3.1. Wind-off

From dispersion relation the duct modes from the rectangular section of the wind tunnel section can be calculated through

(3.1) \begin{equation} f_{mn} = \frac {c}{2} \sqrt {\left ( \frac {m}{a} \right )^2 + \left ( \frac {n}{b} \right )^2} \end{equation}

where, $m$ and $n$ correspond to the mode indices and $c$ the speed of sound. Only the first six theoretical duct modes of the empty test section are shown in table 2 as they were expected to be within the range of the most unstable hydrodynamic instability modes computed in § 5 using linear stability theory (LST). However, in practice these analytical duct modes can vary due to the inhomogeneous characteristic of the test-section, since it was composed of different types of materials. The presence of the wing will also modify the cross-sectional area of the duct locally. Nevertheless, the actual duct modes could be characterised by measuring the response of the duct through excitation of the test section using the tunnel floor disc-exciter. The structural modes of the wing model could be characterised similarly using just the exciter and in both cases the exciter was driven with a broadband input signal of $1$ to $500\,\text{Hz}$ . The resulting strong vibroacoustic interaction within the test section, which could be perceived by unaided ears was then characterised through microphone and accelerometer measurements.

Table 2.

The first six theoretical duct modes referred also as, $D_1$ , $D_2$ , $D_3$ , $D_4$ , $D_5$ and $D_6$ , of the wind tunnel working section calculated using (3.1).

Table 2. Long description

A table with six rows and three columns. The columns are labeled ‘Duct mode’, ‘Theoretical (Hz)’, and ‘Measured (Hz)’. The rows list different duct modes with their corresponding theoretical and measured frequencies. Row 1: Mode (1,0), D1, Theoretical 144 Hertz, Measured 100 Hertz. Row 2: Mode (0,1), D2, Theoretical 216 Hertz, Measured 210 Hertz. Row 3: Mode (1,1), D3, Theoretical 260 Hertz, Measured not available. Row 4: Mode (2,0), D4, Theoretical 288 Hertz, Measured 300 Hertz. Row 5: Mode (2,1), D5, Theoretical 360 Hertz, Measured 360 Hertz. Row 6: Mode (0,2), D6, Theoretical 432 Hertz, Measured not available.

The first wind-off test was conducted by driving the tunnel floor disc-exciter shown in figure 2, with the intention of only forcing the duct modes. The disc-exciter constituted of a 15 mm thick MDF (medium-density fibreboard) sheet and a diameter of 600 mm. It was fixed at four points with spring-loaded bolts on a sturdy steel support resting on the floor of the laboratory. This base and support is usually used to support a model mounted vertically. The surface of the disc was aligned flush with the tunnel floor and a gap of 3–5 mm was ensured between the disc and the tunnel floor when it was housed in the circular insert, as shown in figure 2(b). The intention was to decouple it from the test section to prevent any interaction. However, as can be deduced from figure 2, the wing model was mounted on sidewall supports on the outerside of the test section walls, interfaced to the steel frame of the test section so any perturbation from the model could be transferred to the test section and vice versa.

From the sound pressure level spectra of the microphone measurements in figure 3, for frequencies greater than $100\,\text{Hz}$ , the peaks are quite close to the corresponding theoretically calculated duct modes in table 2, except $D_1$ , which was shifted to approximately 100 Hz instead of 144 Hz, and $D_2$ calculated at 260 Hz was quite weak. The response of the wing captured by the accelerometer, is presented in figure 3(b), and showed spectral content similar to the sound pressure level spectra. The second test involved excitation of the wing by placing the Visaton exciter at the centre of the model, using the same frequency band; however, the amplitude was adjusted to prevent severe vibrations that can compromise the structural integrity of the model. The sound pressure level spectra in figure 3(a) exhibited pretty similar characteristics to those when the duct was excited. However, the accelerometer spectra showed some differences as the overall spectra seem to be shifted by 20–30 Hz to the right-hand side, with the emergence of new peaks corresponding purely to the natural modes of the structure. The structural modes as well have been labelled as $S_n$ , but in this case the first mode $S_1$ is considered to start at $f\gt 100\,\text{Hz}$ to follow the same order as the duct modes. Despite being energetic the modes below 100 Hz and above 500 Hz were not considered to be meaningful, as the LST showed that they were not within the envelope of the most unstable boundary layer modes and for the Reynolds numbers considered here, from our previous study (Jaroslawski et al. Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023) this frequency band was adequate.

Figure 3.

Sound pressure (a) and acceleration levels (b) across different Reynolds numbers. The spectra are offset by 35 dB for distinction. In the legend ‘duct modes’ refers to excitation of the duct modes with the tunnel floor disc-exciter and ‘wing structural mode’ refers to excitation of the wing directly during wind-off testing. The dashed–dotted vertical lines correspond to theoretical duct modes, labelled as, $D_1$ to $D_6$ . Similarly, the important structural modes are labelled from $S_1$ to $S_6$ .

