3.1. Optimal response to inlet perturbation

Figure 4.

Parametric effects on the $N$ -factors of the optimal growth: ( $a$ ) different frequencies with $X_0^* = 44$ mm and $X_1^* = 275$ mm (note that the stationary one is approximated by results at a frequency of 0.01 kHz). ( $b$ ) Streamwise evolution of the component amplitude of the direct (solid lines) and adjoint (dashed lines) states for a frequency of 20 kHz. ( $c$ ) Different inlet locations with a fixed outlet location ( $X^*_1 = 250$ mm) and a frequency of 20 kHz. ( $d$ ) Different outlet locations (marked by the square symbols) with a fixed inlet location ( $X^*_0 = 44$ mm) and a frequency of 20 kHz, along with the normalised wavemaker distribution ( $\max _{\eta ,\phi }\Lambda (\xi ,\eta ,\phi )$ , dashed line).

As a first step we perform a parametric study on the optimal growth regarding the inlet/outlet positions (i.e. the projectors take the form of (2.29)) and the disturbance frequency. The results are displayed in figure 4 from which we can make several observations. First, the optimal disturbances undergo two distinct evolution stages: they initially experience a short-term, yet substantial transient growth (achieving growth of $\delta N\approx 4$ within about 20 mm) (stage I) followed by a relatively long-term moderate growth (stage II). The disturbance frequency seemingly affects the second stage only. The most amplified frequency is around 20 kHz whose amplitude evolutions of the direct and adjoint variables are shown in figure 4( $b$ ). It can be seen that the temperature fluctuation amplitude rapidly grows and soon becomes dominant, followed by the streamwise velocity component ( $\hat u$ ). In contrast, the adjoint variables exhibit an opposite trend, suggesting a strong receptivity of the response to streamwise vortices ( $\hat v, \hat w$ ) in the vicinity of the inlet. Second, there exists an optimal initial position given a frequency and an outlet position where the seeded disturbance can amplify maximally. For a frequency of 20 kHz, the optimal inlet location is at around $X^* = 40$ mm. At last, extending the outlet position with a given initial position does not modify the path but only raises the final energy gain reached by the disturbance. This can be explained by the wavemaker position. By definition (B12), the wavemaker should lie in the overlap region of the direct and adjoint states. Since the adjoint variable amplitude decays at a smaller rate in the vicinity of the inlet than the growth rate of the direct mode near the outlet, the wavemaker concentrates in a small region downstream of the inlet. Therefore, the optimal disturbance characteristics would be substantially preserved as long as the instability core (wavemaker) is included inside the spatial interval.

Figure 5.

Parametric effects on the optimal disturbance profiles depicted by the real part of the temperature component at the inlet and outlet positions (shown in the upper and lower row, respectively, within each subplot). (a) Varying frequency for a fixed spatial domain (same as in figure 4 $a$ ): ( $ai$ ) stationary disturbance, ( $aii$ ) 20 kHz, ( $aiii$ ) 40 kHz. (b) Varying inlet positions for a given frequency (20 kHz) and a fixed outlet position ( $X_1^* = 250$ mm): ( $bi$ ) $X_0^* = 16$ mm, ( $bii$ ) $X_0^* = 43$ mm, ( $biii$ ) $X_0^* = 68$ mm. (c) Varying outlet positions for a given frequency (20 kHz) and a fixed inlet position ( $X_0^* = 44$ mm): ( $ci$ ) $X_1^* = 103$ mm, ( $cii$ ) $X_1^* = 234$ mm, ( $ciii$ ) $X_1^* = 275$ mm. The dashed line denotes the boundary-layer edge.

The parametric effects on the optimal disturbance profiles at the inlet and outlet locations are illustrated in figure 5. One can observe that the optimal disturbances all initially manifest as a tilted wavepacket lying preferentially on the windward side, very reminiscent of the shape function of a typical adjoint mode as displayed in figure 3. At the outlet position, they all tilt in the opposite direction and reside on the leeward side like a crossflow mode. This observation highlights the spatial separation between the highest-receptivity region (windward) and the highest-response region (leeward) for the optimal disturbances. With increasing disturbance frequency, the wavepacket is initially more tilted and ultimately resides closer to the side ray ( $\phi = 90^{\circ}$ ). On the other hand, the initial wavepacket progressively approaches the windward ray for a more upstream inlet, yet appears to be insensitive to the outlet location. This observation is important because one could obtain the optimal initial profile for even the entire model through performing optimal calculation over a relatively small domain where the wavemaker is strong.

Figure 6.

