3.1. Laminar flow pattern

Figure 3($a$) displays the wall pressure distribution on the surface of the HyTRV model, along with some representative streamlines. The corresponding time-averaged flow structures are illustrated in figure 3($b$). One can observe that high-pressure regions form in the vicinity of the shoulder attachment line and the leeward centreline, while low-pressure regions lie in between. Fluid is driven away from high-pressure regions to low-pressure regions where streamwise vortices and mushroom structures form. The mushroom structure in the windward side closely resembles the centreline flow structure on other 3-D configurations, such as an elliptic cone, BoLT or cone at non-zero angle of attack, and is thus expected to be conducive to similar instabilities. In contrast, the mushroom structure in the leeward side loses the azimuthal symmetry, and its stability characteristics have not been studied before. It is convenient to divide the HyTRV model roughly in seven regions, as shown in figure 3($b$), and consider the boundary layer transition separately. This approach is supported by the QDNS results, which show several distinct transitional regions (see figure 8), and by the stability analysis results, which indicate a localized perturbation distribution for all the instabilities. The boundary layers in the last two regions, R6 and R7, are found to be stable, and thereby are not considered hereafter. Moreover, we also omit the results in the windward vortex region (R1) where the transition mechanisms turn out to be essentially the same as that in the shoulder vortex region studied below.

Figure 3. ($a$) Pressure distribution over the entire surface of the HyTRV model, and near-wall streamlines showing motion of the fluid from the attachment line towards the low-pressure regions. The pressure field on the leeward–windward symmetry plane is also displayed, clearly depicting the shock wave structure. ($b$) Contours of axial velocity in several cross-sections ($X^* = 294$ mm, 548 mm, 791 mm, 964 mm, 1136 mm, 1315 mm, 1594 mm). The flow in the region $\phi \in [-{\rm \pi},0]$ (i.e. the left half part) has been replaced by the flow in the right part (fine-grid region). The boundary layer over the HyTRV model is qualitatively divided into seven regions according to the transition characteristics.

It should be emphasized that the partition of the HyTRV model as shown in figure 3($b$) is very general. In calculations of multi-dimensional stability analyses, the computation regions for certain type of instabilities would be adjusted following the principles introduced in § 2.3, as shown in figure 4. Figure 5 displays the evolution of the boundary layer thickness (based on total enthalpy) as a function of the azimuthal angle at several streamwise locations. The boundary layer thickness peaks at the vortex regions, where it increases almost linearly along the axial direction. By contrast, the attachment-line boundary layer thickness is thinnest and changes only slightly with increasing axial stations.

Figure 4. Azimuthal regions utilized in multi-dimensional stability analyses for various instabilities: $\phi \in [86.7^\circ, 180^\circ ]$ for windward cross-flow instabilities; $\phi \in [85.2^\circ, 94.3^\circ ]$ for attachment-line instabilities; $\phi \in [62.3^\circ, 90.9^\circ ]$ for shoulder cross-flow instabilities; $\phi \in [68.0^\circ, 82.9^\circ ]$ for shoulder Mack-mode instabilities; and $\phi \in [22.9^\circ, 50.8^\circ ]$ for shoulder vortex instabilities. The axial velocity slice at $X^*=1000$ mm is also shown.

Figure 5. Variations of boundary layer thickness along the azimuthal direction for several axial stations ($X^* = 600$ mm, 800 mm, 1000 mm, 1200 mm). The edge of the boundary layer is defined as the location for which the local total enthalpy equals 99 % of the freestream total enthalpy.

Since a large-scale vortex is a structure exhibiting distinct transition characteristics in contrast to the ambient slowly varying boundary layer, rapidly determining the approximate location of the vortex without performing expensive numerical simulations is very important to transition predictions. As a vortex generally arises from the fluid concentration driven by spanwise pressure gradients, it is straightforward to associate the vortex location with the pressure valley. The boundary layer pressure depends on the shock wave strength, which can be measured simply by the relative angle, $\psi$, between the incoming flow direction and the wall-normal vector of surface. The definition of $\psi$ can be written as

(3.1)\begin{equation} \psi = {\rm \pi}+ \arccos (\boldsymbol a \boldsymbol{\cdot} \boldsymbol n), \end{equation}

where $\boldsymbol a$ and $\boldsymbol n$ are the unit vector in the incoming flow direction and the unit vector in the wall-normal direction, respectively. Here, $\psi$ varies from 0 to ${\rm \pi} /2$. $\psi = 0$ corresponds to the stagnation-point shock wave case where the shock wave is strongest, while $\psi = {\rm \pi}/2$ is associated with the flat-plate shock wave case with weakest shock wave strength. Therefore, the vortex location is related to the location of the local maximum of the relative angle, which is a purely geometric quantity. Similarly, the local minimum of the relative angle helps to determine the locations of the attachment line.

Figure 6 shows the variations of the relative angle along the half-part of the lifting body for various angles of attack. It can be observed that increasing the angle of attack implies a downward shift of the attachment line (indicated by the triangles) and a upward movement of the shoulder vortex (indicated by the circles) until an 8$^\circ$ angle of attack when the shoulder vortices from two sides meet. On the other hand, as the angle of attack is increased, the locations of the windward vortex and leeward attachment line remain the same, yet the curve of $\psi$ suggests that the windward vortex weakens and the leeward attachment-line flow pattern with diverging streamlines eventually changes to a typical streamwise vortex flow pattern with converging streamlines. The above trend of the flow features with increasing angle of attack is in accord with observations by Chen et al. (Reference Chen, Tu, Wan, Yuan, Yang, Zhuang and Xiang2021). Now we compare the shoulder vortex location predicted by the relative angle with that from QDNS. If we restrict the vortex to be bounded by the up and down edges as shown in figure 7($a$), then the location of $\psi _{{max}}$ does correctly capture the vortex location, as figure 7($b$) indicates. Discrepancies may be attributed to twofold effects, one the immaturity of the vortex in the upstream stage, and the other the self-induction of the vortex that tends to promote the vortex location.

Figure 6. Variations of the relative angle ($\psi$) along the half part of the lifting body for various angles of attack (AOA) (arrow line denoting the increasing direction). Symbols denote the local extremum points that are associated with certain flow regions, i.e. squares indicate the locations of the windward vortex, triangles the locations of the shoulder attachment line, circles the locations of the shoulder vortex and flowers the leeward attachment line.

Figure 7. ($a$) The shoulder vortex illustrated by the axial velocity contour at $X^* = 900$ mm, with the up and down edges of the vortex being marked. ($b$) Comparison of the location of the local maximum of the relative angle ($\psi _{{max}}$) and the shoulder vortex location bounded by the up and down edges.