Figure 3. Long description

Two line graphs depict sound pressure and acceleration levels across different Reynolds numbers. The graphs are offset by 35 decibels for distinction. The x-axis represents frequency in Hertz, ranging from 10 to 1000. The y-axis represents sound pressure level in decibels for the first graph and acceleration level in decibels for the second graph. The spectra are color-coded by Reynolds numbers: 1.5 x 10^5, 2.0 x 10^5, 2.5 x 10^5, and 3.0 x 10^5. Duct modes and wing structural modes are indicated by dashed-dotted vertical lines. The duct modes are labeled D1 to D6, and the wing structural modes are labeled S1 to S6. The graphs show how sound pressure and acceleration levels vary with frequency for different Reynolds numbers, highlighting the interaction between aerodynamic and structural modes.

The disc-exciter on the tunnel floor generated a pure acoustic perturbation in the wind tunnel duct; however, the wing model responded to the forcing as significant activities were picked-up by the accelerometer on the wing mainly linked to the duct modes. Here, the acceleration and sound pressure level spectra were very similar, which would suggest that the wing was responding to the high-amplitude acoustic pressure field inside the wind tunnel working section. When the wing was excited separately, the wing’s acceleration level spectra was slightly different from the sound pressure level spectra, where the latter was richer as it was constituted of both the duct modes and the natural modes of the wing. This shows that the vibration of the wing model translated into noise, too. Although the structure showed a slightly different response to the forcing compared with the duct modes, the modes $D_2$ and $S_2$ , and $D_5$ and $S_5$ , at 210–230 Hz and 360 Hz, respectively, are always expected regardless of whether the forcing is purely acoustic or purely structural, this is potentially due to the wing model being tightly coupled with the duct constituting the test section. Even if $D_6$ was not sufficiently energetic, $S_6$ showed a non-negligible spectral content, since this mode corresponds to the harmonic of $D_2$ and $S_2$ , it can play a role during the secondary nonlinear growth phase.

There is very strong evidence in the literature that during experiments noise is responsible for the presence of naturally excited TS waves even in a specialised wind tunnel with very low background freestream turbulence or acoustic perturbation level. However, to our knowledge only the theoretical work of Goldstein (Reference Goldstein1985) and Ruban et al. (Reference Ruban, Bernots and Pryce2013) have been able to provide a potential explanation on the effect of vibration and/or acoustics on the excitation of TS waves. This falls under the framework of receptivity and the wavelength conversion was proposed to be through the small but finite surface imperfections that are of the order of the scales of the boundary layer. This has raised a major question in the transition community as some schools of thought suggest that the excitation or wavelength conversion can still take place without the presence of surface imperfections, through destabilisation of the Stokes layer. Quantification of the receptivity is outside the scope of the current study due to experimental limitations, however, the aero–vibroacoustic interactions observed here are very likely to play a role in the amplification of instability modes and transition, and will be demonstrated below.

3.2. Wind-on

When the wind tunnel was operated, vibroacoustic interactions could be perceived from just the unaided ears, both in the form of whistling and reverberation noise depending on freestream velocity, and these were picked by the microphone and accelerometer as shown in figure 3, for four selected Reynolds numbers. At ${\textit{Re}}=1.5\times 10^5$ , discrete peaks can be observed in the sound pressure level spectrum and these corresponded to the tonal noise generated by the aerofoil. The sharpest tone is around 120 Hz. When expressed in terms of Strouhal number, $St=fc/U_\infty$ , in figure 4(b), the presence of tonal noise can be confirmed by the ladder-type evolution with increasing Reynolds number, initially reported by Paterson et al. (Reference Paterson, Vogt, Fink and Munch1973) and more recently elucidated by Fosas de Pando et al. (Reference de Pando, Miguel, Peter and Sipp2014). The tonal noise was present for ${\textit{Re}}\lt 2.0\times 10^5$ , as confirmed by both the spectra in figure 3 and the spectrogram in figure 4(b). Further confirmation that these peaks were linked to the tonal noise can be obtained by eliminating them through boundary layer tripping on the suction surface, and this can be observed in figure 4(d). Some part of the ladder-type structure was still recovered at ${\textit{Re}}\lt 1.5\times 10^5$ as at lower Reynolds number the boundary layer was thicker, thus reducing the effectiveness of the tripping device.

From the spectrogram in figure 4(a), within this tonal noise regime a very significant amount of energy can be observed at 210 Hz and this was due to lock-in between tonal noise mode and the duct mode, $TN/D_2$ and lock-in also occurred with duct mode $D_5$ at 360 Hz, but was milder. The presence of lock-in could be confirmed by activating the disc-exciter, where in figure 4(e) this led to excitation of most of the duct modes, namely $D_1$ , $D_2$ , $D_4$ and $D_5$ , as highlighted in the figure. However, as better illustrated by the Strouhal number in figure 4(f), the tonal noise ladder-structure was slightly impacted and this will be revisited below. This lock-in, with $D_2$ , encircled in red in figures 4(a), 4(c) and 4(e) persisted outside the tonal noise regime until ${\textit{Re}}\lt 2.8\times 10^5$ and was also observed in figure 4(c) in the presence of boundary layer tripping which attenuated the tonal noise, but some activities were still present between $1.75\times 10^5\lt Re\lt 2.5\times 10^5$ . This could be an acoustic mode resulting from a purely structural forcing, potentially due to an unsteady loading imposed by the LSB and this constituted a lock-in between the structural mode $S_2$ and the duct mode, $D_2$ , leading to mode $S_2/D_2$ . This will be further elaborated below, during the assessment of the structural response of the wing.

Figure 4.

