactions. Despite this distinct pattern, the interaction map remains largely unchanged, with dominant regions confined to the vicinity of the tip and upstream boundary layer.

Figure 25.

Spatial distribution of bispectral modes and triadic interaction map indicating the region of nonlinearity for difference interactions. Modes (a) BM3 ( $156, -12, 144\,\rm kHz$ ), (b) BM4 ( $156, -138, 18\,\rm kHz$ ) and (c) BM5 ( $306, -150, 156\,\rm kHz$ ) are marked in figure 23.

Since the increased forcing amplitude did not exhibit any distinct peak, sum interaction BM2 ( $138, 18, 156\,\rm kHz$ ) is chosen to assess the spatial distribution and evaluate the effect of high forcing amplitude ( $1\times 10^{-2}$ ) by comparing with BM2 for low forcing amplitude. Figure 26 shows the bispectral mode and interaction intensity distribution for the triad $138, 18, 156\,\rm kHz$ over the cone for forcing amplitude $1\times 10^{-2}$ . Comparing BM2 for forcing amplitude $1\times 10^{-3}$ shown earlier in figure 24(b) with figure 26, it becomes evident that increased inflow amplitude has led to higher amplification within the shock layer. The incoming wave is transmitted through the leading shock, amplified as visible from the increased colour intensity, and extends to the surface of the cone. The developing boundary-layer rollers merge with the amplification coming through the shock. In addition to nosetip nonlinearity as observed for low-amplitude forcing in the interaction map in figure 24(b), the increased amplitude leads to the formation of localised hotspots within the shock layer, shown in the interaction map in figure 26. The sharp gradients in flow variables within the shock layer steepen wavefronts and can convert linear disturbances into nonlinear ones, especially at high amplitudes where the shock layer behaves as a natural amplifier. The presence of localised hotspots in the interaction map signifies strong phase-locked triadic interactions at that specific location (Schmidt Reference Schmidt2020). The concentrated nonlinear energy transfer promotes local mode coupling, leading to enhanced wave–wave interaction.

Figure 26.

Spatial distribution of bispectral mode and triadic interaction map for forcing amplitude $A_0 = 1\times 10^{-2}$ for sum interaction BM2 ( $138, 18, 156\,\rm kHz$ ).

7.2.2. Multidimensional phase-locked interaction: cylinder–flare

To quantify the nonlinear interactions within the separation bubble, figure 27(a) presents the mode bispectrum for the region over the cylinder and flare. The triangular regions, defined by the vertices [( $0,0$ ), ( $f_N,0$ ), ( $f_N/2,f_N/2$ )] and [( $0,0$ ), ( $f_N,0$ ), ( $f_N,-f_N$ )], correspond to sum and difference interactions, respectively. A magnified view focusing on the low-frequency range, where the majority of high-magnitude interactions occur, is shown in figure 27(b). The surface pressure spectra presented in figures 14 and 17 reveal a pronounced amplification in the low-frequency range for the cylinder–flare compared with the cone (Esquieu et al. Reference Esquieu, Benitez, Schneider and Brazier2019; Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020). This trend is consistently reflected in the mode bispectrum, where the triadic interactions over the cylinder–flare exhibit significant amplification at lower frequencies ( ${\leqslant} 120\,\rm kHz$ ) relative to those over the cone ( $\leq 350\,\rm kHz$ ). It underscores the critical role of low-frequency waves in driving nonlinear dynamics within the separation bubble and influencing the reattachment process. Unlike the mode spectrum over the cone, shown in figure 23, where distinct peaks are clearly identifiable, the mode spectrum over the cylinder–flare is comparatively more diffuse, as illustrated in figure 27(b). To identify the dominant triadic interactions, constant frequency lines were selected based on the amplified patterns observed along the frequencies $f_1$ and $f_2$ . Figure 27(c) presents the amplitude variation with $f_1$ for a constant $f_2$ value ( $f_2 = -60, -18, 18\,\rm kHz$ ), indicated by black horizontal arrow lines in figure 27(b). The maximum amplitude is associated with a sum interaction involving the triad $f_1$ , $f_2$ , $f_1+f_2$ $\rightarrow 42, 18, 60\,\rm kHz$ , denoted as BM6 in figure 27(b,c). Similarly, for constant frequency $f_1$ ( $f_1=60,96\,\rm kHz$ ), marked by blue vertical arrow lines in figure 27(b), figure 27(d) shows the amplitude variation with frequency $f_2$ . Two prominent peak interactions are identified as BM7 ( $96, -66, 30\,\rm kHz$ ) and BM8 ( $60, 36, 96\,\rm kHz$ ), corresponding to difference and sum triadic interactions, respectively.

Figure 27.

(a) Mode bispectrum for triadic nonlinear interaction over the cylinder–flare exhibiting sum and difference interactions. (b) An enlarged view of the low-frequency region indicates high nonlinear coupling. The amplified interaction is identified through the line plot in (c) for $f_2 = -60,-18,18\,\rm kHz$ and in (d) for $f_1=60,96\,\rm kHz$ . The peaks are indicated as ‘+’ and marked in (b) as BM6, BM7 and BM8. (e) Mode bispectrum for forcing amplitude $A_0 = 1\times 10^{-2}$ .

