4.1. Effect of the hump on primary stationary CFI

Comparisons between time- and spanwise-averaged spanwise meanflow profiles ( $\bar {w}$ ) and steady disturbance profiles ( $\langle \bar {w} \rangle _z$ , see (2.11)) between DNS and experiments by Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025) are shown in figure 6. Note that the total perturbation profiles from the experimental measurements are used here for comparison with the DNS data. For more details on the experimental data, the reader is referred to Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025, figure 5e). Figure 6(e) shows the chordwise evolution of wall-normal maximum of $\langle \bar {w} \rangle _z$ along the chord. Note that the experimental data along the chord are measured in $z$ – $y_E$ planes, where $z$ is parallel to the leading edge and $y_E$ is normal to the wing surface (at $x/c_x \approx 0.15$ ) in the reference configuration. This direction is held constant along the chord, making the measurement planes quasi-normal to the wing surface downstream, with the $y_E$ origin located on the wing surface.

Figure 6.

(a–d) Profiles of $\langle \bar {w} \rangle _z/w_e$ and time- and spanwise-averaged spanwise velocity component ( $\bar {w}/w_e$ ). (e) Chordwise evolution of wall-normal peak $\langle \bar {w} \rangle _z/w_e$ , for cases A1-C and A1-H. Note that experimental data are plotted along the $\eta _E$ direction, where $\eta _E=y_E/\delta ^*_{{\textit{ref}}}$ . Black and red dashed lines in (e) show the peak value of $\langle \bar {w} \rangle _z/w_e$ obtained from the simulations without unsteady disturbances in the domain for cases A1-C and A1-H, respectively. The grey shaded area shows the region where skin friction starts to increase in DNS for case A1-C (cf. figure 11). Note that DNS and experimental profiles are normalised by local spanwise velocity component in the free stream obtained in DNS and experiments, respectively.

From figure 6, it is clear that both the spanwise-averaged meanflow and primary CF disturbance profiles (figure 6 a–d) and the growth (figure 6 e) in DNS match very well with the experiment. It should be noted that small differences between the profiles obtained by DNS and experiment exist, and are likely caused by a combination of the measurement planes in DNS being normal to local curvature of the wall and the measurement planes in the experiments being quasi-normal to the wall, and also the use of an idealised hump shape for DNS as explained in § 2.2. However, as is shown in later sections, these minor differences do not affect the general conclusions of this work.

Figure 7.

Steady disturbance profiles of chordwise $(u_{\xi })$ , wall-normal $(v_{\eta '})$ and spanwise $(w)$ components for cases A1-C (black) and A1-H (red) in DNS. Note that the disturbance profiles are calculated based on stationary perturbation field, i.e. $\boldsymbol{u}^{\prime}_s$ .

Figure 6(e) clearly shows the stabilisation of the stationary CF disturbances downstream of the hump in both the experiments and the DNS. To better investigate the influence of the hump on the development of CFVs, figure 7 compares the steady disturbance profiles (calculated using stationary perturbation fields $\boldsymbol{u}^{\prime}_s$ from DNS) for all three velocity components ( $\langle {u}^{\prime}_{\xi ,s} \rangle _z$ , $\langle {v}^{\prime}_{\eta ',s} \rangle _z$ and $\langle {w^{\prime}_s} \rangle _z$ ) between cases A1-C and A1-H. A notable effect of the hump is on the shape of the spanwise and chordwise components of the disturbance in the vicinity of the hump. As the flow passes over the hump, it experiences an APG, which results in thickening of the boundary layer and increase of integral length quantities, as shown in figure 3. The effect is evident on the disturbance profiles slightly downstream of the apex of the hump (figure 7Ia,Ib), where the wall-normal location of peak $\langle {w^{\prime}_s} \rangle _z$ and $\langle {u}^{\prime}_{\xi ,s} \rangle _z$ moves away from the wall compared with the reference case. As shown earlier, CF reversal occurs for $0.155\lt x/c_x\lt 0.178$ in the hump case due to the consequent acceleration–deceleration events. In this region, the disturbance profiles ( $\langle {w^{\prime}_s} \rangle _z$ and $\langle {u}^{\prime}_{\xi ,s} \rangle _z$ ) in the hump case become highly distorted compared with the reference case. The disturbance profiles within this region show a double-peak profile, as shown in figure 7(Ib) for $x/c_x=0.172$ . At this station, the stronger peak is located farther away from the wall. Slightly downstream of the CF reversal region (figure 7Ic), the disturbance profiles remain highly distorted compared with the reference case, until $x/c_x\approx 0.19$ where they are approximately recovered to the shape of the reference case. At $x/c_x=0.2$ (figure 7IIa), the wall-normal peak value of $\langle \bar {u}_{\xi } \rangle _z$ is slightly higher in the hump case compared with the reference case. Interestingly, for $x/c_x \gtrsim 0.20$ , a strong stabilisation in the primary CF perturbations occurs in the hump case, as clearly seen in figures 6 and 7(IIb–d). This long-lasting and far-downstream effect of the hump on growth of stationary CF perturbations provides important insights to the core contribution of the hump to the observed transition delay in experiments and DNS (as shown in § 4.3). In the following sections, the physical mechanisms responsible for this stabilisation are further explored.

4.2. Effect of the hump on harmonics of the primary stationary CF perturbation

Spanwise harmonic modes of primary stationary CF perturbation are calculated using spanwise Fourier transform of chordwise perturbation profiles $u^{\prime}_{\xi ,s}$ using (2.8).

The amplitudes of the first four modes, i.e. $A_{\tilde {u}_\xi ,n\beta _0} \,(n \in [1,4])$ , for $0.06\lt x/c_x\lt 0.50$ are plotted in figure 8(a), where the black and red lines correspond to cases A1-C and A1-H, respectively. This figure reveals that the trend in chordwise evolution of the amplitude of the fundamental spanwise mode ( $1\beta _0$ ) is strongly affected in the vicinity of the hump. However, the amplitude itself does not exceed that of the reference case, except slightly upstream of the hump’s apex and for $0.19\lt x/c_x\lt 0.21$ . For $x/c_x\gt 0.21$ , the fundamental mode appears stabilised compared with the reference case. In contrast, higher harmonics behave notably differently in the vicinity of the hump, with a strong increase in their amplitude just downstream of the hump’s apex. Interestingly, after the localised amplification of higher harmonics, their amplitude subsequently decays to well below the corresponding reference case for more downstream locations. It is worth mentioning that the amplification of higher harmonics starts almost at the same location as the CF reversal occurs, and continues until $x/c_x \approx 0.185$ , which is just downstream of the end of the CF reversal region at $x/c_x \approx 0.178$ . Unlike the higher harmonic modes, the fundamental mode $1\beta _0$ behaves differently in the CF reversal region, and its amplitude slightly decreases in that region, as can be seen in the inset of figure 8(a).

Figure 8.

(a) Chordwise evolution of amplitude of the fundamental mode $1\beta _0$ and first three higher harmonics ( $2\beta _0$ – $4\beta _0$ ) of $u^{\prime}_{\xi ,s}$ for cases A1-C (black) and A1-H (red). (b) Ratio of spatial growth rate (cf. (2.10)) of fundamental mode $1\beta _0$ of hump to reference case calculated by chordwise ( $\tilde {u}_\xi$ ) and spanwise ( $\tilde {w}$ ) Fourier coefficients. Subscripts $c$ and $h$ stand for clean (reference) and hump cases, respectively.