Spatial structures of the optimal disturbances of three representative frequencies, visualised by the isosurfaces of the real part of the temperature shape function with amplitude envelopes corresponding to $N_T = 5$ (blue), $N_T = 8$ (red) and $N_T = 11$ (purple). The temperature shape function is normalised by the maximum value at the inlet position. Frequency ( $a$ ) $f^* = 0$ kHz, ( $b$ ) $f^* = 20$ kHz and ( $c$ ) $f^* = 40$ kHz. Streamwise velocity distribution at four equal-distance streamwise slices ( $\eta$ – $\phi$ plane) starting from $X^* = 44$ mm to $X^* = 275$ mm is also shown. The dashed pink line in ( $b$ ) denotes the disturbance amplitude envelope corresponding to $N_T = 2.3$ obtained from PSE3D initiated by modal disturbances at 20 kHz. The spatial structures of the adjoint state (green, real ( $\hat T_p$ ) = 0.1 $|\hat T_p|_{max}$ ), the wavemaker (red, 0.5 $|\Lambda |_{max}$ ) and the edge streamlines are also displayed. ( $d{-}f$ ) The corresponding $N$ -factor isolines (black: $N_{Chu}$ = 5 and green: $N_{Chu} =$ 8) based on the Chu energy in the $X^*$ – $\phi$ plane, along with the LST-eN results (dashed lines).

Figure 6 shows the spatial structures of direct and adjoint states for three frequencies. We observe that the direct disturbance starts on the windward side and moves towards the leeward side, exhibiting a wedge-like amplitude envelope embodying several bent streaks inside; with increasing frequency, the core part of the spatial structure moves progressively towards the windward side, and the streaks are more prominently tilted. By contrast, the adjoint fluctuation, quantifying the high-receptivity region, lies preferentially on the windward side close to the inlet, exhibiting wave angles with respect to the streamwise direction opposite to those of the direct disturbances. Such remarkable spatial separation of the direct and adjoint states is indicative of the streamwise as well as azimuthal non-normality (Chomaz Reference Chomaz2005). Interestingly, the wavemaker marking the structural sensitivity region manifests as a longitudinal structure slightly downstream from the inlet on the windward side, irrespective of the frequency. Consequently, one may expect to employ some control strategy in the wavemaker region for the crossflow instabilities with any frequency.

The area occupied by the amplitude envelope is obviously proportional to the $N$ -factor of the disturbance. By analogy to the classical eN method utilising the 1-D stability analysis (LST), we define the local $N$ -factor distribution on the cone surface based on the temperature disturbance amplitude as (see also Zhang et al. Reference Zhang, Chen, Liu, Wang, Dong, Duan and Chen2025)

(3.3) \begin{equation} N_T(\xi ,\phi ) = \ln (\max _\eta |\hat T(\xi ,\eta ,\phi )|/\max _{\eta ,\phi }|\hat T(\xi _0,\eta ,\phi )|). \end{equation}

Note that a similar formulation was adopted by Araya et al. (Reference Araya2023) with wall pressure as the alternative measure of disturbance amplitude. As such, the amplitude envelope of the optimal disturbance provides an upper bound of the transition front pertaining to a given $N$ -factor in the eN framework. Alternatively, the local $N$ -factor distribution can be depicted by the wall-normal integrated Chu energy as

(3.4) \begin{equation} N_{Chu}(\xi ,\phi ) = \ln \left(\int _0^{\eta _{max}} \boldsymbol {\hat q}^\dagger \boldsymbol M\boldsymbol {\hat q} \textrm {d}{\eta }/E_0 \right), \end{equation}

where $E_0$ as a normalisation factor denotes the maximum of the numerator at the inlet location. Quantitatively, $N_T$ is slightly larger than $N_{Chu}$ by a nearly constant value (about 1) for the present case. Figure 6( $d{-}f$ ) compares the transition fronts corresponding to $N_{Chu} = 5$ and 8 (as long as the targeted $N_{Chu}$ can be reached) predicted by the non-modal analysis and by 1-D modal analysis based on LST (LST-eN). The amplitude envelope obtained by PSE3D for the 20 kHz modal instability (pink dashed line) is also displayed in figure 6( $b$ ) for comparison. The first $N$ -factor ( $N_{Chu} = 5$ ) is typically associated with laminar–turbulent transition of stationary crossflow vortices in the incompressible regime (Bippes Reference Bippes1999), while the second ( $N_{Chu} = 8$ ) is close to the transition $N$ -factor of travelling crossflow waves on the Mach 6 HyTRV model (Chen et al. Reference Chen, Dong, Tu, Yuan and Chen2022). Evidently, the optimal disturbance amplitude envelopes significantly extend in the upstream direction the LST-eN envelope at the same $N$ -factor as well as the modal PSE3D-eN envelope even with a much smaller $N$ -factor ( $N_T = 2.3$ ), highlighting the appreciable non-modal growth. Nevertheless, modal analyses also predict the predominance of the 20 kHz waves, suggesting that non-modal growth does not change the peak frequency. Another interesting observation is that the core part of the downstream optimal structure matches well with the most unstable region predicted by the LST-eN method, but is remarkably far away from the leeward ray compared with the 2-D modal amplitude envelope (pink dashed line). This implies that the optimal disturbance does exploit the crossflow instability, yet not in the same manner as a 2-D local crossflow mode does, which is discussed in detail later.