In the case of increased forcing amplitude of $A_0 = 1\times 10^{-2}$ , the line-like pattern observed for forcing $A_0 = 1\times 10^{-3}$ in figure 27(b) has disappeared as seen in figure 27(e). The amplified frequency bands, representing phase-locked triadic interaction, are found within a range similar to that for the low forcing amplitude, mainly confined below $120\,\rm kHz$ . However, looking at the colourmap range, it is observed that the bicoherence intensity has increased significantly in the case of higher-amplitude forcing, indicative of stronger triadic interaction.

The spatial distribution of the bispectrum modes and interaction maps are illustrated in figure 28 for the cylinder–flare, focusing on the identified peak nonlinear triadic interactions (BM6, BM7, BM8) as shown in figure 27. BM6, displayed in figure 28(a), represents a sum interaction among frequencies $42$ and $18\,\rm kHz$ , yielding $60\,\rm kHz$ . This interaction displays flow structures analogous to those observed at $61\,\rm kHz$ (shown in figure 18), which is identified as the most amplified wave in figure 17. The growth of the most amplified wave ( $61\,\rm kHz$ ) within the separation bubble is likely attributed to the nonlinear triadic interaction among the $42$ and $18\,\rm kHz$ frequencies. However, the distinction between shear-generated and boundary-layer modes within the separation bubble proves challenging due to the dominance of low frequencies and their complex interactions. Both shear-generated travelling waves and boundary-layer modes within the separation bubble exhibit low-frequency characteristics (Balakumar et al. Reference Balakumar, Zhao and Atkins2005; Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2020). Furthermore, Balakumar et al. (Reference Balakumar, Zhao and Atkins2005) demonstrated that boundary-layer modes remain neutrally stable within the separation bubble.

Figure 28.

Spatial distribution of bispectral mode and triadic interaction map indicating the region of nonlinearity for sum interactions BM6 ( $42, 18, 60\,\rm kHz$ ) and BM8 ( $60, 36, 96\,\rm kHz$ ) (a,c) and difference interaction BM7 ( $96,-66,30\,\rm kHz$ ) (b). Dashed black and green lines highlight top edge of the shear layer that reattaches over the flare.

The interaction map for BM6 indicates significant magnitude within the separation bubble, originating downstream of $X=14$ and becoming pronounced in the reattachment zone. Here, prominent wall-normal patterns are evident in the bispectrum mode, extending beyond the reattached boundary layer. The spatial distribution of $30.5\,\rm kHz$ , as in figure 18, is reflected in difference triadic interactions involving frequencies $96$ and $-66\,\rm kHz$ , illustrated in figure 28(b). This frequency is enhanced beyond the separation bubble, particularly above the shear layer, and shows greater amplification compared with the sum interaction BM6, thereby playing a more significant role in shear-dominated instability. Regarding interaction intensity, nonlinearity for BM7 initiates earlier within the separation bubble compared with BM6. However, in the reattachment zone, the difference interaction associated with $30\,\rm kHz$ (shown in figure 28b ) diminishes relative to the sum interaction involving $60\,\rm kHz$ (illustrated in figure 28a ). This observation aligns with the surface amplitude distribution in figure 17, where the amplitude at $30.5\,\rm kHz$ decreases, while that at $61\,\rm kHz$ increases within the reattachment zone over the flare. The early streamwise onset of nonlinearity associated with $30\,\rm kHz$ further underscores its contribution to the triadic interactions that result in its dominance in the separation bubble over the cylinder, as indicated in figure 17.

The bispectrum mode associated with the triadic sum interaction BM8 ( $60, 36, 96\,\rm kHz$ ) is presented in figure 28(c). Notably, finer structures are discernible within the separation bubble compared with BM6 and BM7, which develop into multilayered rollers over the flare beneath the shear layer. The interaction map indicates a predominant presence of these interactions, primarily within the separation bubble and reattachment zone and extending beyond the reattached boundary layer over the flare.

To observe the effect of high-amplitude forcing on the separation bubble, the bispectral mode and the interaction intensity distribution for BM8 are shown in figure 29 for a forcing amplitude of $A_0 = 1\times 10^{-2}$ . Compared with low-amplitude forcing, highly distorted structures appear within the separation bubble for the high-amplitude case. Notably, unlike the low-amplitude case, the bispectral mode amplifies upstream of the oblique expansion wave (originating from the cone–cylinder junction) when the forcing amplitude is increased. This upstream amplification is visible in the interaction map as a black hotspot ahead of the expansion wave, indicating strong triadic interaction. Downstream of the expansion wave, the interaction strength weakens, as reflected by the lower intensity in both the bispectral mode and the interaction map. The expansion induces flow acceleration and diverging streamlines, which cause the disturbance energy to spread, making triadic interactions less localised and less intense. The bispectral mode re-amplifies within the separation bubble, with particularly strong amplification observed over the flare. Here, the triadic interactions associated with BM8 become significantly more intense beyond the reattachment line compared with those under low-amplitude forcing, as shown earlier in figure 28(c).

Figure 29.

Spatial distribution of bispectral mode and triadic interaction map for forcing amplitude $A_0 = 1\times 10^{-2}$ for sum interaction BM8 ( $60, 36, 96\,\rm kHz$ ).