The ratio between the spatial chordwise growth rate (calculated using (2.10)) between cases A1-H and A1-C for the fundamental stationary mode $(\sigma _h/\sigma _c)_{1\beta _0}$ is shown in figure 8(b). It is evident that the growth rate of the fundamental mode downstream of the hump’s apex decreases compared with that of the reference case. In this region ( $0.151\lt x/c_x\lt 0.174$ ), the flow experiences a strong APG (see figure 3 b), which could explain the local stabilisation of the fundamental CF mode. Thus, the fundamental mode in this region is mostly affected by the effects of the pressure gradient, towards stabilising it. However, as soon as the flow experiences the enhanced FPG at $x/c_x=0.174$ , the growth rate of the fundamental mode increases in the hump case. The growth rate then starts to recover to the reference case value, until $x/c_x \approx 0.2$ where $(\sigma _h/\sigma _c)_{1\beta _0} \approx 1$ . More downstream for $0.2\lt x/c_x\lt 0.26$ , the chordwise growth rate of the fundamental mode in the hump case becomes less than the growth rate in the reference case, i.e. $(\sigma _h/\sigma _c)_{1\beta _0} \lt 1$ , which is also observable in figure 6(e). For $x/c_x\gt 0.2$ , the pressure gradient is almost recovered to its values as for the reference case (figure 3 b). Thus, the lower growth rate in the hump case compared with the reference case for $x/c_x\gt 0.2$ , which results in the far-downstream stabilising effect of the hump, can no longer be directly attributed to the local pressure gradient effects. These observations suggest that stationary CF perturbations passing over the hump may undergo topological changes due to the upstream pressure-gradient changeover from FPG to APG. These changes persist far downstream of the hump and lead to a reduced growth rate of CF perturbations compared with that in the reference case, as suggested by Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025). This positive interaction mechanism is investigated in the following section.

4.2.1. Effect of the hump on the topology of primary stationary CF perturbations

Figure 9 depicts the topology of the stationary CF perturbations ( $u_\xi$ component) at selected chordwise stations, for cases A1-C and A1-H. Note that the $z^*$ coordinate is shifted relative to the $z$ direction to align the CFVs across different chordwise stations. As expected for the reference case, the shape and topology of perturbations are almost similar and only their amplitude increases in more downstream locations. However, the perturbations in the hump case undergo severe topological changes. At $x/c_x=0.17$ (figure 9IIa,IVa,VIa) where the CF reversal occurs, $W_{\textit{cf}}=0$ is indicated by the green dashed line. Note that within the CF reversal region, a new inflection point appears in the CF velocity component profile, which might give rise to new inflectional primary instabilities, as shown by Westerbeek et al. (Reference Westerbeek, Casacuberta and Kotsonis2026). Interestingly, at this station for the hump case, CF perturbations near the wall experience a shear in the direction of the reversed CF velocity component (which is from − $z^*$ to + $z^*$ direction), and the perturbations are tilted against their original tilting. This is clearly visible by the zero-phase isolines (i.e. $\phi =\arctan (\operatorname {Im}{(\hat {u}_{\xi })}/\operatorname {\textit{Re}}{(\hat {u}_{\xi })} )=0)$ in figure 9 for the fundamental (row IV) and the first harmonic modes (row VI). The same topology change in the stationary CF perturbations has been observed in the experiments of Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025), and in numerical simulations of Wassermann & Kloker (Reference Wassermann and Kloker2005) investigating the effect of pressure-gradient changeover from FPG to APG. The aforementioned works point towards the formation of a new structure near the wall due to CF reversal (due to pressure-gradient changeover from FPG to APG), and possibly a new stationary inflectional instability which becomes unstable within that region (Westerbeek et al. Reference Westerbeek, Casacuberta and Kotsonis2026). Slightly downstream of the CF reversal region in case A1-H, as shown in figure 9(b) at $x/c_x=0.18$ , the fundamental mode near the wall is still oriented in the direction against the original orientation of CF perturbation due to inertia and upstream history of the flow, but the tilting is less than previous stations.

Figure 9.

(a–d) Chordwise velocity component of total CF perturbation ( $u^{\prime}_{\xi ,s}=\bar {u}_{\xi }-u_{\xi ,0}$ ) (rows I, II), fundamental mode (rows III, IV) and first higher harmonic mode (rows V, VI). Rows (I, III, V) correspond to case A1-C and rows (II, IV, VI) to case A1-H. The black lines show nine equally spaced isolines of the time-averaged chordwise velocity component in the interval of $[0,0.95u_\xi ^{\textit{max}}]$ . Green dashed line at $x/c_x=0.17$ (rows II, IV, VI) denotes $W_{\textit{cf}}=0$ for the hump case. The black dashed lines show constant zero-phase ( $\phi =0$ ) isolines of perturbations. Panels are not drawn to scale.

Figure 10.

(a) Constant-phase ( $\phi =0$ ) isolines of fundamental mode $\hat {u}_{\xi ,1\beta _0}$ for cases A1-C (black) and A1-H (red) at several chordwise locations. (b) Difference between $z$ coordinate of zero-phase isolines of hump and reference cases close to the wall ( $\Delta z^*$ ). Contour of the integrand of the total production term calculated for fundamental mode for case (c) A1-C and (d) A1-H. (e) Chordwise evolution of the wall-normal integral of the integrand of the production term ( $\mathcal{I}_{\mathcal{P}_{1\beta _0}}$ ) for case A1-C and A1-H. Here $\mathcal{I}_{\mathcal{P}_{1\beta _0}}=0$ and $W_{\textit{cf}}=0$ are indicated by green and black dashed contour lines in (d), respectively. The vertical black dashed lines in (b,e) show the boundaries of the CF reversal region, while the hump region is shaded in orange, and the orange vertical solid line marks its apex location. Panels (c,d) are not drawn to scale.

To qualitatively characterise how the tilting of stationary CF perturbations changes as the boundary-layer flow advects over the hump, constant-phase isolines $(\phi =0)$ of $\hat {u}_{\xi ,1\beta _0}$ at several locations are compared between the reference and hump cases. For this comparison, the primary mode isolines for the hump case are shifted in the $z$ direction such that the $z$ coordinate of the zero-phase isoline in the free stream matches that of the reference case. The shifted $z$ coordinates for reference and hump cases are denoted by $z_c^*$ and $z_h^*$ , respectively. The results are shown in figure 10(a). The difference between the $z$ coordinates of the phase profiles at the first grid point off the wall, i.e. $\Delta z^*=(z^*_h-z^*_c)_{(\phi =0,\eta '=0^+)}/\lambda _z$ , is depicted in figure 10(b). As evident in figure 10(a,b), the tilting of the classical shape of the CF perturbation occurs in the hump case over and downstream of the hump. From the initiation of the CF reversal region, $\Delta z^*$ increases until it reaches its peak value at approximately the chordwise location of maximum CF reversal. Downstream of that point, $\Delta z^*$ decreases as the peak value of the CF reversal reduces, and, consequently, the CF perturbation begins to recover its original shape. Notably, downstream of $x/c_x=0.2$ where the pressure gradient is almost recovered to the value of the reference case and $\Delta z^*\approx 0$ , the structure (and orientation) of the stationary CF perturbation continues to change, and only recovers to reference case conditions at $x/c_x \approx 0.25$ . By comparing figures 10(b) and 8, it is clear that downstream of the hump, where $\Delta z^*$ is decreasing, the chordwise growth rate in the hump case decreases until it becomes lower than that in the reference case, i.e. $ (\sigma _h/\sigma _c)_{1\beta _0}\lt 1$ .