Spectrograms of sound pressure levels for different boundary layer trip configurations and acoustic excitation: (a,b) only lower surface tripped; (c,d) both surfaces tripped; (e,f) lower surface tripped with acoustic excitation. Spectrum bars (i) and (ii) refer to the spectral content obtained during wind-off duct and wing structure excitation, respectively; they have been placed for comparison with the reference wind-off condition. The red dash–dot line indicates the frequency of the wind tunnel fan blades.

Figure 4. Long description

The image contains six graphs showing spectrograms of sound pressure levels for different boundary layer trip configurations and acoustic excitation. The graphs are organized in pairs: (a,b) for only lower surface tripped, (c,d) for both surfaces tripped, and (e,f) for lower surface tripped with acoustic excitation. Each pair consists of a frequency (f) versus Reynolds number graph on the left and a Strouhal number (St) versus Reynolds number graph on the right. The sound pressure level is indicated by a grayscale color bar ranging from 30 to 70 decibels. Key features such as TN/D2, S2/D2, D2, D3, and D4 are marked in red on the frequency graphs. Spectrum bars (i) and (ii) refer to the spectral content obtained during wind-off duct and wing structure excitation, respectively, and are placed for comparison with the reference wind-off condition. The red dash-dot line indicates the frequency of the wind tunnel fan blades. The graphs illustrate how different configurations and excitations affect the sound pressure levels across various frequencies and Reynolds numbers.

In figure 3, at ${\textit{Re}}=1.5\times 10^5$ the accelerometer also picked-up some activities related to tonal noise and similarly in figure 5(a). When non-dimensionalised, in figure 5(b) some aspects of the ladder-type evolution with Reynolds number of the tonal noise were also captured by the accelerometer until ${\textit{Re}}\lt 2.0\times 10^5$ . This would suggest that in a confined or hard-walled environment the tonal noise promotes some vibroacoustic interactions and, to our knowledge, this has not been reported previously during wind tunnel experiments. In figure 5 the low frequency structural modes, $S$ at around 50, 70, 100 Hz, $S_1$ at 140 Hz and $S_3$ at 270 Hz also emerged. This confirmed that the aerofoil was forced by an unsteady aerodynamic loading as the response is similar to that when forced artificially with the Visaton exciter during the wind-off test shown in figure 3. These modes also seem to lock-in with the tonal noise modes that were not very clear from the sound pressure level spectra in figure 4, highlighted by the red boxes, where $TN/S_n$ corresponds to the tonal mode lock-in with the corresponding structural mode. Instead of a whistling noise characteristic of tonal noise, a reverberation type of noise was perceived by the unaided ears as the structural modes would also contribute to the noise.

Figure 5.

Spectrograms of the wing’s acceleration levels for different boundary layer trip configurations and acoustic excitation: (a,b) only lower surface tripped; (c,d) both surfaces tripped; (e,f) lower surface tripped with acoustic excitation. Spectrum bars (i) and (ii) refer to the spectral content obtained during wind-off duct and wing structure excitation, respectively; they been placed for comparison with the reference wind-off condition. The red dash–dot line indicates the frequency of the wind tunnel fan blades.

Figure 5. Long description

The image contains three sets of spectrograms labeled (a, b), (c, d), and (e, f). Each set includes two graphs: a frequency (f) versus Reynolds number plot on the left and a Strouhal number (St) versus Reynolds number plot on the right. The first set (a, b) shows data for the lower surface tripped configuration. The second set (c, d) shows data for both surfaces tripped. The third set (e, f) shows data for the lower surface tripped with acoustic excitation. Spectrum bars (i) and (ii) are included for comparison with the reference wind-off condition. The red dash-dot line indicates the frequency of the wind tunnel fan blades. The acceleration levels are represented in decibels (dB).

In the tripped case, from figure 5(c), upon cancellation of the tonal noise, the lock-ins between the tonal noise and the structural modes were also eliminated. However, some energy was still present in the frequency band of 210 Hz and 230 Hz, at $1.75\times 10^5\lt Re\lt 2.5\times 10^5$ , corresponding mainly to the lock-in between the structural and duct mode, $S_2/D_2$ . This demonstrates an unsteady flow and structure type of interaction reported during buffeting. This mode is also recovered in the sound pressure level spectra in figure 4; however, in the current set-up it was not possible to determine whether this was a noise purely from the buffeting or the duct modes, or a combination of both.

The response of the wing shown in figure 5(e) differs slightly from the background acoustic environment figure 4(e) when the duct mode was forced artificially. As mentioned earlier, the tonal noise was also impacted, but to a lesser extent than when tripped by the roughness. This shows that the artificially excited duct mode had an impact on the LSB which would in return affect the tonal noise and the unsteady loading, potentially resulting in the elimination of some of the lock-ins. For instance $TN/S_1$ and $TN/S_3$ , which were present in the naturally excited case in figure 5(a), were not very obvious in figure 5(e), where the tonal noise ladder-structure driving these lock-ins was also eliminated, as more clearly shown when expressed in Strouhal number in figure 5(f). The impact of the wind tunnel test section or the duct on aerofoil tonal noise has been reported in numerous studies, far-field measurements in an openjet-type tunnel shows multiple tones whereas a tunnel with a closed section showed a single tone (Fosas de Pando et al. Reference de Pando, Miguel, Peter and Sipp2014). However, to our knowledge the impact of the duct modes on the near-field tonal noise, as shown here, has not been reported in the literature.