3.2. Transitional flow pattern

Figure 8 presents an overview of the transitional flow pattern. The transition regions are shown clearly by the distribution of skin friction coefficient, $C_f\equiv |2\mu \boldsymbol {\nabla } \bar U|_{{w}}/R$, in figure 8(a,b). The windward vortex region is the first to experience the transition, at $X^* = 1100$ mm, followed by the shoulder vortex region, the windward cross-flow region, and finally the shoulder cross-flow region. Figure 8($c$) illustrates the transition process of the shoulder vortex. The vortical structures are illustrated using the $Q$-criterion (Hunt, Wary & Moin Reference Hunt, Wary and Moin1988). It can be seen that disturbances first set in the head of the vortex, manifesting as crescent-shaped vortices, and rapidly develop into hairpin packets, contaminating the whole bulge. Figure 8($d$) displays the evolution of vortical structures in the shoulder cross-flow region. The vortices initially manifest as oblique rolls with angles relative to the azimuthal direction ranging from 30$^\circ$ to 40$^\circ$. The spacings of these oblique rolls are nearly twice the boundary layer thickness, and the phase velocities (based on the disturbances of the dominant frequency) are around 0.8, indicating that they probably originate in Mack instability. Moreover, these oblique rolls exhibit prominent azimuthal variations as marked by the red longitudinal bands. These longitudinal bands form an angle of approximately 80$^\circ$ with respect to the azimuthal direction. Since time-averaged $C_f$ distributions do not show any visible streaks, or footprints of stationary cross-flow vortices, these longitudinal bands likely arise from travelling cross-flow waves. In the locations of the longitudinal bands, $\varLambda$ vortices gradually emerge riding on the oblique rolls, which further develop into hairpin packets and merge with lateral vortices to form a triangle-shaped transition region. Figure 8($e$) shows the transitional flow pattern in the windward cross-flow region. A triangle-shaped transition region also appears, surrounded by longitudinal streaks with angles of around 70–80$^\circ$. These longitudinal streaks are again likely associated with travelling cross-flow waves, which is apparent from the supplementary movie of the cross-section flow (available at https://doi.org/10.1017/jfm.2021.1125). Small-scale rolls are observed to ride on the longitudinal streaks, which are believed to develop from secondary instabilities of the travelling cross-flow waves. These rolls would further evolve into hairpin vortices. Note that a similar transition pattern has been observed by Tufts et al. (Reference Tufts, Borg, Bisek and Kimmel2020) in numerical simulations of the boundary layer transition on the HIFiRE-5 elliptic cone where the predominance of the travelling cross-flow waves is also highlighted.

Figure 8. An overview of boundary layer transition on the HyTRV model. ($a$) Time-averaged skin friction coefficient distribution on the windward side. ($b$) Time-averaged skin friction coefficient distribution on the leeward side. ($c$) Instantaneous vortical structures in the shoulder vortex region visualized by the isosurface of the $Q$-criterion ($Q=0.001$) coloured by the axial velocity, and the isosurface of the axial velocity ($U = 0.7$), with some velocity slices also displayed. ($d$) Instantaneous vortical structures (same settings as panel ($c$)) in the shoulder cross-flow region. ($e$) Instantaneous flow pattern in the windward cross-flow region, depicted by a wall-normal slice (at the 50th grid point from the wall) coloured by axial velocity, along with the corresponding vortical structures ($Q=0.001$) coloured by $Q$. The transition locations of different regions are also marked, which are defined where the time-averaged skin friction coefficient reaches 50 % of its maximum value in the late-transition stage along a constant azimuthal-angle ray. Note that the coarse-grid part of the model has been replaced by the mirror region of the fine-grid part.

3.3. Shoulder vortex region

Figures 3 and 8($c$) illustrate the progression from formation to breakdown of the vortex. A typical flow profile is displayed in figure 7($a$), depicting a ‘flat-top’-shaped structure. Interestingly, it looks very similar to the boundary layer modulated by low-azimuthal-wavenumber cross-flow vortices on a yawed cone as studied by Moyes et al. (Reference Moyes, Paredes, Kocian and Reed2017b), and (one-half of) the centreline mushroom structure in BoLT as studied by Knutson et al. (Reference Knutson, Thome and Candler2019). To help to determine if the vortex is conducive to inviscid instability, an inflection point calculation was completed. The $\phi$-direction and wall-normal gradients at $X^* = 872$ mm are shown in figure 9, where figure 9(a) would depict the outer mode instability, concentrating on the top shear layer, and figure 9(b) would depict the inner mode instability, residing in the inner shear layer.

Figure 9. Eight isolines (solid black) of axial velocity $U$ (from 0.1 to 0.8) and isocontours at $X^* = 872$ mm of normalized gradients ($a$) $\rho \,\partial U/\partial \delta$ and ($b$) $\rho \,\partial U/\partial \phi$.

Figure 10 shows the BiGlobal instability results at multiple axial locations along the vortex path from $X^* = 534$ mm to $X^* = 1000$ mm. At the first station, the Mack mode peaks at around 65 kHz and is the most unstable. In comparison, a shear-layer mode, denoted as mode 1, emerges in a slightly lower-frequency region. The Mack mode resides in the flat body part, whereas mode 1 concentrates on the head part, in accord with the observation by Moyes et al. (Reference Moyes, Paredes, Kocian and Reed2017b) for secondary instabilities of cross-flow vortices with similar shape. An important distinction between the Mack mode and the shear-layer mode corresponds to the wall-normal distribution of the temperature fluctuations. Whereas the Mack mode induces significant fluctuations near the wall, the fluctuations associated with the shear-layer mode are localized within the shear layer in the vicinity of the boundary layer edge. As the vortex develops, the Mack mode decreases in frequency with the increasing boundary layer thickness, and eventually disappears before $X^*=872$ mm. In comparison, mode 1 shifts towards higher-frequency regions and is substantially destabilized. Multiple shear-layer modes denoted by mode 2 to mode  5 also emerge in the meantime. Modes 1–3 cover a broader frequency band, $f^*\in (0,200)$ kHz, in the spectra and possess much higher phase velocities, $c\in (0.7,0.8)$, than modes 4 and 5, which have frequencies below 40 kHz and phase velocities from 0.3 to 0.55.

Figure 10. Variations of growth rates and phase velocities of unstable modes for the shoulder vortex at four axial locations: (a,b) $X^* = 534$ mm, (c,d) $X^* = 630$ mm, (e,f) $X^* = 872$ mm, and (g,h) $X^* = 1000$ mm. The real parts of the temperature eigenfunctions for the most unstable Mack mode (66 kHz) and mode 1 (50 kHz) are shown at the first station, along with the base flow depicted by the axial velocity (isolines).

The typical mode structure for each shear-layer instability is displayed in figure 11. As expected, all these modes are localized in regions of high shear. In particular, modes 1–3 locate at the outer shear layer and can be referred to as outer modes, whereas the other two modes concentrate on the inner shear layer and can be classified as inner modes. Coexistence of the inner and outer shear-layer modes has also been identified in stability analyses for leeward streamwise vortices of a yawed cone (Chen et al. Reference Chen, Chen, Dong, Xu and Yuan2020; Li et al. Reference Li, Chen, Huang, Yang and Xu2020b) and for centreline streamwise vortices of an elliptic cone (Li et al. Reference Li, Zhang, Liu, Huang, Luo and Zhang2018) and the BoLT model (Li et al. Reference Li, Choudhari and Paredes2020a).