4.3. Effect of the hump on secondary CFI modes and laminar breakdown

Figure 11.

(a) Time- and span-averaged wall skin-friction coefficient for cases A1-C and A1-H. The blue and grey shaded areas indicate the approximate transition region in the experiment of Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025, figure 2) and in DNS for the reference case, respectively. Time-averaged skin-friction coefficient for cases (b) A1-C and (c) A1-H in the range of $[0{-}3]\times 10^{-3}$ . The orange shaded area in (a) shows the extent of the hump, while the orange solid line marks the location of the hump’s apex. Note that, in (b,c) two duplicate spanwise periods of the simulation domain are used for ease of visualisation. Panels (b,c) are not drawn to scale.

Figure 11(a) depicts the time- and spanwise-averaged wall skin-friction coefficient $C_{\kern-1.5pt f}$ along the chord for cases A1-C and A1-H, whereas figure 11(b,c) depicts the time-averaged skin-friction coefficient. The blue shaded area indicates the approximate transition region measured in the experiments of Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025) for the reference case, which shows a close agreement with the transition region in the DNS for case A1-C (grey shaded region in figure 11 a). Note that the transition region in the DNS is identified from figure 11(a), where $ C_{\kern-1.5pt f}$ begins to increase (see the grey shaded region in figure 11 a). In case A1-H, in agreement with the experiments, no transition to turbulence is observed in the simulation domain for $x/c_x\lt 0.69$ , which shows the success of the hump in delaying the transition for the present conditions.

Figure 12 shows the intensity of temporal fluctuations (i.e. standard deviation) of $u_\xi$ for case A1-C (row I) and A1-H (row II) at several selected $z{-}\eta$ planes along the chord. Time signals are collected at a fixed sampling frequency of 50 kHz from the simulations, and total numbers of 4352 and 3584 samples are collected for reference and hump cases, respectively. Initially, the stationary CF perturbations are relatively weak, causing only slight distortion of the boundary layer, and fluctuations can only be seen close to the wall. At $x/c_x=0.2$ in the hump case (figure 12IIa), the amplitude of fluctuations close to the wall is noticeably higher compared with the reference case (figure 12Ia); however, still very weak ( ${\approx} 0.4 \,\%$ of $Q_{{\textit{ref}}}$ ). Further downstream, the boundary layer becomes highly distorted as stationary CF perturbations become stronger. At these stations, in case A1-C, at the inner side of upwelling region of the CFV (IU in figure 12Ib,c), strong fluctuations can be observed; however, for case A1-H (figure 12IIb,c) these are much weaker. At $x/c_x=0.4$ , velocity fluctuations in case A1-C are getting stronger at the outer side of the upwelling region of the CFV (OU in figure 12Ic). Also, weaker fluctuations can be observed at the upper part of the CFV and away from the wall. However, in case A1-H (figure 12IIc), no notable fluctuations are observed either at the outer side of the upwelling region of the vortex, or at its upper part away from the wall. Finally, the boundary-layer flow becomes transitional in case A1-C (figure 12Id), as indicated by the increase in wall friction coefficient shown in figure 11(a). However, for case A1-H, flow remains laminar with no significant sign of velocity fluctuations in regions rather than close to the wall (figure 12IId).

Figure 12.

(a–d) Temporal fluctuations of chordwise velocity component ( $u_{\xi ,{\textit{STD}}}/Q_{{\textit{ref}}}$ ) for case A1-C (row I) and case A1-H (row II). The white lines show nine equally spaced isolines of the time-averaged chordwise velocity component in the interval of $[0,0.95u_\xi ^{\textit{max}}]$ . Panels are not drawn to scale.

The presence of high velocity fluctuations at the outer side of the upwelling region of the CFV and away from the wall are indicators of type I ( $z$ -type) and type II ( $y$ -type) (Malik et al. Reference Malik, Li and Chang1996, Reference Malik, Li, Choudhari and Chang1999; Wassermann & Kloker Reference Wassermann and Kloker2003) secondary CFI modes, respectively. Thus, the observed differences between velocity fluctuations in the two cases, especially on the outer side of the upwelling region of the CFV, suggest that in case A1-H, the interaction between the hump and the stationary CFVs leads to a substantial modification in the development of high-frequency secondary CFI modes. In the following section, the development of secondary CFI modes in both cases is analysed in detail.

4.3.1. Spatial organisation of secondary instabilities

To segregate the different types of secondary CFI modes which develop in the present flow, spectral proper orthogonal decomposition (SPOD) (Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018) is used. This method aids in isolating the temporally and spatially correlated coherent structures from the unsteady simulation data. For case A1-C, 4352 three-dimensional snapshots of all three velocity components, i.e. $u_\xi ,\ v_{\eta '}$ and $w$ (collected at a sampling frequency of 50 kHz), for $0.355\lt x/c_x\lt 0.468$ (slightly upstream of transition location) are divided into 16 blocks (with 50 % overlap), resulting in a frequency bin ( $\Delta f$ ) equal to 97.7 Hz. For case A1-H, 3584 snapshots within the range $0.43\lt x/c_x\lt 0.6$ are divided into 13 blocks, with the same overlap and frequency bin as for the reference case. Figure 13 shows the leading SPOD mode energy spectrum. The chosen number of snapshots, blocks and overlaps of the blocks leads to converged SPOD spectra for both cases. The low-frequency parts of the spectrum in both cases are very similar and show high energy in the frequency band of ${\textit{BP}}_{\textit{III}} \in [350{-}750]$ Hz. However, for the energy associated with high-frequency parts of the spectrum, significant differences can be observed. In the reference case, two clear regions of high energy can be observed in frequency bands of ${\textit{BP}}_{I} \in [4000{-}5500]$ Hz and ${\textit{BP}}_{\textit{II}} \in [7000{-}8500]$ Hz. The increase in energy can be observed in the hump case for ${\textit{BP}}_{\textit{II}} \in [7000{-}8500]$ , although it is much less pronounced.

Figure 13.

Leading SPOD mode spectra (solid lines) for cases A1-C and A1-H. The dashed lines represent the sum of the energy of all SPOD modes for each frequency. The purple shaded areas indicate frequency bands of ${\textit{BP}}_{\textit{III}}\in [350{-}750]$ , ${\textit{BP}}_{I}\in [4000{-}5500]$ and ${\textit{BP}}_{\textit{II}}\in [7000{-}8500]$ Hz.

The dominant frequency bands obtained by SPOD are used to filter the velocity fluctuations to identify the secondary instability mode corresponding to each frequency band. To do this, following Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), chordwise velocity fields ( $u_\xi$ ) are band-pass-filtered for both the reference and hump cases using a zero-phase eighth-order Butterworth filter, where the three frequency bands found using SPOD are used.

Figure 14.