4.1. Mean flow topology

From figure 6(a), the presence of the much-expected laminar separation and turbulent reattachment could not be easily interpreted from the distribution of the pressure coefficient, $C_p$ , due to the coarseness in the resolution of the pressure taps towards the trailing edge of the aerofoil, where the LSB was expected from preliminary XFOIL analysis. However, a discontinuity becomes more apparent at $0.65\lt x/c\lt 0.90$ when zooming only in the rear-half of the aerofoil, in figures 6(b) and 6(c), another small difference can be found between the ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ cases at similar chordwise location. To compensate for the poor resolution in pressure taps, a streamwise hot-wire scan, at a fixed wall normal position, was conducted to identify the rapid increase in velocity fluctuations expected in the vicinity and inside the LSB, as seen at $0.65\lt x/c\lt 0.90$ . The location of the turbulent reattachment would be normally be indicated by a saturation in velocity fluctuations. However, at low Reynolds number, the nonlinear amplification and breakdown is slower and extends over a larger streamwise extent, therefore equilibrium and fully developed turbulent conditions are not attained very rapidly, leading to the formation of multiple saturation zones as seen in figure 6. We revisited this issue very recently (Prieto, Gowree & Forte Reference Prieto, Gowree and Forte2026) as it was found to make identification of the onset of boundary layer transition more challenging. To compensate for some of these limitations, an additional optical measurement technique was employed and further XFOIL calculations were conducted for a range of N-factors until the best match with the experimental pressure distribution was obtained.

Figure 6.

The pressure distribution on the suction side of the wing and the evolution of $u_{rms}$ of the hot-wire along the streamwise direction at a fixed wall normal position, y = 1.2 mm, with and without the excitation by the disc-exciter. The pressure distribution is also compared with the results from XFOIL for critical factor, $N_{crit}$ of 6, 7, 8 and 9. The solid red lines represent the start and end of the LSB.

Figure 6. Long description

The image contains three graphs. The first graph shows the pressure distribution on the suction side of the wing and the evolution of the hot-wire along the streamwise direction at a fixed wall normal position, y = 1.2 millimeters, with and without the excitation by the disc-exciter. The pressure distribution is compared with the results from XFOIL for critical factors of 6, 7, 8, and 9. The solid red lines represent the start and end of the laminar separation bubble (LSB). The second and third graphs provide detailed views of the pressure distribution and hot-wire evolution for Reynolds numbers of 2.0 times 10 to the power of 5 and 2.5 times 10 to the power of 5, respectively. These graphs highlight the first and second critical factors and their impact on the flow characteristics. All values are approximated.

Referring back to figure 6(a), in the baseline unforced case, as mentioned earlier the sharp increase in velocity fluctuation at $x/c\gt 0.6$ is due to the rapid amplification of the unstable hydrodynamic modes. This is in line with Marxen & Henningson (Reference Marxen and Henningson2011) who demonstrated from direct numerical simulations (DNS) that both the viscous and inviscid instability modes underwent rapid growth even slightly upstream of the LSB. This hinders the identification of the actual position of separation, but rapid linear growth in LSB is clearly depicted, as well as the saturation, which happens very quickly within the LSB due to the secondary nonlinear growth and breakdown. The turbulent reattachment is expected to be at the beginning of the second growth stage after the first saturation. Therefore, in the baseline case, from figure 6 the separation corresponds to $x/c\approx 0.75$ at both Reynolds numbers, but unexpectedly the turbulent reattachment, indicated by the end of the first peak was slightly upstream at $x/c\approx 0.8$ at ${\textit{Re}}=2.0\times 10^5$ and, at $x/c\approx 0.85$ at ${\textit{Re}}=2.5\times 10^5$ , providing the first indication that the bubble was longer at a higher Reynolds number.

From the experimentally captured pressure distribution, in figure 6(a), the position of the LSB was not easily detectable. But, from the XFOIL calculations in figures 6(b) and 6(c), conducted for a range of critical N-factor, $N_{crit}$ , the localised discontinuity in the pressure distribution is more remarkable, indicating the position of the LSB. For the baseline case, the best match between the experiment and XFOIL is obtained with $N_{crit}\approx 6$ at ${\textit{Re}}=2.0\times 10^5$ and $N_{crit}\approx 7$ in the case of ${\textit{Re}}=2.5\times 10^5$ . An increase in Reynolds number would normally lead to a reduction in $N_{crit}$ , as transition is expected to occur earlier. But once again this was not the case, confirming that the bubble stretching with Reynolds number can be indirectly demonstrated from the pressure distribution too. Also, despite the higher growth rate in the LBS the saturation in $u_{rms}$ is further delayed too. The effect of artificial forcing is more evident on the growth rate of the instability shown by the $u_{rms}$ in figures 6(b) and 6(c). No significant effect was observed at ${\textit{Re}}=2.0\times 10^5$ except on the saturation corresponding to a delay in turbulent reattachment and at ${\textit{Re}}=2.5\times 10^5$ the initial growth rate was modified, but the turbulent reattachment was not significantly impacted.

Figure 7.