Figure 11. Temperature disturbance structures associated with the locally most unstable disturbance frequency of each mode: ($a$) mode 1, $f^* = 90$ kHz; ($b$) mode 2, $f^* = 70$ kHz; ($c$) mode 3, $f^* = 60$ kHz; ($d$) mode 4, $f^* = 10$ kHz; ($e$) mode 5, $f^* = 18$ kHz. Axial velocity isosurfaces ($U$) are displayed with values of 0.7 for ($a$–$c$) and 0.4 for (d) and (e). The normalized temperature eigenfunction is also shown with the velocity base flow contour in the start slice in (d) and (e).

Next, we adopt PSE3D to trace the evolution of single-frequency disturbances. The $N$-factors (i.e. the logarithmic amplification ratios relative to the neutral station) of outer modes for a range of frequencies (40,130) kHz are shown in figure 12($a$). The inner modes are found to be substantially less unstable than the outer modes, thus the results are not displayed here. It can be seen that the component of 70 kHz is dominant until $X^* = 1000$ mm, beyond which the component of 90 kHz achieves slightly larger $N$-factors. The maximal $N$-factor is around 12 near the transition point, slightly smaller than the transition $N$-factor (about 15) for the centreline streamwise vortices of the HIFiRE-5 vehicle at a flight condition (Choudhari et al. Reference Choudhari, Li and Paredes2020). The temperature isosurface of this frequency is shown in figure 12($b$), which highlights how the regular modal structure stretches to form the crescent-shaped structure in the late stage. The structure pattern closely resembles that of QDNS results preceding transition in figure 8($c$), indicating that the outer-mode instability is indeed dominant in the transition process. The Mack-mode perturbation (not shown here) undergoes only a short-distance moderate amplification, hence it is unlikely to trigger the breakdown of the vortex.

Figure 12. Downstream evolution of disturbances predicted by PSE3D: ($a$) $N$-factors for a range of frequencies of (50,130) kHz with step 10 kHz, the thick redline representing 70 kHz; ($b$) spatial structure of frequency 70 kHz, illustrated by the isosurface of the real parts of temperature disturbances. The isosurface value is prescribed to be 200 times the initial value (i.e. $N = 5.3$). The base flow is also visualized by isosurface $U=0.7$ and contours ($U$) at several axial stations.

Figure 13 further displays root mean square (r.m.s.) distributions of disturbances and the corresponding spectra at the peak r.m.s. locations for several axial stations. At the first station, fluctuations concentrate on the inner high-shear layer. The peak frequency is around 3 kHz, which is far below the peak inner-mode instability frequency (around 10 kHz) from BiGlobal analysis. One possible explanation for the appearance of such low-frequency disturbances is the non-normal instability mechanism (Schmid Reference Schmid2007) by which very-low-frequency disturbances may attain a rapid growth. Although these low-frequency disturbances can amplify and reach a large amplitude downstream, the low-frequency spectrum seems not to expand and the inner shear layer is not as remarkably distorted as the outer shear layer, indicating that they are unlikely to be the essential factors inducing transition. On the other hand, high-frequency disturbances centred at approximately 70 kHz manifest themselves in the outer shear layer since $X^*=900$ mm, and quickly become dominant components in the r.m.s. distribution before $X^*=1000$ mm. At the last station, the spectrum in the outer shear layer has been broadened rapidly, indicating the onset of turbulence there. In comparison, a weak peak at around 20 kHz, likely originating in inner-mode instabilities, appears in the spectrum of the sampling point in the inner shear layer. The amplitude evolution of the 70 kHz component is shown in figure 14, clearly depicting the exponential growth and the subsequent saturation stages. Unless otherwise stated, the amplitude of the QDNS results is defined as the maximal value of the temperature fluctuations in the cross-sections. Good agreement between QDNS and PSE3D results initiated by mode 1 can be observed, which is a verification of both approaches. The QDNS values depart from the PSE3D results for $X^*>900$ mm, most likely due to nonlinear modulations, which are known to tend to stabilize the shear-layer instabilities as well as decrease the peak mode frequency (Chen et al. Reference Chen, Huang and Lee2019b, Reference Chen, Chen, Dong, Xu and Yuan2020; Li et al. Reference Li, Chen, Huang, Yang and Xu2020b). Indeed, PSE3D results based on the time-averaged basic states of the transitional flow, which take the nonlinear modulations into account, show a better agreement with QDNS results, as shown in figure 14. The BiGlobal result from integration of growth rates of mode 1 is also displayed. The growth rate predicted by BiGlobal is notably larger than those of PSE3D and QDNS. Such a discrepancy is likely because of non-parallel effects.

Figure 13. Normalized disturbance root-mean-square (r.m.s.) distribution from QDNS at four presentative stations: ($a$) $X^*= 800$ mm, ($c$) $X^*= 900$ mm, ($e$) $X^*= 1000$ mm, and ($g$) $X^* = 1100$ mm, along with isolines of the time-averaged axial velocity $U$ (from 0.1 to 0.8). The sampling points are marked by filled symbols, and the corresponding spectra are displayed in the right-hand column (b,d, f,h).

Figure 14. Comparison of amplitude evolutions of disturbances with frequency 70 kHz from QDNS and theoretical predictions. The initial amplitudes of PSE3D and BiGlobal are prescribed to be equal to that of PSE3D at $X^* = 700$ mm.

3.4. Shoulder cross-flow region

Low-frequency cross-flow instabilities and high-frequency Mack instabilities can coexist in the shoulder cross-flow region. Figure 15(a,b) displays the eigenvalue spectra of Mack modes at $X^* = 1000$ mm from BiGlobal. Multiple Mack modes are present, of which the six most unstable ones are shown. For the sake of clarity, we denote these modes as modes 1–6, with an ordering of ascending phase velocities at frequency 125 kHz. Interestingly, the spacings in the phase velocity between adjacent modes are almost the same for all the modes and remain nearly unchanged with varying frequency. Moreover, phase velocities decrease almost linearly with increasing frequency. From mode 1 to mode 6, the peak growth rate decreases; the left-hand part of the spectrum along with the peak frequency moves towards the high-frequency region, while the right-hand part changes only slightly. Figures 15($c$–$f$) show the real part of the temperature eigenfunction of the first four modes at their peak frequencies. For the purpose of highlighting the spatial scale of the mode, the arc length, $S^*$, starting from the left azimuthal boundary of the BiGlobal computation region, is introduced. A dual-peak structure across the boundary layer can be seen clearly for all the modes, as a prominent feature of Mack-mode instabilities. From mode 1 to mode 4, the azimuthal oscillation appears to be more compact. It is convenient to further divide the boundary layer into three subregions, i.e. the expanding region in the left side connecting to the shoulder vortex region, the contracted region in the right side adjacent to the side attachment-line region, and in between the plateau region where the boundary layer changes slowly. Then one immediately finds that all the modes concentrate mainly in the plateau region, which is probably due to the sensitivity of Mack-mode instability to the azimuthal variations of the boundary layer thickness (Mack Reference Mack1984). This feature also holds for other frequencies. In the research of the HIFiRE-5 elliptic cone by Paredes et al. (Reference Paredes, Gosse, Theofilis and Kimmel2016), Mack modes are found to be the extension of attachment-line instabilities, and their frequencies and growth rates decrease as the location of the mode shape peak moves from the vicinity of the attachment line towards the centreline. In our study, however, Mack modes in the shoulder cross-flow region appear to have no connection with the attachment-line instabilities, since their mode shapes reside only in the plateau region.