Contours of normalised root mean square of band-pass-filtered chordwise velocity fluctuation fields for (a) case A1-C: ${\textit{BP}}_{\textit{III}}$ (I), ${\textit{BP}}_{I}$ (II), ${\textit{BP}}_{\textit{II}}$ (III) at $x/c_x=0.4$ , and $f\gt 12\,000$ Hz at $x/c_x=0.458$ (IV) corresponds to high-frequency fluctuations. For (b) case A1-H: ${\textit{BP}}_{\textit{III}}$ (I), ${\textit{BP}}_{I}$ (II), ${\textit{BP}}_{\textit{II}}$ (III) at $x/c_x=0.55$ . The green lines show isolines of $0.6(\partial u_{\xi } / \partial z)^{\textit{max}}$ in (Ia,Ib), $0.4(\partial u_{\xi } / \partial z)^{min}$ in (IIa,IVa) and $0.05(\partial u_{\xi } / \partial \eta ')^{\textit{max}}$ in (IIIa,IIb,IIIb). The superscripts ‘max’ and ‘min’ denote the extrema at each respective location. The white lines show nine equally spaced isolines of the time-averaged chordwise velocity component in the interval of $[0,0.95u_\xi ^{\textit{max}}]$ . Two repeat spanwise periods of the simulation domain are used for visualisation. Panels are not drawn to scale.

Figure 14 shows the root mean square of band-pass-filtered $u_\xi$ fluctuation field (i.e. $u^{\prime}_{\xi ,BP}$ ), for case A1-C at $x/c_x=0.4$ (figure 14Ia,IIa,IIIa) and case A1-H at $x/c_x=0.55$ (figure 14Ib,IIb,IIIb). To better visualise the spatial structure of the fluctuations, each panel is normalised by its respective maximum value of the velocity fluctuation. From figure 14(Ia) it is clear that for the reference case, the velocity fluctuations corresponding to the low-frequency mode ( ${\textit{BP}}_{\textit{III}}$ ) are located at the inner side of upwelling region (IU in figure 14Ia) of the CFV, and corresponds to type III secondary CFI mode (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). This region coincides with the region of local maximum spanwise shear of the flow, as shown in figure 14(Ia) by the green solid line (i.e. $0.6(\partial u_{\xi } / \partial z)^{\textit{max}}$ ). The velocity fluctuations corresponding to ${\textit{BP}}_{I}$ are located at the outer side of the upwelling (OU in figure 14IIa) region of the CFV, and corresponds to type I secondary CFI mode. This region coincides with the region of local minimum of spanwise shear, as shown in figure 14(IIa) by the green solid line (i.e. $0.4(\partial u_{\xi } / \partial z)^{min}$ ). Finally, the velocity fluctuations corresponding to ${\textit{BP}}_{\textit{II}}$ are located at the top of the CFV, and are associated with type II secondary CFI. It appears in the region of local maximum of wall-normal shear, as shown in figure 14(IIIa) by the green solid line. Under similar (though not identical) free-stream conditions and for the same wing model, Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) identified frequency bands of (5000–6000), (7000–8000) and (350–550) Hz for type I, type II and type III secondary CFI modes, respectively. The frequency bands determined using SPOD in the present study align closely with the identified frequency bands obtained in the experiments of Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). To identify the mode responsible for the onset of the laminar–turbulent transition in the reference case, following Rius-Vidales & Kotsonis (Reference Rius-Vidales and Kotsonis2022), velocity fields are high-pass-filtered, with a cutoff frequency of $f_c=12$ kHz. The root mean square of high-pass-filtered $u_\xi$ fluctuation field is depicted in figure 14(IVa) at $x/c_x=0.458$ , for case A1-C. The presence of very high-frequency fluctuations (indicative of turbulent fluctuations) in the outer part of the upwelling region of the CFV, coinciding with a local region of minimum spanwise gradients, suggests that the breakdown of type I secondary CFI mode triggers the transition to turbulence in case A1-C. This is in line with the previous measurements on the same swept-wing model, under similar (though not identical) flow conditions (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016).

Due to the strong stabilising effect of the hump and the resulting delay in the transition process, in case A1-H, the topology of secondary CFI modes is qualitatively inspected at a more downstream location of $x/c_x=0.55$ . In the hump case, the fluctuations corresponding to ${\textit{BP}}_{\textit{III}}$ are located at the inner region of the upwelling part of the CFV, indicative of type III secondary CFI mode. However, the fluctuations corresponding to both ${\textit{BP}}_{I}$ and ${\textit{BP}}_{\textit{II}}$ in the hump case are located at the top of the vortex and away from the wall. This suggests that only type II and type III secondary CFI modes are present in the hump case, and type I secondary instability appears to be stabilised as a result of the interaction of the CFVs with the hump.

4.3.2. Chordwise evolution of secondary instabilities

To obtain the local amplitude for each secondary CFI mode and trace the evolution of the amplitudes along the chord, the method outlined in White & Saric (Reference White and Saric2005), Downs & White (Reference Downs and White2013) and Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) is used. The amplitude corresponding to each secondary CFI mode ( $A_{\textit{II}}^{\textit{BP}}$ ) is calculated as

(4.1) \begin{equation} A_{\textit{II}}^{\textit{BP}}=\frac {1}{\bar {\delta }}\frac {1}{\lambda _z} \int _{0}^{\bar {\delta }} \int _{0}^{\lambda _z} t_{\textit{rms}}\left\{\frac {u^{\prime}_{\xi ,BP}}{Q_{{\textit{ref}}}}\right\}\,{\rm d}z\,{\rm d}y. \end{equation}

Here, $\bar {\delta }$ and $\lambda _z$ are the local boundary-layer thickness (obtained from time- and spanwise-averaged meanflows) and the spanwise size of the domain ( $7.5$ mm), respectively, and $t_{\textit{rms}}\{\dots\}$ is the temporal root-mean-square operator. Figure 15(a–c) shows the amplitudes of individual secondary CFI modes, while figure 15(d) presents the amplitude of the high-frequency fluctuations, computed using (4.1) for cases A1-C (black) and A1-H (red).

Figure 15.

(a–c) Chordwise evolution of amplitudes of band-pass-filtered and (d) high-pass-filtered temporal chordwise velocity $(u_{\xi })$ fluctuations for case A1-C and case A1-H, computed using (4.1). The grey shaded area corresponds to the grey area shown in figure 11(a) and indicates the transitional and breakdown regions for case A1-C in DNS. The extent of the hump is shaded in orange, the vertical orange solid line marks the location of the hump’s apex and the vertical black dashed lines mark the boundaries of the CF reversal region. In (b–d), the solid black and red lines start from the neutral point of the secondary CFI modes obtained by DNS.

Starting with the low-frequency type III mode ( ${\textit{BP}}_{\textit{III}}$ ; figure 15 a), it is clear that the amplitude of this mode increases until saturation in the reference case. For the hump case, the growth rate of fluctuations corresponding to ${\textit{BP}}_{\textit{III}}$ increases significantly from the beginning of the CF reversal region (marked by the black vertical dashed line in the figure). Consequently, $A_{\textit{II}}^{BP_{\textit{III}}}$ exceeds that of the reference case up to $x/c_x \approx 0.25$ . In § 4.1, it was shown that within the CF reversal region in the hump case, the amplitude of fundamental stationary CF perturbation decreases slightly due to the strong APG. However, a strong overshoot in the amplitude of higher harmonics was observed. The same behaviour as the growth of higher harmonics of the fundamental mode is seen in the evolution of the fluctuations corresponding to ${\textit{BP}}_{\textit{III}}$ in the hump case. Furthermore, the tilting of stationary primary CF modes and the appearance of an additional inflection point in local $W_{\textit{cf}}$ profiles within the CF reversal region are shown to lead to the existence of a new stationary primary instability (Westerbeek et al. Reference Westerbeek, Casacuberta and Kotsonis2026). Analogously, new low-frequency secondary instability modes (structurally similar to the typical type III mode) can similarly be expected to become destabilised within the CF reversal region. While the destabilised type III, and, possibly, the new low-frequency secondary instability modes in the present case do not appear to directly influence the eventual laminar breakdown, a more detailed analysis of this hypothesis (using, for example, local stability analysis) is needed, which is beyond the scope of the present study.