Oil droplets visualisation, for the case of ${\textit{Re}}=2.0\times 10^5$ in (a) and (c) and ${\textit{Re}}=2.5\times 10^5$ in (b) and (d), respectively; panels (a) and (b) are the naturally excited case and (c) and (d) are the artificially excited cases. The flow is from left to right. The solid red lines represent the start and the end of the LSB.

Figure 7. Long description

The image contains four separate panels labeled (a), (b), (c), and (d), each showing oil droplets visualization in a flow from left to right. Panels (a) and (b) represent the naturally excited case, while panels (c) and (d) represent the artificially excited cases. The solid red lines in each panel indicate the start and end of the LSB. The flow direction is consistent across all panels, moving from left to right. The visualization highlights the presence and behavior of oil droplets under different excitation conditions.

Given the limitation in the pressure and hot-wire techniques a very simple optical diagnostic technique, constituting the oil droplet visualisation, played a crucial role here. From figure 7, the oil droplets are stretched in the region of attached flow by the wall shear, as this is zero at separation and reattachment, and very low throughout the LSB the droplets underwent a negligible change in this region. An initial stretching is experienced by the droplets upon deposition on the surface due to gravity and the surface curvature of the wing; therefore, they are not perfectly circular even in the separated flow regions. However, this effect was considered as negligible. The unexpected longer LSB at ${\textit{Re}}=2.5\times 10^5$ can be further confirmed when comparing figures 7(a) and 7(b), where, at ${\textit{Re}}=2.0\times 10^5$ the region of unstretched droplets was between $0.725\lt x/c\lt 0.8$ and slightly more extended, $0.75\lt x/c\lt 0.90$ , at ${\textit{Re}}=2.5\times 10^5$ . This could be further quantified by calculating the length or chordwise extension of the droplets during wind-off and wind-on conditions and their ratio can be expressed as a stretch ratio shown in figure 8. The stretch ratio reduces significantly and remains constant within the LSB and increases rapidly in the turbulent reattachment region. Although the reduction in stretch ratio occurred at very close chordwise positions, in the baseline case at ${\textit{Re}}=2.0\times 10^5$ , the rapid increase started at $x/c\approx 0.825$ , whereas at ${\textit{Re}}=2.5\times 10^5$ , this occurred at $x/c\approx 0.875$ , confirming the earlier transition to turbulence at lower Reynolds number. It is also interesting to note that upon excitation at ${\textit{Re}}=2.0\times 10^5$ the increase in stretch ratio is further delayed downstream than in the baseline case, indicating that turbulent reattachment has been delayed again, but for the case of ${\textit{Re}}=2.5\times 10^5$ , no significant change was observed.

Figure 8.

Chordwise variation of the ratio of the final length of the oil droplet along the streamwise direction over the initial length during wind-off conditions.

Figure 8. Long description

Two scatter plots compare the stretch ratio along the chordwise direction for different Reynolds numbers. The top plot represents a Reynolds number of 2.0 times 10 to the power of 5, while the bottom plot represents a Reynolds number of 2.5 times 10 to the power of 5. Each plot features two data series: one for baseline conditions marked by black circles and one for excitation conditions marked by red squares. The x-axis is labeled as x over c, ranging from 0.55 to 0.95, and the y-axis is labeled as stretch ratio, ranging from 0 to 6. The plots show variations in stretch ratio along the chordwise direction under wind-off conditions, highlighting differences between baseline and excitation scenarios.

4.2. Unsteady flow

From the flow topology analysis above, the global effect on the mean flow and the impact on transition and turbulent reattachment encouraged further analysis of the unsteadiness in the flow. The $u_{rms}$ profiles in figure 9(a–d), at ${\textit{Re}}=2.0\times 10^5$ , are typical of the eigenfunction of TS-waves, with the maximum amplitude near the wall and is equal to 1 when normalised by the local maximum, ${u_{rms}}_{max}$ . In the presence of an adverse pressure gradient, a third intermediate lobe emerges around $y/\delta _1=1$ , which Marxen, Lang & Rist (Reference Marxen, Lang and Rist2012) referred as the ‘outer’ mode. The ‘inner’, near-wall mode, is the most unstable mode and could undergo a viscous TS or inviscid KH instability. At ${\textit{Re}}=2.5\times 10^5$ , in figure 9(e–h), for the baseline case, even upstream of the LSB at $x/c=0.55$ and $x/c=0.6$ , the energy was concentrated in the ‘outer’ mode and was non-negligible even if it gradually shifted to the inner modes at $x/c=0.625$ and $x/c=0.65$ . The dominance of the outer mode at the upstream stations at ${\textit{Re}}=2.5\times 10^5$ indicates a stronger influence of the LSB, however, the emergence of the TS modes at $x/c\gt 0.6$ shows that both a KH and TS mode coexisted.

Figure 9.

The root-mean-square (r.m.s.) of the fluctuating velocity signal, $u_{rms}$ , normalised by the local $u_{{rms}_{max}}$ at $x/c=0.55$ , $x/c=0.6$ , $x/c=0.65$ and $x/c=0.7$ . Panels (a)–(d) and (e)–(h) correspond to ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively. The excited case is obtained by driving the disc-exciter with a frequency band of 1–500 Hz.