Figure 15. Spectra of Mack mode instabilities from BiGlobal calculations at $X^* = 1000$ mm for the shoulder cross-flow region: ($a$) spatial growth rates and ($b$) phase velocities as a function of frequency, and normalized temperature disturbance (real part) distribution in the wall-normal distance ($\delta ^*$)–arc length ($S^*$) plane for the most unstable component of mode 1 ($c$), mode 2 ($d$), mode 3 ($e$) and mode 4 ($f$), at $X^* = 1000$ mm. The position of $S^*=0$ corresponds to the left azimuthal boundary of the computation region in BiGlobal. The temperature base flow (black lines) is also plotted with contour level increments in intervals of 0.5.

Another interesting observation is that all the modes are essentially 3-D or oblique. The obliqueness can be illustrated clearly by the spatial structures of these modes at frequency 128 kHz (which is close to the peak frequency of each mode), as shown in figure 16. The mode structures manifest as two-layer oblique rolls with certain phase lags. Because the upper-layer roll is dominant, the wave angle is estimated from the upper-layer roll. From mode 1 to mode 4, the wave angle decreases from 35$^\circ$ for mode 1 to 24$^\circ$ for mode 3 and then increases to 28$^\circ$ for mode 4. Moreover, the mode with the lower phase velocity tends to reside in locations slightly closer to the shoulder attachment line. Figure 17 further compares spatial structures of mode 1 for three frequencies. The wave angle increases from 20$^\circ$ at frequency 115 kHz to 37$^\circ$ at frequency 135 kHz. Since the phase velocity also decreases with increasing frequency, the wave angle of the single mode is also negatively correlated with the mode phase velocity. The azimuthal locations of mode shapes appear to be directly associated with the phase velocity too, as they gradually shift towards the attachment line with decreasing phase velocity. This observation holds qualitatively for other modes.

Figure 16. Spatial structures of the most amplified component for four modes at frequency 128 kHz: ($a$) mode 1, ($b$) mode 2, ($c$) mode 3 and ($d$) mode 4.

Figure 17. Spatial structures of mode 1 at three frequencies: ($a$) 115 kHz, ($b$) 125 kHz and ($c$) 135 kHz.

Figure 18 displays downstream evolutions of the first three modes with frequency 125 kHz. It can be observed that the growth rates are close between modes and exhibit a parabolic shape peaking at $X^*\approx 1030$ mm. Moreover, the phase velocities all decrease monotonically along the axial direction. The PSE3D results initiated by mode 1 are also shown for comparison. The growth rate from PSE3D shows a shape similar to that for the BiGlobal results, but it is notably larger than the latter especially for the right-hand half of the parabolic shape. As a result, the amplification region from PSE3D extends almost 50 mm downstream of the neutral location predicted by BiGlobal. The phase velocity from PSE3D decreases more slowly, and its trace crosses the phase velocities of all three modes. This suggests that the disturbances in PSE3D would excite multiple modes due to non-parallel effects when they travel downstream, and behave as a combination of multiple modes rather than as a single mode.

Figure 18. Downstream evolutions of the first three Mack modes with frequency 125 kHz: ($a$) growth rates and ($b$) phase velocities. The PSE3D results initiated by mode 1 are shown for comparison.

The spatial structures of mode 1 from BiGlobal are depicted in figure 19 for three representative axial locations. Obviously, in the axial direction, the wave angle increases and the mode structures move towards the shoulder attachment line, again implying a strong correlation with phase velocity (recall that the phase velocity also decreases along the axial direction).

Figure 19. Spatial structures of mode 1 with frequency 125 kHz at three axial locations: ($a$) $X^* = 952$ mm, ($b$) $X^* = 1048$ mm and ($c$) $X^* = 1144$ mm.

Downstream evolution of the Mack-mode instability for various frequencies by PSE3D are further illustrated in figure 20. Note that various inlet modes will lead to various evolution routes, and the most amplified one of each frequency is shown in figure 20($a$). The dominant frequency decreases with increasing axial locations, in accord with the growth of boundary layer thickness. The largest $N$-factor is approximately 4.5 at $X^* = 1250$ mm by frequency 125 kHz. The corresponding mode shapes at four axial stations, along with the base flow, are displayed in figures 20($b$–$e$). The azimuthal wavelength ($\lambda_\phi$ defined as the width between adjacent peaks of the mode shape) remarkably decreases from around 20 mm at the first slice to nearly 10 mm at the last slice, resulting in increases of the wave angle. This is consistent with the decrease of the phase velocity as shown in figure 18($b$). As the Mack-mode wave travels downstream, the main part of the mode structure remains in nearly the same range of the azimuthal angles (i.e. $\phi \in (74,83)^\circ$), rather than moving towards the shoulder attachment line as implied by BiGlobal.

Figure 20. The PSE3D results for Mack-mode disturbances. Downstream evolution of $N$-factors of representative frequencies ($a$). Distribution of disturbance of 125 kHz at four axial stations: ($b$) $X^* = 945$ mm, ($c$) $X^* = 1017$ mm, ($d$) $X^* = 1100$ mm and ($e$) $X^* = 1185$ mm, illustrated by contours of the real part of the normalized temperature fluctuations, along with the temperature base flow.

The cross-flow instabilities are caused by the inflectional three-dimensional boundary layer profile formed over the surface of the leeward zone between the shoulder attachment line and the shoulder vortex. Therefore, their shape functions are expected to occupy a much wider region in the azimuthal direction than the Mack-mode instabilities do. As a result, large resolutions are required to solve these modes using the BiGlobal analysis. Numerical tests show that cross-flow instabilities are relatively insensitive to the wall-normal resolution. Hence we fixed the wall-normal grids number $N_\delta$ to 101 and vary the azimuthal grids number $N_\phi$ from 160 to 2000. It turns out that the eigenvalues undergo an overall decrease in growth rate with increasing azimuthal resolution, but they do not converge, as shown in Appendix C. This implies that the eigenvalue problem for the cross-flow instabilities are extremely sensitive to the numerical fluctuations that likely arise from truncation and interpolation errors (Trefethen & Embree Reference Trefethen and Embree2005). Paredes et al. (Reference Paredes, Gosse, Theofilis and Kimmel2016) seemed to have also noted the bad convergence of this eigenvalue problem, but they did not show the spectra so we cannot make a comparison.