In contrast to the type III mode, chordwise evolution of type I ( ${\textit{BP}}_{I}$ ; figure 15 b) and type II ( ${\textit{BP}}_{\textit{II}}$ ; figure 15 c) secondary CFI modes follows a different scenario. In the reference case A1-C, each mode is amplified from its corresponding neutral point, i.e. type I at $x/c_x \approx 0.32$ and type II at $x/c_x \approx 0.31$ . Note that the neutral point for secondary CFI modes is approximated as the chordwise location where the mode amplitude begins to increase (i.e. the starting point of the solid lines in figure 15 b,c). After that, the amplitudes of both modes start to increase, until transition to turbulence occurs due to the breakdown of type I secondary instability as shown in figure 14(IVa). In the hump case, the amplitudes of fluctuations corresponding to ${\textit{BP}}_{I}$ and ${\textit{BP}}_{\textit{II}}$ start to increase from much more downstream, i.e. $x/c_x\approx 0.38$ , and remain very small throughout the simulation domain. Consequently, transition to turbulence does not occur for case A1-H. Note that in the hump case, as shown in figure 14, the classical type I secondary instability mode appears to be stabilised, and the amplitude for both ${\textit{BP}}_{I}$ and ${\textit{BP}}_{\textit{II}}$ corresponds to the type II secondary CFI mode.

The weakening of the type I secondary CFI mode can be explained by investigating the peak negative spanwise gradients along the chord, as this secondary CFI mode is driven by the spanwise shear (Malik, Li & Chang Reference Malik, Li and Chang1996). Figure 16 shows the maximum and minimum values of spanwise gradients of $u_\xi$ for cases A1-C (black) and A1-H (red). The spanwise gradients in the hump case are in general weaker than in the reference case, except downstream of the hump for $0.18 \lesssim x/c_x \lesssim 0.21$ . The weaker gradients in the hump case are clearly visible by comparing the peak derivative depicted by the dashed lines, which show the gradients in the absence of the unsteady noise in the simulation, and are not affected by the growth of secondary CFI modes in the flow. For $0.2\lesssim x/c_x \lesssim 0.22$ , the peak gradients do not increase in case A1-H. The relatively large plateau region in the spanwise gradients ( $\Delta x/c_x \approx 0.05$ ) results in significantly lower spanwise shear over the wing in the hump case compared with the reference case. This plateau region in case A1-H coincides with the region where stationary primary CFI experiences a reduced growth rate compared with the reference case (cf. figure 8 b).

Figure 16.

(a) Peak positive and (b) peak negative spanwise shear of time-averaged $u_{\xi }$ component of velocity for cases A1-C (black) and A1-H (red). The grey shaded area corresponds to the grey area shown in figure 11(a) and indicates the transitional and breakdown regions for case A1-C in DNS. The black and red dashed lines show the same gradients (as for the solid lines in each panel) from the simulations without unsteady noise in the simulations. The orange shaded area denotes the extent of the hump, the vertical orange solid line marks the location of the hump’s apex and vertical black dashed lines indicate the boundaries of the CF reversal region.

Overall, the underlying hump-driven mechanisms leading to transition delay can be summarised as follows. Within the CF reversal region downstream of the hump apex, the CF perturbation begins to tilt near the wall. This tilting of the CF perturbation persists even downstream of the hump, where the pressure gradient has recovered to the reference case value. In the region where the CF perturbation is regaining its reference organisation, the lift-up mechanism is weakened. Consequently, the energy transfer from the base flow to the perturbation field becomes less effective compared with the reference case (cf. figure 10). As a result, the primary CFI exhibits a reduced chordwise growth rate. In the same region, the spanwise gradients within the boundary layer do not increase. Consequently, the neutral point of the high-frequency secondary CFI modes shifts further downstream relative to the reference case. This shift prevents these fluctuations from reaching sufficiently high amplitudes to trigger the laminar–turbulent transition for the present conditions.

5.1. Effect of the hump on primary stationary CF perturbations

Steady disturbance profiles $\langle \bar {w} \rangle _z/w_e$ , obtained by DNS and total perturbation fields from the experiments are compared in figure 18, confirming also their close agreement as in the low-amplitude case. For case A2-H, the amplitude of stationary primary perturbations increases significantly compared with case A2-C downstream of the hump for $x/c_x \gtrsim 0.178$ , until it reaches its maximum at $x/c_x \approx 0.2$ . In an artificial scenario when the unsteady noise is not present in the boundary layer, the amplitude of stationary CF perturbation in case A2-H starts to decrease from almost the same location that stabilisation in case A1-H was observed, i.e. $x/c_x\approx 0.2$ , as shown by the red dashed line in figure 18(e). Thus, even in case A2-H and in the absence of unsteady noise, the stationary CF perturbation is stabilised by the hump, relative to case A2-C. This confirms the observations of Westerbeek et al. (Reference Westerbeek, Casacuberta and Kotsonis2026), stemming from purely stationary harmonic Navier–Stokes simulations on a similar flow. However, in the presence of the unsteady noise, which is the experimental condition, transition to turbulence in the hump case appears to occur prior to any significant stabilisation of stationary CF perturbations being registered. It must be further noted here that in both cases A2-C and A2-H with the presence of unsteady noise in figure 18(e), a strong reduction of the amplitude of the stationary CF perturbation is evident downstream of the transition location. This is not to be taken as actual stabilisation of the primary CFI, but rather a reduction of the stationary spanwise modulation of the flow due to the ensuing turbulent mixing.

Figure 18.

(a–d) Profiles of $\langle \bar {w} \rangle _z/w_e$ and time- and spanwise-averaged spanwise velocity component ( $\bar {w}/w_e$ ). (e) Chordwise evolution of wall-normal peak $\langle \bar {w} \rangle _z/w_e$ for cases A2-C and A2-H. Note that experimental data are plotted along the $\eta _E$ direction, where $\eta _E=y_E/\delta ^*_{{\textit{ref}}}$ . Black and red dashed lines in (e) show the peak value of $\langle \bar {w} \rangle _z/w_e$ obtained from the simulations without unsteady disturbances in the domain for cases A2-C and A2-H, respectively. The grey shaded areas show the region where skin friction starts to increase in DNS for cases A2-C and A2-H (cf. figure 17). Note that DNS and experimental profiles are normalised by the local spanwise velocity component in the free stream obtained in DNS and experiments, respectively.

Figure 19 compares all three components of steady disturbance profiles ( $\langle {u}^{\prime}_{\xi ,s} \rangle _z$ , $\langle {v}^{\prime}_{\eta ',s} \rangle _z$ and $\langle {w^{\prime}_s} \rangle _z$ ) obtained with DNS for cases A2-C and A2-H at several stations. For case A2-C, the profiles are qualitatively similar to those for case A1-C (see figure 7). Nevertheless, the secondary lobe in the profiles signifying the rise of nonlinear interactions appears notably earlier than in case A1-C. For case A2-H, perturbation profiles lack a distinct double-peak structure within the CF reversal region (unlike case A1-H; cf. figure 7). For $x/c_x\gt 0.18$ , the maximum amplitude of the chordwise and spanwise disturbance profiles in case A2-H significantly exceeds that of case A2-C, contrasting with the trends observed for cases A1-C and A1-H in figure 7.