3.1.1. Growth mechanism

Figure 7.

Optimal structure evolution in the first stage (44 mm $\lt X^*\lt 64$ mm) depicted by ( $a$ ) the isosurfaces of the streamwise vorticities (red: positive; blue: negative) and the axial vorticity distribution on the cross-sections at three successive axial stations: ( $b$ ) $X^* = 44$ mm, ( $c$ ) $X^* = 54$ mm and ( $d$ ) $X^* = 64$ mm. The cross-section coordinates ( $Y,Z$ ) have been enlarged twofold for facilitating comparison. ( $e{-}g$ ) The corresponding streamwise velocity disturbance profiles at these three axial stations overlayed with the vector plot comprising the wall-normal and crossflow components. The pink line denotes the boundary-layer edge. The rightmost column shows the mean crossflow components at the peak location of the optimal disturbance.

Figure 7 clearly illustrates how the optimal profile initiates a crossflow-instability wavepacket, from which an important insight into the physical mechanisms in stage I can be gained. Initially, the optimal profile gives rise to immediately downstream alternating positive and negative longitudinal vortical rolls with axes aligned along the axial direction, which are well known in shear-layer flows to be able to optimally leverage the lift-up effects. The lift-up effect can also be inferred from the observation that the initially dominant vortex components ( $\hat v,\hat w$ ) are soon taken over by the streak component ( $\hat u$ ) in amplitude, as displayed in figure 4( $b$ ). The initially strong vortex components are directly driven by the adjoint components ( $\hat v_p, \hat w_p$ ). Despite weakening rapidly due to viscous dissipation, these longitudinal vortices induce counter-rotating vortices below, forming two-cell vorticity structures with axes being tilted with respect to the axial direction. Such vorticity structures are no longer optimal in the sense of the lift-up effect because of the counteracting forces from two opposite-orientation vorticity layers, yet are typical of crossflow instabilities (Wassermann & Kloker Reference Wassermann and Kloker2003) of which they can take advantage to sustain the downstream growth.

At the same time, the apparent change of the vorticity inclination in the cross-section as shown in figure 7( $b$ – $g$ ) is indicative of the Orr mechanism that can enhance the lift-up effect as well as the near-wall vorticity induction by strengthening the primary vorticity. By analogy to the analysis of Butler & Farrell (Reference Butler and Farrell1992), the Orr mechanism in the present case is rooted in the production term, $-\textrm {Re}(\hat w^\dagger \hat v) ({\partial \bar w}/{\partial \eta })$ , stemming from the azimuthal momentum equation as is discussed later. When this term is positive, or equivalently the disturbance structure is tilted against the shear of the crossflow, the disturbance is able to gain energy from the mean flow. Otherwise, the disturbance would lose energy. Figure 7( $e$ – $g$ ) reveals that the crossflow profile exhibits two distinct regions across the boundary layer with opposite sign pertaining to the shear term ( ${\partial \bar w}/{\partial \eta }$ ). As a result, if the disturbance gains energy in the upper region, it would return energy back to the base flow in the lower region with the same tilted direction. At the initial position, the disturbance structure well extends into the free stream and is tilted against the upper shear, favouring energy gain. The slanted structure soon turns upright at the next station and tilts along the opposite direction further downstream. Beyond the second station, the disturbances penetrate more deeply into the boundary layer, occupying almost equally the two shear regions of the crossflow, and the total energy transfer pertaining to the Orr mechanism should thus be smaller thereafter.