Now we consider the mode shape of a typical cross-flow mode with frequency 20 kHz as shown in figure 21. It can be seen clearly that the cross-flow mode exhibits a wave packet distribution with the oscillation amplitude vanishing towards the azimuthal boundaries. The negligibly small values in the vicinity of the azimuthal boundaries indicate that the cross-flow instability is insensitive to the azimuthal boundary conditions. The rapid oscillation of the mode shape translates into a small azimuthal wavelength and a large wave angle. In contrast, the envelope of the mode shape is much smoother. Furthermore, the azimuthal wavelength changes little along the azimuthal direction. Therefore, we can seek a Wentzel–Kramers–Brillouin–Jeffrey-like expansion (Bender & Orszag Reference Bender and Orszag1978) of the mode shape as

(3.2)\begin{equation} \hat{\boldsymbol q}(\delta,\phi) = \tilde{\boldsymbol q}(\delta,\phi)\exp({\rm i}4{\rm \pi}\beta \tilde S), \end{equation}

where $\tilde S \in [0,1]$ is the normalized form of the arc length $S^*$, so that the rapid variations are absorbed into the exponential term (much akin to the PSE methodology). As a result, the equation (2.9) reduces to

(3.3)\begin{equation} \left(\begin{array}{cc} \boldsymbol{0} & \boldsymbol{I} \\ -\tilde{\boldsymbol A_0} & -\tilde{\boldsymbol A_1} \end{array}\right) \left(\begin{array}{c} \tilde{\boldsymbol q} \\ \alpha \tilde{\boldsymbol q} \end{array}\right) = \\ \alpha \left(\begin{array}{cc} \boldsymbol{I} & \boldsymbol{0} \\ \boldsymbol{0} & \tilde{\boldsymbol A_2} \end{array}\right) \left(\begin{array}{c} \tilde{\boldsymbol q} \\ \alpha \tilde{\boldsymbol q} \end{array}\right), \end{equation}

where the azimuthal wave number $\beta$ has been included in the modulated linear operators with a tilde ($\,\tilde{}\,$) over it. It should be noted that the reduced eigenvalue equation (3.3) is completely equivalent to the original one (2.9) since no additional assumptions have been introduced. Similarly, this methodology can be applied straightforwardly to PSE3D with a fixed $\beta$ along the axial direction.

Figure 21. Comparison of results from the conventional eigenvalue problem with 1000 azimuthal grid points and the reduced eigenvalue problem with 160 azimuthal grid points for a typical cross-flow mode with frequency 20 kHz. ($a$) Temperature disturbance illustrated by its real part (filled contours from (2.9)) and by its amplitude function (contour lines from (3.3)); ($b$) wall pressure disturbance illustrated by its real part (blue line from (2.9)) and by its amplitude function (red line from (3.3)).

In calculation, we can choose an appropriate wavenumber $\beta$, which can be a constant or can vary slowly with $\phi$, in order to make the amplitude term, $\tilde {\boldsymbol q}(\delta,\phi )$, a slowly varying function of $\phi$. As a benefit, the azimuthal grid points required to obtain a smooth solution can be largely reduced.

Figure 22 shows the convergence test of the reduced eigenvalue problem for 20 kHz. For $N_\phi \ge 240$, the eigenvalues concentrate in a parabolic-like band but exhibit small shifts with increasing azimuthal grid points. The magnitudes of such shifts do not show a decreasing trend with increasing resolution, indicating that the reduced eigenvalue problem can hardly converge too. Nevertheless, the eigenvalue fluctuations with increasing resolution are much smaller than those obtained from the conventional equation (2.9) with the same grid points, as shown in figure 46. It is noteworthy that varying $\beta$ in a certain range, which acts as perturbations to the eigenvalue equation, would lead to shifts of the eigenvalues in a manner similar to changing grid resolution, as shown in Appendix D. Figure 22($b$) compares the fluctuation amplitude distribution of the mode calculated from three resolutions as marked by the ellipse in figure 22($a$). As expected, the small eigenvalue shift between two resolutions corresponds to a small shift of the mode shape distribution. Unless otherwise stated, the azimuthal grid points used in multi-dimensional stability calculations for the cross-flow instability are 240 hereafter.

Figure 22. Convergence test for the cross-flow instability (20 kHz) with varying azimuthal resolutions: ($a$) spectra at $X^* = 1000$ mm; ($b$) contours of the amplitude term magnitudes, $|\hat q|$ for the modes enclosed by the ellipse curve in ($a$), along with the temperature base flow. The chosen wavenumber $\beta$ is prescribed to be an integer and decreases linearly from 26 to 8 as the phase velocity increases.

Spectra variations with frequency at $X^*=1000$ mm are displayed in figure 23. It can be observed that the travelling cross-flow instability is more unstable than the stationary counterpart, and the maximum growth rate occurs at approximately 35 kHz; the phase velocity, ranging from 0.2 to 0.6, increases with frequency, which agrees with the experimental results obtained by Borg et al. (Reference Borg, Kimmel, Hofferth, Bowersox and Mai2015) for the HIFiRE-5 elliptic cone.

Figure 23. Spectra for the cross-flow instability with various frequencies at $X^* = 1000$ mm: (a) spectra in the $-\alpha _i^*$–$c$ plane, (b) spectra in the $c$–$f^*$ plane.

Now we consider the spatial structures of the cross-flow instability. Figure 24 shows the real part of the shape function, $\hat q(\delta,\phi )$, for three modes with the same frequency (20 kHz) but different phase velocities, highlighting the streaky structures arising from the crossflow modes. The shape function (and all such terms hereafter) is recovered by first interpolating the amplitude term onto a finer grid and then multiplying by the phase term, ${\rm e}^{4\,{\rm i}{\rm \pi} \beta \tilde S}$. With increasing phase velocity, the shape function moves towards the shoulder vortex region with larger azimuthal wavelengths. The wave angles of the streaks are estimated to be around 80$^\circ$. With increasing frequency the opposite trend is seen on the modal structure, as shown in figure 25, although the phase velocity also increases. This indicates that the frequency is a more important factor than the phase velocity in determining the behaviour of the modal structure.

Figure 24. Isosurfaces of the normalized real part of the temperature eigenfunction ($\hat T_r = 0.5$) for three modes with the same frequency 20 kHz: ($a$) $c = 0.27$, $\beta = 23$,($b$) $c=0.35$, $\beta = 16$ and ($c$) $c = 0.43$, $\beta = 10$. Twice the axial wavelength of each mode is shown.

Figure 25. Isosurfaces of the normalized real part of the temperature eigenfunction ($\hat T_r = 0.5$) for the most amplified mode of three frequencies: ($a$) $f^* = 10$ kHz, $c = 0.23$, $\beta = 15$, ($b$) $f^* = 25$ kHz, $c=0.37$, $\beta = 18$ and ($c$) $f^* = 45$ kHz, $c = 0.49$, $\beta = 20$. Twice the axial wavelength of each mode is shown.