Figure 19.

(a–d) Steady disturbance profiles for chordwise ( $u_{\xi }$ ), wall-normal ( $v_{\eta '}$ ) and spanwise ( $w$ ) velocity components for case A2-C (black) and case A2-H (red) in DNS. Note that the disturbance profiles are calculated based on stationary perturbation field, i.e. $\boldsymbol{u}^{\prime}_s$ .

Figure 20(a–d) compares the chordwise fundamental disturbance profiles, i.e. $\langle \tilde {u}_{\xi ,1\beta _0}\rangle _z$ , for cases A1-H and A2-H with the disturbance profile obtained by a linear simulation for the hump case. Note that, to enable comparison of disturbance shapes, each profile is normalised by its respective wall-normal peak value. The relative growth of the amplitude of fundamental mode ( $A_{\tilde {u}_{\xi ,1\beta _0}}$ ) compared with the amplitude at $x/c_x=0.1$ is shown in figure 20(e). The linear evolution of the stationary CF perturbation is obtained by solving the linearised Navier–Stokes equations, where $\boldsymbol{u}_0$ is used as the base flow. At the inlet of the domain for the linear simulation, the same fundamental CFI mode $(1\beta _0)$ as used for the nonlinear DNS of case A1-H is imposed. Although the amplitude of the perturbation in case A2-H is larger than that in case A1-H, the amplitude of the first higher harmonic $(2\beta _0)$ at the inlet location ( $x/c_x=0.055$ ) remains negligible ( $A_{\tilde {u}_{\xi ,2\beta _0}}/Q_{{\textit{ref}}} \approx 10^{-5}$ ). As a result, the shape of the inlet profile for the fundamental mode $(1\beta _0)$ is effectively identical between cases A1-H and A2-H, and the linear simulation is therefore conducted only using the inlet profile of case A1-H. Upstream of the hump, the normalised DNS profiles and the growth of perturbations match the linear solution in both cases A1-H and A2-H, suggesting the dominance of linear mechanisms. At the apex of the hump, slight deviations of the profiles and growth rate from the linear solution can be observed for case A2-H. On the other hand, downstream of the hump and within the CF reversal region, only the profiles and growth of perturbations for case A1-H match with the linear solution. The deviations from the linear solution for case A2-H indicate the importance of nonlinear effects downstream of the hump when the amplitude of incoming CF perturbation is high.

Figure 20.

(a–d) Comparison of $u_{\xi }$ component of the fundamental disturbance profile, i.e. mode $1\beta _0$ , for cases A1-H and A2-H with the disturbance profile obtained by a linear simulation for the hump case. (e) Comparison of relative growth of amplitude of perturbations with respect to their amplitudes at $x/c_x=0.1$ in cases A1-H and A2-H with the linear growth. The shaded grey area in (e) marks the transition region for case A2-H (see figure 17).

Figure 21(a) shows the chordwise evolution of the amplitude of spanwise Fourier modes for cases A2-C and A2-H calculated using (2.8) and (2.9). The overall trend in the evolution of Fourier mode amplitudes along the chord for case A2-H is similar to that observed for case A1-H (see figure 8), except for the fundamental mode $1\beta _0$ . In case A2-H, the amplitude of the fundamental stationary mode $(1\beta _0)$ exceeds case A2-C values within the range $0.18 \lt x/c_x \lt 0.24$ , which was not observed for cases A1-C and A1-H. This effect is also evident in figure 18, where the CF perturbation amplitude increases significantly in case A2-H compared with its reference counterpart.

Figure 21.

(a) Chordwise evolution of amplitude of the fundamental mode $1\beta _0$ and the first three higher harmonics ( $2\beta _0$ – $4\beta _0$ ) of chordwise stationary perturbation ( $u^{\prime}_{\xi ,s}$ ) for cases A2-C (black lines) and A2-H (red lines). (b) Ratio of the amplitude of the first ( $n=2$ ) and second ( $n=3$ ) harmonics to the fundamental mode for cases A1-C, A1-H, A2-C and A2-H. The shaded grey areas in (a) mark the transition region for cases A2-C and A2-H, and that in (b) marks the transition region for case A2-H (see figure 17).

Downstream of the hump, within the CF reversal region, the amplitude of higher harmonics in case A2-H exhibits a notable increase similar to that seen in case A1-H. However, the amplitudes in case A2-H are substantially higher. For example, the amplitude of the first harmonic (mode $2\beta _0$ ) at $x/c_x = 0.185$ reaches $A_{\tilde {u}_\xi } \approx 7\,\%$ (of $Q_{{\textit{ref}}}$ ) in case A2-H, compared with only $A_{\tilde {u}_\xi } \approx 1\,\%$ (of $Q_{{\textit{ref}}}$ ) in case A1-H. The high amplitude of higher harmonics in the CF reversal region in the hump case suggests the importance of nonlinear interactions when the amplitude of incoming CF perturbation is high, as shown in figure 20.

Figure 21(b) shows the ratio of the first ( $n=2$ ) and second ( $n=3$ ) harmonic mode amplitudes to the fundamental mode amplitude, i.e. $A_{\tilde {u}_\xi ,n\beta _0}/A_{\tilde {u}_\xi ,1\beta _0}$ , for all four cases within the range $0.1\lt x/c_x\lt 0.25$ . In cases A1-H and A2-H, $A_{\tilde {u}_\xi ,2\beta _0}/A_{\tilde {u}_\xi ,1\beta _0}$ increases significantly within the CF reversal region (region between the vertical dashed black lines in figure 21 b). In case A2-H this ratio reaches values of ${\approx} 47 \,\%$ , compared with ${\approx} 25 \,\%$ for case A1-H. In case A2-H, $A_{\tilde {u}_\xi ,3\beta _0}/A_{\tilde {u}_\xi ,1\beta _0}$ also reaches relatively large values of ${\approx} 22 \,\%$ , while it remains very small in case A1-H. The disproportional destabilisation of higher harmonics in case A2-H due to the interactions between the fundamental and first harmonic modes is notable. An increase in higher Fourier harmonics of the fundamental primary CFI mode can manifest in the physical domain as a localisation of regions of streaky structures, high shears and complex flow patterns. Due to their stationary nature, such regions can potentially sustain new secondary instabilities in the presence of unsteady noise inside the boundary layer. In the following, the mentioned hypothesis is investigated.

Stationary chordwise CF perturbations at $x/c_x=0.185$ are shown for cases A2-H and A1-H in figure 22. Significant differences are observed between the shape of stationary CF perturbations. At the examined chordwise location ( $x/c_x=0.185$ ), which lies downstream of the CF reversal region, a new near-wall structure is formed in case A2-H. This manifests itself as a localised positive perturbation in the total perturbation field as marked by the green rectangle in figure 22(Ia). An examination of the fundamental mode ( $\hat {u}_{\xi ,1\beta _0}$ ; figure 22IIa) and its first harmonic ( $\hat {u}_{\xi ,2\beta _0}$ ; figure 22IIIa) reveals that this new structure is formed dominantly within the first harmonic (see the green dashed rectangular region in figure 22 a) and is located close to the wall. Note that this structure is not observed in case A1-H. The new structure in case A2-H is formed in chordwise stations where the amplitudes of higher harmonics of the primary mode are strongly increased, reaching amplitudes comparable to that of the fundamental mode ( $A_{\tilde {u}_\xi ,1\beta _0}$ ; see figure 21).