Figure 9. Long description

The image contains eight scatter plots arranged in two rows and four columns. Each plot compares the root-mean-square of the fluctuating velocity signal normalized by the local maximum velocity at different chordwise positions. The x-axis represents the normalized fluctuating velocity, and the y-axis represents the normalized height. The first row of plots (a-d) corresponds to one set of conditions, while the second row (e-h) corresponds to another. Each plot includes two datasets: baseline (black circles) and excitation (red circles). The excitation case is obtained by driving a disc-exciter with a frequency band of 1500 Hz. The plots show how the fluctuating velocity changes with height at different chordwise positions under baseline and excitation conditions.

Figure 10.

The r.m.s. of the fluctuating velocity signal, $u_{rms}$ , at $x/c=0.55$ , $x/c=0.6$ , $x/c=0.65$ and $x/c=0.7$ . Where (a)–(d) and (e)–(h) correspond to ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively. The excited case is obtained by driving the disc-exciter with a frequency band of 1–500 Hz.

Figure 10. Long description

Eight scatter plots compare baseline and excitation data for fluctuating velocity signals at different conditions. Each plot shows the root mean square of the fluctuating velocity signal on the x-axis and the normalized wall-normal position on the y-axis. The baseline data is represented by black circles, while the excitation data is represented by red circles. The excitation case is obtained by driving the disc-exciter with a frequency band of 1500 Hz. Plots (a) to (d) correspond to one set of conditions, and plots (e) to (h) correspond to another set of conditions. The scatter plots show how the fluctuating velocity signals vary with wall-normal position under baseline and excitation conditions.

Upon artificial excitation, negligible difference was observed for ${\textit{Re}}=2.0\times 10^5$ , but at ${\textit{Re}}=2.5\times 10^5$ the energy is shifted mainly in the inner mode, resulting in profiles with dominant TS mode characteristics. Since artificial excitation accentuated the duct modes, this confirmed that they were driving the TS modes. The effect of excitation is more evident when the $u_{rms}$ is expressed in dimensional form, in figure 10 once again no important effects are seen at ${\textit{Re}}=2.0\times 10^5$ confirming the dominance of the TS modes, even if the KH mode was expected. However, at ${\textit{Re}}=2.5\times 10^5$ in figure 10(e–h) an interesting scenario is observed, where TS modes are significantly boosted as they attained higher peaks in $u_{rms}$ . However, as will be shown later in § 5.3, even if the TS modes were boosted upstream they were not able to overtake the KH mode in this case, as it grew rapidly in the vicinity and within the LSB.

Figure 11.

Power spectral density (PSD) of the fluctuating velocity signal at the wall normal position corresponding to the local $u_{{rms}_{max}}$ . The spectra, shifted vertically for clarity, correspond to the chordwise position between $x/c=0.55$ and $x/c=0.8$ , at intervals of $x/c=0.05$ , for both the baseline naturally and artificially excited case. Panels (a) and (b) are at ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively. The excited case is obtained by driving the disc-exciter with a frequency band of 1–500 Hz.

Figure 11. Long description

Two line graphs compare power spectral density (PSD) of fluctuating velocity signals at the wall normal position. The spectra are shifted vertically for clarity and correspond to the chordwise position between 0.4 and 0.8, at intervals of 0.05, for both the baseline naturally and artificially excited case. Panel (a) is at a Reynolds number of 2.0 times 10 to the power of 5, and panel (b) is at a Reynolds number of 2.5 times 10 to the power of 5. The excited case is obtained by driving the disc-exciter with a frequency band of 1500 hertz. The graphs show PSD of u prime over u bar squared on the y-axis and frequency in hertz on the x-axis. The baseline data is represented in black, while the excitation data is in red. Duct modes are indicated with dashed lines.

From the spectra of the hot-wire signal shown in figure 11, there is a very clear response of the boundary layer to the perturbations generated by the aero–vibroacoustic interaction. In the baseline case, at ${\textit{Re}}=2.0\times 10^5$ , upstream of the bubble at $x/c\lt 0.60$ , the narrowband peak between $200$ and $220\,\text{Hz}$ corresponds to the lock-in between duct mode and structural mode $D_2/S_2$ . The emergence of another mode between $400$ and $440\,\text{Hz}$ coinciding with the duct mode $D_6$ , indicates the potential coexistence of other modes; however, it can also be perceived as a harmonic mode of the boundary layer instability. For ${\textit{Re}}=2.5\times 10^5$ , at similar chordwise position the mode between $200$ and $220\,\text{Hz}$ is again excited, but another mode also emerged between $350$ and $370\,\text{Hz}$ coinciding purely with the duct mode $D_5$ , and here stronger evidence of the coexistence of the two modes is observed. Since these three modes match perfectly those identified in § 3, the impact of the perturbation generated by the aero–vibroacoustic interaction from the buffeting aerofoil on the hydrodynamic instability can be confirmed.

Referring back to figure 11, upon artificial excitation with a wider frequency band of 1–500 Hz, similar modes to the baseline unexcited cases were recovered, together with the mode coinciding with mode $D_5$ that was barely present in the baseline forcing at ${\textit{Re}}=2.0\times 10^5$ . Excitation at ${\textit{Re}}=2.5\times 10^5$ appeared to boost mode $D_5$ significantly which also underwent rapid secondary growth due to the presence of superharmonics, already at $x/c=0.55$ . At this Reynolds number, from figures 9 and 10(e–h), the clear change in the eigenfunction profile between the baseline and the artificial excitation confirms the shift in the dominant instability mode. The amplitude of the outer lobe is reduced and the profiles are more reminiscent of a TS mode as at ${\textit{Re}}=2.0\times 10^5$ . However, unlike at ${\textit{Re}}=2.0\times 10^5$ , the growth of this TS mode might be slightly delayed downstream. To sum up, the presence of several distinctive peaks shows the coexistence of several instability modes which can be confirmed by artificial excitation. Therefore, further stability analysis is required to track the dominant one at each Reynolds number, in order to understand the reason behind the bubble shrinkage at ${\textit{Re}}=2.0\times 10^5$ .