This vortex evolution process is representative of optimal disturbances, the quantitative difference being that the inclined angle of the two-cell vorticity structures increases with the frequency. As per the observations made so far, we might thus conclude that the non-modal growth of the optimal disturbance in stage I primarily stems from a synergistic combination of the lift-up effect and the Orr mechanism, which conforms with the optimal growth results for the infinite swept wing boundary layer (SWBL) (Tempelmann et al. Reference Tempelmann, Hanifi and Henningson2010, Reference Tempelmann, Hanifi and Henningson2012).

Figure 8.

( $a$ ) Comparison of $N$ -factor of the optimal disturbance at a frequency of 20 kHz and the crossflow modal disturbances obtained from PSE3D initiated by different crossflow modes at $X^* = 100$ mm at the same frequency (grey lines). The results from the local transient growth analysis (solid blue lines) and the most unstable mode obtained by 2-D LST (dashed blue lines) for two representative stations ( $X^* = 134$ and 200 mm) are displayed, with an upward shift to facilitate the comparison. ( $b$ ) Contributions of individual 2-D local modes to the local optimal growth at $X^* = 134$ mm, visualised by the circles sized proportional to the corresponding expansion coefficients $\kappa _k$ . The energy gain of the disturbance superimposed by a few most unstable modes (delimited by the dashed box) is also shown in ( $a$ ) (dotted blue line).

In the following, we consider whether the crossflow-instability wavepacket initiated by the optimal profile is governed by a dominating unstable mode as in the SWBL (Tempelmann et al. Reference Tempelmann, Hanifi and Henningson2010). Figure 8( $a$ ) compares the $N$ -factor of the optimal disturbance of 20 kHz and those calculated by PSE3D with initial profiles provided by different crossflow modes (see figure 3 $a$ for spectrum) at $X^* = 100$ mm (deemed close to the neutral location). Obviously, the optimal disturbance at stage II amplifies at a substantially larger rate than the modal disturbances do, indicating that it could not be well represented by a single mode. This is not surprising because the 2-D local crossflow instability spectrum as shown in figure 8( $b$ ) contains multiple unstable modes without a predominant one, in stark contrast to the SWBL case where generally only one unstable crossflow mode exists given a frequency and a wavenumber. The (linear) non-modal combinations among these similar-looking modes may yield a much more rapid energy growth than the most unstable mode, which could have been exploited by the optimal disturbance. To justify such speculation, we have conducted a local optimal growth analysis, (2.16)–(2.19), seeking an optimal combination of modes that yields the maximum energy growth at the specific distance downstream, for two representative stations.

The spectrum used in calculating the local optimal energy gain for $X^* = 134$ mm is displayed in figure 8( $b$ ). It contains a total of 208 modes, but only a small portion of the spectrum primarily comprising significantly unstable modes plays an important role. The corresponding optimal energy gain, shown in figure 8( $a$ ), clearly reveals the initial algebraic growth stage and the ensuing quasi-exponential growth stage. Here, the local optimal $N$ -factor curves have been shifted in the vertical direction so that they are tangent to the global optimal $N$ -factor curves. It should be noted that the local optimal growth analysis fails to capture initial energy gain due to the Orr and lift-up mechanisms (see Liu et al. (Reference Liu, Chen, Wan, Zhang and Chen2024) for more details) which might need much more modes than presently used (Bagheri et al. Reference Bagheri, Åkervik, Brandt and Henningson2009). Nevertheless, the growth rate pertaining to the quasi-exponential growth stage is substantially larger than that of the most unstable mode, and bears a striking resemblance to that of the global optimal disturbance for an appreciably long distance. The optimal energy gain based on a few most unstable modes (delimited by the dashed box) exhibits a diminishing algebraic growth, yet quantitatively the same exponential growth as the fuller spectrum. This indicates that the quasi-exponential growth stage is governed by the few most unstable modes while the less unstable modes would affect the initial algebraic growth stage only. Such observations seem to hold at other stations as well, justifying the speculation that the optimal disturbance may have exploited the non-modal combination among the few most unstable modes locally to achieve the global optimal growth. Indeed, the global transient analysis on 2-D flow configurations has shown that superposition of non-normal global (even decay) modes recovers the convective (exponential) instability (Cossu & Chomaz Reference Cossu and Chomaz1997; Bagheri et al. Reference Bagheri, Åkervik, Brandt and Henningson2009; Nichols & Lele Reference Nichols and Lele2011), during which the optimal disturbance is convected from upstream to downstream (Ehrenstein & Gallaire Reference Ehrenstein and Gallaire2005; Alizard & Robinet Reference Alizard and Robinet2007; Åkervik et al. Reference Åkervik, Ehrenstein, Gallarire and Henningson2008). Analogously, superposition of 2-D crossflow modes leads to an azimuthal shift of the disturbance structure from the windward side to the leeward side as the optimal disturbance travels downstream in the present 3-D configuration.