Figure 22.

Total stationary CF perturbation of chordwise velocity component ( $u^{\prime}_{\xi ,s}=\bar {u}_{\xi }-u_{\xi ,0}$ ) at $x/c_x=0.185$ for cases A2-H (Ia) and A1-H (Ib), fundamental mode ( $1\beta _0$ ) for cases A2-H (IIa) and A1-H (IIb) and first harmonic mode ( $2\beta _0$ ) for cases A2-H (IIIa) and A1-H (IIIb). Panels are not drawn to scale.

Figure 23.

(a–d) Stationary chordwise perturbation field for case A2-H. In-plane (normal to CFV) flow streamlines are plotted as black lines with arrows, while the green dashed lines depict negative values of the $\lambda _2$ criterion in (a,b). The counterclockwise (CCW)- and clockwise (CW)-rotating vortices are marked in (a,b). (e) The $\lambda _2$ visualisation of the time-averaged meanflow. Isosurfaces of $\lambda _2=- {0.001Q^2_{{\textit{ref}}}}/{\lambda ^2_z}$ are coloured by spanwise velocity. A $z{-}\eta$ plane at $x/c_x = 0.174$ and a $\xi {-}\eta$ plane depict the total chordwise perturbation field, with red indicating positive and blue indicating negative perturbations. Panels (a–d) are not drawn to scale.

Figure 23 presents in detail the topology of the CFVs in the vicinity of the CF reversal region in case A2-H. Stationary chordwise CF perturbations at four different $z{-}\eta$ planes are depicted in figure 23(a–d). A three-dimensional visualisation of vortical structures in the time-averaged flow field (for several repeated spanwise periods of the simulation domain) is shown in figure 23(e) by means of the $\lambda _2$ criterion (Jeong & Hussain Reference Jeong and Hussain1995). From figure 23(a,b) it is clear that a counterclockwise (CCW)-rotating vortex, referred to as the hump-induced vortex, is formed within the CF reversal region where the localised near-wall structure is formed. Upon formation within the CF reversal region at $x/c_x \approx 0.166$ , the hump-induced vortex coexists with the classical clockwise-rotating CFV, and remains in the flow until slightly downstream of the CF reversal region, i.e. until $x/c_x \approx 0.196$ . This pair of counter-rotating vortices is marked in figure 23(a,b,e). In the following sections, the possible role of this pair of counter-rotating vortices in triggering the early transition observed in the experiments of Rius-Vidales et al. (Reference Rius-Vidales, Morais, Westerbeek, Casacuberta, Soyler and Kotsonis2025) and DNS (figure 17) for case A2-H is examined.

It should be noted that a CCW-rotating vortex has been observed even in the absence of the surface hump; see e.g. Wassermann & Kloker (Reference Wassermann and Kloker2002, figure 7). However, such vortices are in general very weak and have been identified to be unimportant for transition to turbulence (Wassermann & Kloker Reference Wassermann and Kloker2002). In the present case, downstream of the hump, the sign of the CF velocity component reverses close to the wall. Consequently, a new inflection point appears in the CF velocity profiles of the base flow (see figure 5). In this case, the vorticity of the CCW vortex is congruent with the vorticity component of the base flow (in the direction of the inviscid streamline) at the location of the newly formed inflection point closer to the wall. Thus, the growth of the CCW vortex is now supported by the base flow. Consequently, when the amplitude of the incoming CF disturbance is large enough, as in case A2-H, a strong CCW vortex forms within and slightly downstream of the CF reversal region.

5.2. Effect of the hump on secondary CFI modes and laminar breakdown

To analyse secondary CFI modes and the laminar–turbulent transition, similar to the analysis for cases A1-C and A1-H, SPOD is used. For SPOD analysis, 3072 three-dimensional snapshots of velocity for case A2-C ( $0.278\lt x/c_x\lt 0.378$ ) and 3840 snapshots for case A2-H ( $0.16\lt x/c_x\lt 0.186$ ) are collected at a sampling frequency of 50 kHz, and are divided into 11 and 14 blocks (with 50 $\,\%$ overlap), respectively. This results in a frequency bin of $\Delta f = 97.7$ Hz. The downstream end of the domain used for SPOD calculations for case A2-C is selected upstream of the transition region to avoid turbulent fluctuations biasing the SPOD mode ordering. For case A2-H, part of the region where the pair of counter-rotating vortices exists is used for SPOD. Figure 24 presents the resulting leading SPOD mode spectrum. From the SPOD spectrum it is clear that for case A2-C similar frequency bands to those for case A1-C become amplified (purple shaded regions). However, for case A2-H, there is a distinct increase in the energy of the leading mode within the frequency band of $900\lt f\lt 1500$ Hz (green shaded area in the figure), with maximum energy at $f=1171.88$ Hz. This frequency band is distinctly segregated from the expected frequency of secondary CFI modes pertinent to the present flow as mentioned in the previous sections.

Figure 24.

Leading SPOD mode spectra (solid lines) for cases A2-C and A2-H. The dashed lines represent the sum of the energy of all SPOD modes for each frequency. The purple shaded areas mark the same three frequency bands as for case A1-C (see figure 13). The green shaded area marks $f \in [900{-}1500]\,\rm Hz$ .

To isolate individual secondary CFI modes, similar to the analysis performed in § 4.3.1, unsteady velocity fields are band-pass-filtered using the frequency bands obtained with SPOD. The chordwise evolution of the amplitude of secondary CFI modes corresponding to each frequency band is shown in figure 25 for case A2-C (solid lines) and A2-H (dashed lines). For case A2-C, the spatial growth of secondary CFI modes is similar to that of case A1-C, except that the neutral points for type I ( $f_{BP_{I}} \in [4000{-}5500] \,\rm Hz$ ) and type II ( $f_{BP_{\textit{II}}} \in [7000{-}8500] \,\rm Hz$ ) are shifted notably more upstream compared with case A1-C values (i.e. $x/c_x \approx 0.25$ for A2-C versus $x/c_x \approx 0.31$ for A1-C). This is largely expected due to the higher amplitude of stationary CFVs and the subsequent formation of strong spanwise shears. Furthermore, the high-frequency fluctuation fields for case A2-C ( $f_c\gt 12000$ Hz, not shown here) confirm that similar to case A1-C, breakdown of type I secondary CFI mode causes the transition to turbulence. For case A2-H, initially only the amplitude of the modes corresponding to ${\textit{BP}}_{\textit{III}}\in [350{-}750]$ Hz is increasing. However, as soon as the CCW-rotating vortex appears downstream of the apex of the hump (i.e. $x/c_x\approx 0.166$ ), the mode corresponding to $f \in [900{-}1500]$ Hz becomes unstable, and its amplitude increases by almost two orders of magnitude over a very short distance ( $\Delta x/c_x \approx 5 \,\%$ ), after which breakdown to turbulence occurs. Figure 26 presents the band-pass-filtered velocity fluctuation fields for ${\textit{BP}}_{\textit{III}} \in [350{-}750]$ Hz (figure 26I–IIIa) and ${\textit{BP}}\in [900{-}1500]$ Hz (figure 26I–IIIb), alongside the total fluctuation field ( $u_{\xi ,{\textit{STD}}}$ ; figure 26I–IIIc), across three different chordwise locations. Hereafter, the band pass ${\textit{BP}}\in [900{-}1500]$ Hz is referred to as ${\textit{BP}}_{{\textit{CCW}}_V}$ . All panels within each row are normalised by the maximum total fluctuation, i.e. $u^{\textit{max}}_{\xi ,{\textit{STD}}}$ , at that corresponding chordwise location.