5.1. The base flow

The baseflow was calculated using ONERA’s 3C3D code that solves Prandtl’s boundary layer equations (Houdeville & Malecki Reference Houdeville and Malecki1996) while using the experimental pressure distribution as input. The major limitation is marching through the LSB as the boundary layer equation formulation assumes the absence of separation. Therefore, the calculation was conducted for the chordwise positions before a separation was detected, which was at $x/c \approx 0.68$ , where the code will start to fail. Even if this approach did not allow for the calculation of the dominant instability generated by the shear layer from the LSB, figure 11 showed that the unstable modes excited by the aero–vibroacoustic interaction were already active upstream of the LSB. From figure 12, the excellent agreement between the experiment and the calculation shows the ability of the numerical method to reliably resolve the base flow necessary for the stability analysis.

Figure 12.

The mean streamwise velocity, $U$ , normalised by the mean velocity at the edge of the boundary, $U_e$ , measured by the hot-wire, ‘ $\circ$ ’, and calculated by the numerical tool 3C3D, ‘dashed line’, at chordwise positions $x/c = 0.55$ , $0.60$ , $0.625$ and $0.65$ . Panels (a)–(d) and (e)–(h) correspond to ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively.

Figure 12. Long description

Eight line graphs compare experimental and numerical data of mean streamwise velocity at different chordwise positions. The x-axis represents the normalized velocity U/Ue, and the y-axis represents the normalized height y/delta1. Each graph shows data points from experiments and dashed lines from numerical calculations. Panels (a) to (d) correspond to x/c values of 0.550, 0.600, 0.625, and 0.650, respectively, for one condition. Panels (e) to (h) correspond to the same x/c values for another condition. The graphs illustrate how the mean streamwise velocity varies with height at different chordwise positions, highlighting the agreement or discrepancy between experimental and numerical results.

The slight mismatch with the experiment very close to the wall was due to uncertainty introduced by radiative heat transfer between the model’s surface and the hot-wire probe as it was located very close to the wall. In fact, the first measurement, which was discarded, was obtained by resting the hot-wire prongs on the model’s surface. Another source of uncertainty could be due to the low velocity very close to the wall, which fell outside the hot-wire calibration range. At $x/c\gt 0.6$ the numerical velocity profile tends towards an inflection point at the wall, which is an indication that a separation was imminent. At ${\textit{Re}}=2.0\times 10^5$ the boundary layer calculations were stopped at $x/c=0.68$ and at a similar position at ${\textit{Re}}=2.5\times 10^5$ , as the position of separation is not expected to be significantly affected within this Reynolds number range. However, the turbulent reattachment will be expected to occur earlier in the latter’s case because of the higher amplification rate as shown in figure 13, and thus a slightly shorter bubble. But once again, this was not the case during the experiment, therefore the linear stability analysis below was expected to provide further insights.

5.2. Linear stability analysis

The stability analysis was conducted using ONERA’s stability and transition prediction code CASTET (Perraud (Reference Perraud1997)). The code solves the linearised Navier–Stokes equations with parallel flow approximation or the Orr–Sommerfeld equations in a spatial formulation and therefore the growth rate of the convective instability modes can be expressed as N-factor,

(5.1) \begin{equation} N(x) = \int _{x_0}^{x} -\alpha _i(x') \, \text{d}x', \end{equation}

where $\alpha _i$ is the imaginary part of the wavenumber and $x_0$ is the neutral point after which the disturbance starts to grow. From this modal approach, the growth rate of the instability modes corresponding to a frequency range can be calculated. In figure 13, the N-factors of the full envelope corresponding to a frequency range of 100–890 Hz are presented, together with individual modes corresponding to 210 Hz, 360 Hz and 420 Hz labelled as $D_2$ , $D_5$ and $D_6$ (or harmonic of $D_2$ ) during both the aeromechanical interaction and flow response analysis in § 3 and § 4, respectively. Note that the calculations were conducted right before separation, where the base flow calculation would stop. For both Reynolds numbers rapid growth starts at $x/c\gt 0.4$ ; however, earlier the modes attained slightly higher overall amplitude at ${\textit{Re}}=2.5\times 10^5$ . Here, the individual modes are compared with the experimental values obtained by integrating the streamwise velocity fluctuations, $u'$ , over a narrow band of $\pm 10\,\text{Hz}$ centred at 210 Hz, 360 Hz and 420 Hz, which corresponded to the most amplified modes identified from the spectra in figure 11. The experimental results have been adjusted to match the calculated N-factor, but as will be shown later, this could be misleading, as the initial amplitude right after receptivity cannot be obtained experimentally, and this depends significantly on the level of background forcing. Nevertheless, the aim of the N-factor calculation was to track how the unstable modes evolved and check if they were susceptible to undergo substantial growth and destabilisation upstream of the bubble.