Next we examine how the cross-flow instability changes with axial locations. Without loss of generality, take the modes of 20 kHz, for example. The spectra at three stations, i.e. $X^* = 800$ mm, 1000 mm and 1200 mm, are shown in figure 26 along with the shape function of the most unstable mode at each station. The cross-flow instability is seen to be enhanced from $X^* = 800$ mm to 1200 mm, whereas the phase velocity of the dominant mode changes little. It is evident from figure 26($b$–$d$) that the dominant mode moves towards the shoulder attachment line.

Figure 26. Spectrum variation with axial locations ($\square$ $X^* = 800$ mm, $\triangle$ $X^* = 1000$ mm, $\bigcirc$ $X^* = 1200$ mm) for the cross-flow instability of 20 kHz. The real part of the temperature shape function of the most unstable mode at each station (marked by the circle) is displayed in ($b$–$d$).

However, because the eigenvalue spacing is very small, disturbances will always be composed of multiple modes rather than being dominated inevitably by any single mode as they travel downstream. As a result, accurately tracking a single mode in the axial direction is nearly impossible and also of insignificance in practice. In comparison, PSE3D is able to capture adequately evolution of instabilities in the context of multiple modes (Chen et al. Reference Chen, Chen, Yuan, Tu and Zhang2019a; Choudhari et al. Reference Choudhari, Li and Paredes2020), and turns out to be less sensitive to the grid resolution, as shown in Appendix E. Therefore, we utilize PSE3D to follow evolutions of cross-flow instabilities in this paper. Figure 27(a,b) show the evolution of $N$-factors and phase velocities obtained by PSE3D with frequency 30 kHz but different initial conditions, say different inlet modes or different inlet locations. It is obvious that various initial conditions lead to different routes. Hence it is necessary to seek the optimal initial condition that produces the largest $N$-factor at the outlet in order to estimate the transition criterion. Principally, the optimal disturbances can be determined by solving iteratively the PSE3D and its adjoint counterpart just as in the one-dimensional case (see, for example, Paredes et al. Reference Paredes, Gosse, Theofilis and Kimmel2016), which is, however, beyond the scope of this work. Another important observation is that the phase velocities of all the routes substantially increase in the axial direction, whereas the BiGlobal analysis shows that the most unstable mode remains at nearly the same phase velocity along the axial direction. Figure 27(c,d) further compare the most amplified route of four frequencies. One can observe that the dominant frequency is 30 kHz for $X^*<1200$ mm and changes to 40 kHz before the outlet. The largest $N$-factor near the transition location (1297 mm) is slightly above 5.

Figure 27. The PSE3D results for the cross-flow instability of 30 kHz with various initial conditions (a,b) and of the most amplified one for each frequency (c,d). The most amplified route of 30 kHz is denoted by the thick line. (a,c) $N$-factor, (b,d) phase velocities.

Figure 28 illustrates the real parts of the temperature shape functions at five successive axial stations for the most amplified route of frequency 20 kHz. For routes with higher (lower) initial phase velocity, the shape functions would exhibit a larger (smaller) azimuthal wavelength and lie closer to (farther away from) the shoulder vortex region. In this case, the main part of the shape functions from PSE3D remains in the plateau region, rather than moving towards the attachment line as indicated by the BiGlobal results above. The distribution of fluctuations of 20 kHz from QDNS are also shown for comparison. Clearly, regular cross-flow wave profiles also appear and reside almost exclusively in the plateau region. After the fourth station, $X^* = 1100$ mm, the cross-flow wave profile is obviously distorted in a manner such that its left-hand part is notably enhanced and substantially stretched in the wall-normal direction. Such distortion is likely caused by nonlinear interactions with the Mack-mode instability. Furthermore, the azimuthal wavelength ($\approx 6$ mm) is slightly larger than the PSE3D results ($\approx 5$ mm) and also does not exhibit any prominent variations along the axial direction.

Figure 28. Comparison of evolution for the real parts of the normalized temperature disturbances from PSE3D results (left column) initiated by modes with phase velocity $c=0.37$ and QDNS results (right column) at five successive stations: $X^* = 800$ mm, 900 mm, 1000 mm, 1100 mm and 1200 mm. The disturbance frequency is fixed at 20 kHz.

A more realistic disturbance pattern can be obtained by taking the amplitude growth into account. Figure 29 shows the spatial evolution for the travelling cross-flow wave with frequency 20 kHz and for the Mack-mode wave with frequency 125 kHz, using isosurfaces at a value of 10 times the initial amplitude. Whereas the cross-flow wave first enters the exponential growth phase, the Mack-mode wave undergoes a faster growth so that both types of instability start to manifest themselves at nearly the same axial location. The cross-flow waves form longitudinal streaks with wave angles of around 84$^\circ$, while the Mack-mode wave appears as oblique rolls with a wave angle of 30–40$^\circ$. Because PSE3D marching stops at the neutral point where the growth rate of disturbance becomes negative, Mack-mode structures are invisible far downstream. Downstream increase of the magnitude leads to a triangle-shaped pattern for both kinds of instability. The overlap of the distribution regions indicates possible interactions between two kinds of instabilities, as has been observed in inclined hypersonic cone boundary layers (Munoz, Heitmann & Radespiel Reference Munoz, Heitmann and Radespiel2014; Dong et al. Reference Dong, Chen, Yuan, Chen and Xu2020). Inviscid streamlines are also displayed in figure 29. The relative angles of the travelling cross-flow wavefronts and the Mack-mode wavefronts with respect to the edge streamlines are estimated to be approximately 11$^\circ$ and 58$^\circ$, respectively.

Figure 29. The structure evolution, illustrated by the real part of the temperature disturbance, for the cross-flow instability with frequency 20 kHz (the black isosurface) and the Mack-mode instability with frequency 125 kHz (the red isosurface). The isosurface value is prescribed to be 10 times the initial value (i.e. $N = 2.3$). The edge streamlines are also shown (blue lines) for reference.

Figure 30 shows the fluctuation distributions from QDNS for three axial stations. The disturbance spectra are evaluated at locations where the r.m.s. amplitude is at its peak. The spectrum at the first station exhibits a prominent low-frequency peak at approximately 23 kHz and a much weaker high-frequency peak at approximately 190 kHz. The low-frequency peak corresponds to the travelling cross-flow instability, while the high-frequency peak emerges from the Mack-mode instability, as shown by the above analyses. Further downstream, at $X^* = 1000$ mm, a dual-peak r.m.s. amplitude structure is manifest in the plateau region. The fluctuation amplitude in the upper layer is larger and displays two isolated peaks. The spectra indicate that the Mack-mode wave amplitude appears to be larger in the left peak, while the travelling cross-flow instability dominates in the right peak. This is consistent with the distribution feature of these two kinds of instabilities. The Mack-mode fluctuations prevail at the last station. The appearance of its first harmonic implies that self-interactions are present. Owing to the increase of the boundary layer thickness, the Mack-mode instability decreases in frequency as it travels downstream.