Figure 25.

Chordwise evolution of amplitude of band-pass-filtered chordwise velocity fluctuations ( $A_{\textit{II}}^{\textit{BP}}$ ) for case A2-C (solid lines) and A2-H (dashed lines). The grey shaded areas correspond to the grey areas shown in figure 17(a) and indicate the transition regions in DNS for cases A2-C and A2-H. The orange shaded area shows the extent of the hump, the vertical orange solid line marks the location of the hump’s apex and vertical black dashed lines indicate the boundaries of the CF reversal region.

Figure 26.

(a,b) Band-pass-filtered velocity fluctuation fields and (c) total fluctuations of $u_\xi$ at three chordwise locations (I–III) for case A2-H. Green and blue solid isolines represent $40\,\%$ of $(\partial u_\xi / \partial z)^{{min}}$ and $60\,\%$ of $(\partial u_\xi / \partial z)^{{\textit{max}}}$ , respectively. Dashed black lines indicate negative $\lambda _2$ -criterion regions. (IV) Peak value of spanwise gradients for time-averaged meanflow for case A2-C and case A2-H. The red dashed lines show the peak spanwise gradients in case A2-H, in the absence of unsteady noise in the simulations. (V) Maximum band-pass-filtered ( ${\textit{BP}}_{{\textit{CCW}}_V}$ ) fluctuation amplitudes for regions B (green) and C (blue), and high-pass-filtered fluctuation amplitude (purple dashed). Inset in (V) shows high-pass-filtered fluctuations. The vertical green dashed lines show the region where the counter-rotating vortex pair exists. Panels in (I–III) and the inset in (V) are not drawn to scale.

At $x/c_x=0.174$ (figure 26I), the total fluctuation field shows contribution of the modes corresponding to both ${\textit{BP}}_{\textit{III}}$ and ${\textit{BP}}_{{\textit{CCW}}_V}$ . At this location both modes have comparable amplitudes (see figure 25). Fluctuations corresponding to ${\textit{BP}}_{{\textit{CCW}}_V}$ are evident at two distinct locations. Firstly, region B (figure 26Ib), which coincides with the region of minimum spanwise gradient in the flow (the green solid isolines depict $40\,\%$ of $(\partial u_\xi / \partial z)^{{min}}$ ). This region is where the CCW vortex exists (see the region of negative values of $\lambda _2$ criterion, shown by dashed black lines in figure 26Ia,b, and also the inset in figure 23 e). Secondly, region C which coincides with the region of maximum spanwise gradient in the flow (the blue solid isolines depict $60\,\%$ of $(\partial u_\xi / \partial z)^{{\textit{max}}}$ ). The hump-induced CCW vortex strongly deforms the boundary layer. Consequently, the peak negative spanwise gradient of $u_\xi$ (and $w$ ; not shown here) is split into two regions, A and B, with the maximum negative spanwise gradient localised near the CCW vortex location (region B). This region is notably different compared with the typical spatial location of peak negative spanwise gradient in a classical CFV, which is at the outer side of the upwelling part (shoulder) of the vortex (region A in figure 26Ib) (Bonfigli & Kloker Reference Bonfigli and Kloker2007). Figure 26(IV) illustrates an evident increase in both positive and negative spanwise gradients in case A2-H (compared with case A2-C) in the region where the hump-induced CCW vortex exists (the region between the vertical green dashed lines in figure 26IV), and further downstream. The spanwise gradients in the absence of unsteady noise in the simulation (i.e. three-dimensional base flow) for case A2-H are shown by the dashed red lines in figure 26(IV). For $x/c_x \lesssim 0.2$ , the peak negative gradients of the meanflow and three-dimensional base flow for case A2-H are almost identical. This similarity further confirms that the increase in the peak gradients in case A2-H is a feature of the stationary flow itself, rather than a result of flow modifications caused by the growth of secondary CFI modes. Further downstream at $x/c_x=0.195$ (figure 26II) and $x/c_x=0.21$ (figure 26III), the CCW vortex no longer exists. However, the localised high-shear (both positive and negative) regions feed the fluctuations within regions B and C. At these chordwise locations, the amplitude of the fluctuations corresponding to ${\textit{BP}}_{{\textit{CCW}}_V}$ is one order of magnitude larger than the amplitude of the fluctuations corresponding to ${\textit{BP}}_{\textit{III}}$ (see figure 25). This can also be seen from figure 26(II), where no significant fluctuations (relative to the intensity of total fluctuations) can be seen within ${\textit{BP}}\in [350{-}750]$ Hz.

It is worth mentioning that the fluctuations within regions B and C are similar to the classical type I and type III secondary CFI modes, in the sense that they appear at regions of high negative (type I) and high positive (type III) spanwise shear. However, the frequency of these fluctuations in case A2-H is distinctly different compared with the typical frequency of type I and type III secondary CFI modes, as shown earlier in this work. To further elucidate the nature of fluctuations in regions B and C, the maximum value (at $z{-}\eta$ planes) of band-pass-filtered ( ${\textit{BP}}_{{\textit{CCW}}_V}$ ) fluctuations in these regions are traced along the chord. This is depicted in figure 26(V), where the green line shows the maximum of fluctuations in region B and the blue line shows the maximum fluctuations in region C. The purple dashed line shows the maximum value of fluctuations for high-pass-filtered ( $f_c=12$ kHz) velocity fields. Within the region where the counter-rotating vortex pair exists (i.e. the region between the two vertical green dashed lines for $0.166 \lesssim x/c_x \lesssim 0.196$ ), the peak amplitude of fluctuations in both regions (B and C) is comparable and grows with almost identical growth rates. However, for $x/c_x \gtrsim 0.196$ , the amplitude of the fluctuations in region C saturates prior to the onset of the transition. In contrast, the amplitude corresponding to the fluctuations in region B (i.e. the region of local minimum of spanwise gradient) increases until breakdown to turbulence occurs. This can also be seen from figure 26(IIIb), where the intensity of fluctuations at region B is considerably higher than that at region C. Thus, even though the secondary CFI mode corresponding to ${\textit{BP}}_{{\textit{CCW}}_V}$ can be seen as one single secondary CFI mode with mixed properties of the classical type I and type III secondary CFI modes, its spatial evolution shows a complicated behaviour.

Finally, to identify the spatial location for the onset of CFV breakdown in case A2-H, the inset in figure 26(V) shows the fluctuations corresponding to the high-pass-filtered velocity fields. Specifically, the inset corresponds to the chordwise location $ x/c_x = 0.217$ , marked by a red circle on the purple dashed line in figure 26(V). This location lies slightly upstream of the location where the skin-friction coefficient $ C_{\kern-1.5pt f}$ begins to increase (i.e. within the shaded grey region; see also figure 17 a). The fluctuations in the inset are normalised by their local maximum value.

At this station, the influence of the CCW vortex on the local spanwise shear distribution within the boundary layer has nearly vanished. Consequently, the negative shear region is no longer split into subregions A and B. It is evident from the inset in figure 26(V) that CFV breakdown initiates from the shoulder of the vortex.