Figure 9.

Convergence process of the optimal disturbance at 20 kHz in the streamwise range of $(44,275)$ mm initiated by a random profile: ( $a$ ) the $N$ -factor evolution, ( $b$ ) the ratio of the outlet $N$ -factor to the optimal one and ( $c$ – $f$ ) streamwise velocity profiles at the inlet and outlet for the second iteration to the fifth iteration, normalised by the cross-velocity components ( $\sqrt {\vphantom{{\hat{v}^{2^2}}}\hat v^2 + \hat w^2}$ ) with the dashed line denoting the boundary-layer edge.

The above results reveal three active mechanisms beneath the optimal growth, i.e. the Orr mechanism, the lift-up effect and the crossflow instability mechanism manifesting in both the streamwise and azimuthal directions. A significant insight regarding the role played by these mechanisms can be gained from the typical convergence process of the optimal growth, as displayed in figure 9. We observe that after just one iteration the inlet disturbance structure already retrieves all the aforementioned mechanisms, yielding nearly 80 % of the final optimal energy gain (figure 9 $b$ ): firstly, its streamwise velocity component is much smaller than the vortex component in amplitude, an indicator of the lift-up effect; secondly, the slanted structures in the $\eta$ – $\phi$ plane suggest the Orr mechanism; finally, the outlet profile exhibiting typical crossflow-instability features implies the crossflow-instability mechanism. After one more iteration, the inlet disturbance structure entirely leans against the crossflow shear such that it can take full advantage of the Orr mechanism; moreover, the spatial separation is more pronounced, enhancing the disturbance energy gain along the azimuthal direction through crossflow instability. Further iteration drives the inlet disturbance to the windward side progressively, yet barely improves the optimal gain, indicating that the azimuthal crossflow-instability growth is relatively weak compared to other mechanisms. In summary, the iteration process tends to first distil the lift-up effect and the streamwise crossflow instability mechanism, which can thus be viewed as most important for the optimal growth, and then fully exploits the Orr and azimuthal crossflow-instability mechanisms to achieve the maximum amplification.

3.1.2. High-frequency optimal growth

Figure 10.

Results for optimal disturbances of high frequencies optimised over a streamwise interval corresponding to $x\in (44,212)$ mm. ( $a$ ) Evolution of $N$ -factor of various frequencies. The dashed line denotes the result for 200 kHz optimised for the restricted azimuthal region $\phi \in (120^{\circ}, 180^{\circ})$ . ( $b$ – $d$ ) Contours of the normalised temperature of the optimal disturbance at 200 kHz displayed in the inlet ( $b$ ) and outlet ( $c$ ) as well as in the $X$ – $\eta$ plane at the windward ray ( $\phi$ = 0°). The temperature components at ( $b$ , $c$ ) are normalised by the total velocity amplitude, while those at ( $d$ ) are normalised by the maximum temperature amplitude. The blue–red contours and the green–purple contours denote the optimal structure optimised for the entire azimuthal region and for the restricted azimuthal region, respectively. The dashed and dashed-dotted lines indicate the edge of the boundary layer, $\delta _h$ , and the edge of the entropy layer, $\delta _S$ , respectively. ( $e$ ,f) Spatial structure evolution for the optimal disturbance at 200 kHz, optimised for the entire azimuthal region and for the restricted azimuthal region, respectively, depicted by the isosurfaces of the temperature disturbance (red: positive; blue: negative; amplitude corresponding to $N_T$ = 3.2).