Figure 30. Normalized disturbance r.m.s. amplitude distribution from QDNS at three representative stations: ($a$) $X^* = 800$ mm, ($c$) $X^* = 1000$ mm and ($e$) $X^* = 1200$ mm, along with the temperature base flow. The sampling points are marked by filled circles, and the corresponding spectra are displayed in ($b$), ($d$) and ($\,f$).

Figure 31 compares the amplitude evolutions of Mack-mode waves with frequency 125 kHz and cross-flow waves with frequency 20 kHz from QDNS and PSE3D results. The QDNS results indicate that the Mack-mode wave starts to amplify after $X^*=900$ mm and reaches saturation before $X^* = 1200$ mm. The cross-flow wave is observed to grow at least before $X^* = 700$ mm, and becomes saturated after $X^*=1300$ mm. The Mack-mode wave amplitude grows more rapidly and soon reaches the same amplitude level as the cross-flow wave at $X^* = 1100$ mm. The PSE3D results based on the most amplified case can capture adequately the exponential stage of both instabilities, but underestimate the initial growth rates of disturbances. This discrepancy is likely owing to the transient growth of the disturbances in QDNS. Moreover, a notable deviation between QDNS and PSE3D results is observed for the cross-flow wave amplitude for $X^*> 1000$ mm, which is likely attributed to the nonlinear effects from the Mack-mode waves, in accord with the distortion of shape functions as shown in figure 28.

Figure 31. Comparison of axial evolution of amplitudes of the low-frequency cross-flow waves with frequency 20 kHz and the high-frequency Mack-mode waves with frequency 125 kHz from QDNS and PSE3D. The PSE3D results based on the time-averaged profiles of the transitional state are not shown because they are almost indistinguishable from those based on the laminar state in this case.

3.5. Windward cross-flow region

In this subsection, we will consider boundary layer stability in the windward crossflow region. The stability calculations are performed with the same methodology as in the shoulder cross-flow region. The Mack-mode instability is found to be only marginally unstable, i.e. its growth rate is one order of magnitude smaller than the maximal growth rate of the cross-flow instability, hence its stability characteristics will not be discussed further. The cross-flow instability in this region closely resembles that in the shoulder cross-flow region but also exhibits some distinct features. Figure 32 shows the cross-flow instability spectra at $X^* = 1000$ mm. It is seen clearly that multiple unstable modes are present for each frequency; modes of each frequency occupy a certain band of phase velocity, and the band increases in value but shrinks in width with frequency. The most unstable frequency lies between 15 kHz and 25 kHz, with the maximal growth rate being approximately 15 m$^{-1}$, compared to the case of the shoulder cross-flow region, the peak frequency being lower and the peak growth rate being larger.

Figure 32. Spectra for the cross-flow instability with various frequencies at $X^* = 1000$ mm in the windward cross-flow region: ($a$) spectra in the $-\alpha _i^*$–$c$ plane and ($b$) spectra in the $c$–$f^*$ plane.

Next we examine the mode shapes of the cross-flow instability. Figure 33 compares the spatial structures of modes at a fixed frequency with increasing phase velocity. It can be seen that the travelling cross-flow wave lies far away from both the shoulder attachment line and the windward centreline, implying insensitivity to the azimuthal boundary conditions, as is also noted by Lakebrink et al. (Reference Lakebrink, Paredes and Borg2017). The isosurface of the temperature eigenfunction exhibits streaky structures that are inclined towards the windward vortex region. From the attachment line to the windward centreline, the local azimuthal wavelength slightly increases in accord with the growth of the boundary layer thickness, accompanied by a moderate decrease of wave angle. In comparison, the crossflow modal structure in the shoulder crossflow region does not show prominent azimuthal variations because the boundary layer thickness of the shoulder crossflow region possesses a plateau region. Moreover, compared to the crossflow mode of the same frequency in the shoulder crossflow region, the azimuthal wavelength is notably larger and the wave angle is smaller in the case of the windward crossflow region. With increasing phase velocity, the azimuthal wavelength increases and the wave angle decreases, but the spatial location does not change. This trend is somewhat different from the case of the shoulder crossflow region.

Figure 33. Isosurfaces of the normalized real part of the temperature eigenfunction ($\hat T = 0.5$) for three modes with the same frequency 20 kHz in the windward cross-flow region: ($a$) $c = 0.4$, $\beta = -18$, ($b$) $c=0.44$, $\beta = -14$, ($c$) $c = 0.49$, $\beta = -8$. Twice streamwise wavelengths of each mode are shown. The averaged azimuthal wavelength and the wave angles at two sides are also displayed.

The frequency effects on the mode shape are highlighted in figure 34, showing spatial structures of most unstable modes of three frequencies. With increasing frequency, the azimuthal wavelength decreases and the spatial distribution shifts towards the shoulder attachment line, similar to the case of the shoulder crossflow region. The wave angle also decreases as frequency increases.

Figure 34. Isosurfaces of the normalized real part of the temperature eigenfunction ($\hat T = 0.5$) for the most unstable modes of three frequencies in the windward crossflow region: ($a$) $f^* = 5$ kHz, $c = 0.20$, $\beta = -12$, ($b$) $f^*=20$ kHz, $c = 0.44$, $\beta = -14$, ($c$) $f^* = 40$ kHz, $c = 0.56$, $\beta = -17$. Twice streamwise wavelengths of each mode are shown. The averaged azimuthal wavelengths and the wave angles at two sides are also displayed.

The downstream evolution of cross-flow instability is studied with PSE3D. Like the case of the shoulder cross-flow region, the $N$-factor curve (not shown here) is very sensitive to the initial conditions. The maximal $N$-factor value is around 7 preceding the transition point for the frequency range 15–25 kHz. The spatial structure is relatively more robust to the initial conditions and is thus more helpful to characterize the cross-flow mode. Figure 35 shows the spatial evolution of the travelling cross-flow wave with frequency 15 kHz using isosurfaces. As the wave amplitude increases downstream, the number of streaks increases substantially, forming a triangle-shaped spatial distribution. In the direction away from the attachment line, the azimuthal wavelength increases while the wave angle decreases. Take the outlet location, for example: the azimuthal wavelength increases from around 7 mm to 16 mm, while the wave angle decreases from around 82$^\circ$ to 65$^\circ$.

Figure 35. The structure evolution from PSE3D for the cross-flow instability with frequency 15 kHz, illustrated by the real part of the temperature disturbance. The isosurface value is prescribed to be 10 times the initial value. The edge streamlines are also shown (blue lines) for reference.