In addition to the low-frequency optimal disturbances, we also calculated optimal disturbances at high frequencies which exhibit totally different features from the former. Figure 10 depicts the evolution of $N$ -factors of frequencies ranging from 100 to 800 kHz for a specified streamwise interval and the temperature contour of one representative frequency. One can observe moderate transient growths, albeit much weaker than the low-frequency optimal growth, experienced by a broadband disturbance far from the modal instability frequencies (around 1 MHz for Mack mode instabilities). The relative differences of the peak $N$ -factors among various frequencies are less than 1, indicating a relatively flat frequency response. The optimal shape function, exemplified by the frequency of 200 kHz, concentrates in the domain between the boundary-layer edge and the entropy-layer edge of the windward side throughout the entire streamwise interval. Following the work of Paredes et al. (Reference Paredes, Choudhari and Li2020), the entropy-layer edge, $\delta _S$ , is defined as the location where the local entropy increment is 0.25 times the entropy increment at the wall ( $\Delta \bar S(\eta =\Delta _S) = 0.25\Delta \bar S_{wall}$ ). The entropy increment, $\Delta \bar S$ , is defined with respect to the free-stream value, i.e. $\Delta \bar S = 1004 \ln (\bar T)-287\ln (\bar P)$ . The fact that the disturbance remains in the vicinity of the windward ray indicates that the aforementioned azimuthal shift of the low-frequency optimal disturbances is solely due to the crossflow instability, which is absent here. The mode shapes at the inlet and outlet positions exhibit opposite-slanted structures in both the azimuthal and streamwise directions, suggesting an Orr-like mechanism responsible for the transient growth. As opposed to the classical Orr mechanism which mainly amplifies the velocity components, the temperature component of the high-frequency perturbation is the prime benefactor (see figure 10 $b,c$ ) from algebraic amplification where the temperature gradient within the entropy layer is a driving factor, as is shown later. Finally, the spatial evolution of the optimal disturbance at 200 kHz is illustrated in figure 10(e), featuring a wave-train structure in the vicinity of the windward ray with a preferential wave angle on either side.

It is noteworthy that Paredes et al. (Reference Paredes, Choudhari, Li, Jewell and Kimmel2019) have also discovered entropy-layer travelling disturbances that experience appreciable non-modal growth on a hypersonic blunt cone at zero AoA. They (Paredes et al. Reference Paredes, Choudhari and Li2020) later showed that a pair of oblique non-modal waves could induce onset of transition via oblique breakdown. Our work extends their results to a 3-D configuration, showcasing the optimal entropy-layer disturbances concentrating in the vicinity of the windward ray. Because the optimal disturbance on either side of the windward ray comprises exclusively components with wave angles of the same sign, prohibiting interactions between components with opposite wave angles, the classical oblique breakdown mechanism is not active here (see Chen et al. (Reference Chen, Dong, Tu, Yuan and Chen2023) for detailed discussion). It is interesting to assess whether high-frequency disturbances can undergo transient growth in other places than the windward ray as well. A straightforward method is to calculate the optimal disturbance within a restricted azimuthal domain. As inferred from figure 10(a,b,c,f), the optimal disturbance restricted in the leeward region, exhibiting an essentially 2-D structure, turns out to enjoy a commensurate energy amplification as well through exploiting an Orr-like mechanism in the streamwise direction (not shown here).

3.2. Optimal response to external forcing in the entire domain

Figure 11.

( $a$ ) The $N$ -factor of the optimal response to external forcings at frequency ranging from $f^* = 0$ to $f^* = 50$ kHz, together with the optimal response to inlet profiles at 20 kHz (dashed line) for comparison. ( $b$ ) Streamwise component amplitude evolution of the normalised optimal response (solid lines) and the normalised optimal forcing (dashed lines) for 20 kHz. The normalised resolvent wavemaker distribution ( $\max _{\eta ,\phi }\Lambda (\xi ,\eta ,\phi )$ ) is also displayed. The corresponding (c) optimal forcing and (d) response shape functions at the inlet (upper row) and outlet (lower row) positions, illustrated by $\hat f_u$ and $\hat u$ .

In this subsection, we present results for the optimal response to the external forcing in the entire domain. Without loss of generality, we set $\boldsymbol Q = \boldsymbol I$ in measuring the forcing energy. Because the inlet response is prescribed null for this case, the conventional $N$ -factor would yield an infinitely large value. To circumvent such difficulty, we consider the integrated growth rate between the outlet and a position slightly downstream of the inlet. Figure 11 shows the results for a representative spatial interval $X^*\in (44,215)$ mm. The $N$ -factors starting from $X^* = 47$ mm in figure 11( $a$ ) feature an algebraic growth stage followed by a long-term (quasi-exponential growth) phase, and peak at 20 kHz, closely resembling the optimal growth results. Direct comparison between the dominant $N$ -factor evolution for the optimal forcing case and the optimal inlet-profile case indicates that their differences lie primarily in the algebraic growth stage wherein the former is remarkably enhanced. Interestingly, the algebraic growth is more pronounced with increasing frequency, in stark contrast to the optimal inlet-profile case wherein it almost coincides for different frequencies.

The streamwise distribution of the component amplitude of the optimal forcing and the response for 20 kHz is displayed in figure 11( $b$ ). As expected, the forcing amplitudes exhibit an almost opposite trend against the response counterpart; they peak within the algebraic growth stage and rapidly diminish downstream. Forcing terms on the $\phi$ - and $\eta$ -direction momentum equations are dominant, implying that the optimal forcing tends to leverage the lift-up effect to prompt the energy growth by enhancing the streamwise vorticity. Moreover, the optimal forcing shape function shown in figure 11( $c$ ) is always tilted against the crossflow shear, suggesting that the optimal forcing exploits the Orr mechanism as well. The resulting response shown in figure 11( $d$ ) is very reminiscent of that in the optimal inlet-profile case. To conclude, the optimal disturbance under external forcing undergoes essentially the same evolution process with the same growth mechanisms as in the optimal inlet-profile case; the optimal forcing mainly acts on the algebraic growth stage (stage I) by constantly enhancing the lift-up and Orr mechanisms, yet barely affects the ensuing stage where the crossflow instability prevails.

Figure 12.

Spatial structure of the optimal forcing (green, $\textrm {Re}$ ( $\hat f_T$ ) $=10^{-5}$ ), the optimal response (blue, $\textrm {Re}$ ( $\hat T) = 50\,000$ ) and the resolvent wavemaker (red, 0.5 $|\Lambda |_{max}$ ).

As inferred from figures 12 and 6, the optimal forcing is exerted on the same position of the adjoint state except for the vicinity of the inlet where it manifests as longitudinal structures aligned with the axial direction. Since axially aligned vortices (with nearly zero streamwise wavenumber) are most favourable for the lift-up effect (Farrell & Ioannou Reference Farrell and Ioannou1993), the optimal forcing presumably takes more advantage from the lift-up effect, thus enhancing the algebraic growth. Moreover, the wavemaker almost coincides with that of the optimal inlet-outlet growth problem, implying the same localised feedback mechanisms for both cases.

3.3. Energy transfer analysis

To further understand the energy growth mechanisms of the optimal disturbances, it is instructive to perform an energy transfer analysis. Following the work of Chen et al. (Reference Chen, Zhu and Lee2017, Reference Chen, Dong, Tu, Yuan and Chen2023), the energy transfer under external forcings but without nonlinear interactions can be conveniently divided into four parts as

(3.5) \begin{equation} \frac {1}{E}\frac {\textrm D}{{\textrm D}t}E = \varPi + \varUpsilon + \varXi + \varPhi . \end{equation}

The production reads

(3.6) \begin{align} \varPi =-\frac {1}{E}\text {Real}\left [\iint \underbrace {\bar \rho {\tilde u_i}^\dagger \tilde u_j\frac {\partial \bar u_i}{\partial x_j}}_{\varPi _{u_iu_j}^U}+\underbrace {\frac {2\bar \rho }{\gamma (\gamma -1)M_\infty ^2\bar T}\tilde T^\dagger \tilde u_i\frac {\partial \bar T}{\partial x_i}}_{\varPi _{u_i}^T} + \underbrace {\frac {2\bar T}{\gamma M_\infty ^2\bar \rho }\tilde \rho ^\dagger \tilde u_i\frac {\partial \bar \rho }{\partial x_i}}_{\varPi _{u_i}^\rho }{\textrm d}\eta \,{\textrm d}\phi \right ],\nonumber\\ \end{align}

(3.7) \begin{eqnarray} (\tilde u_{i})_{i=1,2,3}=(\tilde u, \tilde v, \tilde w), \quad (\bar u_{i})_{i=1,2,3}=(\bar u, \bar v, \bar w), \quad (x_{i})_{i=1,2,3}=(\xi , \eta , \phi ), \quad \end{eqnarray}

representing the energy transfer from the mean flow to the disturbance. Note that the production contains three parts, i.e. the Reynolds stress work ( $\varPi ^U$ ), the energy transfer owing to the temperature gradient ( $\varPi ^T$ ) and the energy transfer owing to the density gradient ( $\varPi ^\rho$ ). The energy transfer directly driven by the external forcing, which vanishes in the inlet–outlet optimal growth problem, takes the form of

(3.8) \begin{equation} \varPhi = \frac {1}{E}\text {Real}\left [\iint {\tilde u_i}^\dagger \tilde f_{u_i} + \frac {2\bar \rho }{\gamma (\gamma -1)M_\infty ^2\bar T}\tilde T^\dagger \tilde f_T + \frac {2\bar T}{\gamma M_\infty ^2\bar \rho }\tilde \rho ^\dagger \tilde f_d {\textrm d}\eta \,{\textrm d}\phi \right ], \end{equation}

where $\tilde f_{u_i}$ , $\tilde f_{T}$ and $\tilde f_{d}$ denote the forcing on the (three) momentum equations, the energy equation and the continuity equation, respectively. Here $\varUpsilon$ is the energy transfer associated with the viscous and heat conduction terms and $\varXi$ represents the remainder terms. Here we focus on the production terms and energy transfer pertaining to the external forcing only in an attempt to unveil the important driving factor during the energy growth of the optimal disturbances.