Figure 36 shows the distributions of normalized r.m.s. amplitude of the temperature fluctuations from QDNS for several axial locations. The spectra calculated at the locations where the r.m.s. value is maximum are also displayed. The results indicate that low-frequency components arising from travelling cross-flow waves (peaking at approximately 17 kHz before the last location) dominate the transition process, which is consistent with the stability analyses results. High-frequency disturbances, likely originating in the Mack-mode instabilities, emerge after $X^* = 1000$ mm and amplify downstream due to the modulation of the cross-flow waves. The onset of turbulence is manifest at the last station, $X^* = 1300$ mm, where the base flow is significantly distorted, and the spectrum is filled up with a scaling law, $|T'|\sim f^{-4}$, describing the behaviour of the temperature fluctuation amplitude in the high-frequency region.

Figure 36. Normalized temperature fluctuation distributions from QDNS at four representative stations: (a) $X^* = 900$ mm, ($c$) $X^* = 1000$ mm, ($e$) $X^* = 1200$ mm and ($g$) $X^* = 1300$ mm, along with the temperature base flow. The sampling points are marked by $\bullet$ with the corresponding spectra being displayed in the right-hand column. Note that the r.m.s. in the vicinity of the windward centreline has been omitted. A zoom-in plot is displayed in ($d$) to highlight the high-frequency peak. A log-log plot is added in ($h$) to indicate the scaling law of the spectrum. The green and black triangles denote the locations of the attachment line and the windward centreline, respectively.

Figure 37 further compares the amplitude evolutions of the travelling cross-flow waves with frequency 15 kHz from QDNS and PSE3D results. Good agreement can be observed except for the initial transient stage and the late nonlinear saturation stage, where the linear amplification seems to be weakened by the mean flow distortion.

Figure 37. Comparison of streamwise amplitude evolutions of the low-frequency cross-flow waves with frequency 15 kHz from QDNS and PSE3D based on either the laminar state or the transitional state.

3.6. shoulder attachment line

The spatial eigenvalue spectrum for the shoulder attachment-line instability from BiGlobal calculations at $X^* = 1000$ mm is displayed in figure 38. In contrast to previous studies on the attachment-line instability where alternatively arranged symmetrical and antisymmetrical instabilities exist, only one branch of instability is identified in the present case and it no longer possesses perfect spanwise symmetry due to the symmetry breaking of base flow. The peak growth rate is slightly larger than that of the boundary layer instabilities in cross-flow regions, but smaller than that of the shear-layer instabilities in the vortex regions. Because of the small boundary layer thickness (${\sim}0.5$ mm here), this instability lies in an extremely high frequency range ($600\ {\rm kHz} < f^*<700\ {\rm kHz}$), which corresponds to a very small axial wavelength. Such high temporal and spatial oscillations are beyond the present numerical resolution. Therefore, the attachment-line boundary layer remains laminar throughout the whole model in the simulation. The real parts of the normalized temperature eigenfunctions of three representative modes, along with their spatial structures, are shown in figure 39. These eigenfunctions all exhibit dual-peak structures across the boundary layer, which is strong evidence of the Mack-mode instability. The smallest frequency mode resides almost exclusively in the windward side, and behaves like an oblique Mack mode with wave angle approximately $21^\circ$. The middle one, which is also the most amplified one of the branch, is distributed nearly equally in both the windward and leeward sides. Its windward-side part has wave angle approximately $14^\circ$, while its leeward-side part is less oblique, with the opposite wave direction. The largest frequency mode, mainly concentrating in the leeward side, is essentially planar. The attachment-line instability does not extend towards the leeward side as far as it does towards the windward side. This is likely due to the asymmetry of the boundary layer as shown in figure 39($g$). The boundary layer changes slowly in the vicinity of the $Y^*=0$ plane, but grows fast in both sides. Fast variations of boundary layer in the azimuthal direction likely inhibit the Mack-mode instability, as observed in the shoulder region. Moving away from the $Y^*=0$ plane, the boundary layer thickness in the leeward side increases faster than that does in the windward side, hence the Mack-mode instability is expected to decay faster in the leeward side than in the windward side.

Figure 38. Spatial spectrum of the shoulder attachment-line instability from BiGlobal calculations at $X^* = 1000$ mm. ($a$) Spatial growth rate as a function of frequency; ($b$) phase velocity as a function of frequency.

Figure 39. Real part of the normalized temperature eigenfunction from BiGlobal for three representative modes at $X^* = 1000$ mm: ($a$) 615 kHz, ($b$) 655 kHz, ($c$) 695 kHz. The corresponding spatial structures, including eight axial wavelengths, are shown in ($d$–$f$), illustrated by isosurfaces of real parts of temperature at value 0.5. The boundary layer in the vicinity of the attachment line is shown in ($g$).

The spatial evolutions of fixed frequency disturbances are evaluated with the help of PSE3D. Figure 40 displays the axial evolutions of $N$-factors for disturbance frequencies from 610 to 690 kHz. As the streamwise coordinate increases, the boundary layer thickens and the frequency of the Mack mode decreases. The mode with frequency 610 kHz achieves an $N$-factor of 15 near the rear part of the model, which could potentially lead to laminar–turbulent transition (Tu et al. Reference Tu, Wan, Chen, Yuan and Chen2019). Without loss of generality, we consider in detail the spatial structure evolution for the mode $f^* = 650$ kHz, whose $N$-factor first reaches 10. Figure 41($a$) depicts the whole disturbance structure illustrated by the isosurface of the real part of the temperature component. It can be observed that the attachment-line disturbances first manifest themselves in the windward side. This coincides with the above BiGlobal results since fixed-frequency disturbances tend to be located in the low-frequency part of the unstable branch in the upstream boundary layer, thereby being expected to appear in the windward side. Immediately downstream of their emergence, the attachment-line disturbances quickly form a nearly symmetrical crescent structure, as shown clearly in figure 41($b$), similar to the structure of the locally dominant mode. The slice in the $X^*$–$Z^*$ plane depicted in figure 41($c$) also exhibits a dual-peak disturbance structure, as expected. Further downstream, the disturbances gradually move towards the leeward side, as shown in figure 41($d$). This is again consistent with the BiGlobal results. Another important observation is that the disturbances virtually do not expand in the azimuthal direction as they travel downstream, likely because there is a fast increase of the boundary layer thickness away from the attachment line. This also suggests that the attachment-line instability may not interact with or excite boundary layer instabilities in the outboard region.

Figure 40. Streamwise evolution of $N$-factors of attachment-line instabilities from PSE3D for various frequencies from 610 kHz to 690 kHz.

Figure 41. Spatial evolution of the attachment-line wave with frequency 650 kHz from PSE3D. ($a$) The overall structure depicted by the isosurface of the real part of temperature perturbation; the isosurface value is prescribed to be 10 times the initial value. ($b$) Zoomed-in image of the spatial structure corresponding to the middle squared region in ($a$). ($c$) The normalized real part of the temperature disturbance in the $X^*$–$Z^*$ plane marked by the yellow line ($Y^*=0$). ($d$) Zoomed-in image of the spatial structure corresponding to the rear squared region in ($a$), with the isosurface value of 5000 times the initial value to highlight the feature of disturbance distribution.