Figure 13.

Streamwise evolution of the production terms normalised by the maximum value for the energy transfer of representative optimal disturbances: the low-frequency optimal disturbance at 20 kHz (with a zoom of the inlet) without ( $a$ ) and with ( $b$ ) distributed external forcing, and ( $c$ ) the high-frequency optimal disturbance at 200 kHz without distributed forcing. Significant terms are labelled while others are plotted in grey.

Figure 13 displays the streamwise evolution of the normalised production terms for representative optimal disturbances. It can be seen that the three terms

(3.9) \begin{eqnarray} \varPi _{uv}^U = -\frac {1}{E}\text {Real}\left [\iint \bar \rho {\tilde u}^\dagger \tilde v\frac {\partial \bar u}{\partial \eta }\right ], \quad \varPi _{v}^T = -\frac {1}{E}\text {Real}\left [\frac {2\bar \rho }{\gamma (\gamma -1)M_\infty ^2\bar T}\tilde T^\dagger \tilde v\frac {\partial \bar T}{\partial \eta }\right ], \end{eqnarray}

(3.10) \begin{eqnarray} \varPi _{v}^\rho = -\frac {1}{E}\text {Real}\left [\frac {2\bar T}{\gamma M_\infty ^2\bar \rho }\tilde \rho ^\dagger \tilde v\frac {\partial \bar \rho }{\partial \eta }\right ] \end{eqnarray}

are significantly more prominent than others for all the cases. For the low-frequency candidate initiated by the inlet perturbation, these three terms all immediately undergo a sharp rise near the inlet, followed by a rapid decay towards a moderate level further downstream. The Reynolds tress term $\varPi _{uv}^U$ encompassing the lift-up effect remains the largest as expected, while the energy transfer associated with the temperature wall-normal gradient is also non-negligible. Interestingly, another Reynolds stress term, $\varPi _{wv}^U \sim \rho \tilde w^\dagger \tilde v \partial \bar w/\partial \eta + \text{c.c.}$ , which is associated with the Orr mechanism in the azimuthal direction, manifests itself in the vicinity of the inlet and quickly vanishes downstream. This observation indicates that the azimuthal Orr mechanism, clearly observable in the change of the inclined direction of the streamwise vortices, only moderately promotes the energy growth initially in a direct manner. Nevertheless, the indirect, yet positive influence exerted by the azimuthal Orr mechanism on the lift-up effect through enhancing the crossflow velocities should not be underestimated. Exactly quantifying such indirect effect of the Orr mechanism requires further investigation, which is, however, beyond the scope of the present work.

Clearly, the external forcing tends to primarily transfer energy into the vortex components (especially the azimuthal velocity component) by

(3.11) \begin{equation} \varPhi _w = \frac {1}{E}\text {Real}\left [\iint {\tilde w}^\dagger \tilde f_{w}{\textrm d}\eta \,{\textrm d}\phi \right ], \quad \varPhi _v = \frac {1}{E}\text {Real}\left [\iint {\tilde v}^\dagger \tilde f_{v}{\textrm d}\eta \,{\textrm d}\phi \right ], \end{equation}

which in turn indirectly enhance other primitive components through the lift-up/Orr mechanisms. The contribution of the direct energy transfer from the external forcing rapidly vanishes downstream of the inlet such that the energy transfer distribution closely resembles that of the inlet–outlet optimal growth problem.

Contrarily, for the high-frequency candidate, the Reynolds stress term initially dominates while the temperature energy transfer term $\varPi _v^T$ rapidly takes over, peaking midway between the inlet and the outlet. The density production term, attaining appreciably large values, evolves similarly to the temperature counterpart. Close inspection of the flow structure displayed in figure 10( $d$ ) reveals the coincidence between the peak location of the temperature/density production and the position when the structures turn upright, indicating that these production terms are associated with the (streamwise) Orr mechanism. Intriguingly, although the azimuthally tilted structures as shown in figure 10( $b,c$ ) are an indicator of the azimuthal Orr mechanism, the corresponding production term $\varPi _w^T$ ( $\varPi _w^\rho$ ) remains negligibly small. Therefore, whether the azimuthal Orr mechanism is (dynamically) active in the energy growth of the high-frequency optimal disturbance is still unclear.