Figure 13.

The N-factor envelope (100–890 Hz) and evolution of the three modes identified during the aeromechanical interactions analysis. The experimental results correspond to the maximum in $u'$ during the boundary layer traverse, at each chordwise station. The solid red lines represent the start and end of the LSB for the case of ${\textit{Re}}=2.0\times 10^5$ and only the beginning of the LSB for ${\textit{Re}}=2.0\times 10^5$ .

Figure 13. Long description

Two line graphs compare the N-factor evolution at different Reynolds numbers. The left graph represents a Reynolds number of 2.0 times 10 to the power of 5, while the right graph represents a Reynolds number of 2.5 times 10 to the power of 5. Each graph includes data for three frequencies: 210 hertz, 360 hertz, and 420 hertz, represented by different symbols and colors. The envelope of the N-factor is shown as a solid black line, while experimental results are marked with circles. Theoretical predictions from Linear Stability Theory (LST) are indicated with dashed lines. The solid red lines denote the start and end of the Laminar Separation Bubble (LSB) for the case of Re_c equals 2.0 times 10 to the power of 5, and only the beginning of the LSB for Re_c equals 2.5 times 10 to the power of 5. The graphs illustrate how the N-factor evolves along the chordwise station, with the experimental results corresponding to the maximum N-factor during the boundary layer traverse at each station.

Figure 14.

Fluctuating velocity profile, $u'$ , integrated between 200 and 220 Hz and normalised by $u'_{max}$ and the eigenfunction corresponding to the 210 Hz mode obtained from the linear stability analysis at $x/c=0.55$ , $x/c=0.60$ , $x/c=0.625$ and $x/c=0.65$ . Where (a)–(d) and (e)–(h) correspond to ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively.

Figure 14. Long description

The image contains eight line graphs comparing fluctuating velocity profiles at different chordwise positions. Each graph plots the normalized fluctuating velocity (u’/U_max) against the normalized wall-normal coordinate (y/delta1). The graphs are arranged in two rows, with four graphs in each row. The top row (a-d) and the bottom row (e-h) correspond to different conditions. Each graph shows experimental data as circles and linear stability theory (LST) predictions as dashed lines. The frequency range of interest is between 200 and 220 Hz. The x/c values for the graphs are 0.550, 0.600, 0.625, and 0.650. The graphs illustrate how the velocity profiles vary at these positions, highlighting differences between experimental results and theoretical predictions.

At ${\textit{Re}}=2.0\times 10^5$ , from figure 13 the LST shows that the mode linked to $D_2$ at $210\,\text{Hz}$ can amplify upstream of the LSB, where the growth rate becomes more significant at $x/c\gt 0.4$ and an excellent match can be seen between the numerical and experimental results within the limits of the computation. This is followed by a saturation at $x/c\approx 0.8$ that corresponds to the trend in $u_{rms}$ in figure 6, while attaining an $N\approx 6$ , again corresponding to the N-factor that gave a best match between the experimental pressure distribution and that calculated by XFOIL. The mode shapes can be assessed from the eigenfunctions or the fluctuating velocity profile corresponding to a specific mode and from figure 14(a–d), the 210 Hz mode shows characteristics of a TS wave experiencing an adverse pressure gradient due to the presence of the middle lobe at $y/\delta _1\approx 1$ . In general a good match can be observed with respect to the experimental results and this demonstrates that at ${\textit{Re}}=2.0\times 10^5$ , the instability can be through amplification of a TS mode driven by duct mode $D_2$ or the mode resulting from the lock-in between $S_2/D_2$ .

After an initial growth upstream, the 360 Hz mode linked with $D_5$ remains damped until $x/c\gt 0.6$ , where it undergoes another minor growth towards the end of the numerical domain. The experimentally measured amplitudes were superposed on top of the numerical results and the former showed rapid growth at $x/c\gt 0.65$ , which corresponds to the beginning of the LSB. The numerical eigenfunctions of this mode in figure 15(a–d) are those of a TS mode, however, experimentally these profiles were not captured, as the real amplitudes of this mode were very low at $x/c\lt 0.65$ , as shown in figure 16. In fact, at the end of the experimental domain, this mode was approximately $O(10^3)$ lower than the one excited by $D_2$ at 210 Hz, so it was very unlikely to have an important contribution. The $420\,\text{Hz}$ mode in figure 13 is actually damped, therefore, the emergence of this mode experimentally in figure 11, is not necessarily a response from the duct mode $D_5$ and in fact it is a harmonic of the TS mode that started growing rapidly even upstream of the LSB as shown in figure 16.

Figure 15.

Fluctuating velocity profile, $u'$ , integrated between 350 and 370 Hz and normalised by $u'_{max}$ and the eigenfunction corresponding to the 360 Hz mode obtained from the linear stability analysis at $x/c=0.55$ , $x/c=0.60$ , $x/c=0.625$ and $x/c=0.65$ . Where (a)–(d) and (e)–(h) correspond to ${\textit{Re}}=2.0\times 10^5$ and ${\textit{Re}}=2.5\times 10^5$ , respectively.

Figure 15. Long description

The image contains eight graphs arranged in two rows and four columns. Each graph plots the fluctuating velocity profile normalized by the maximum velocity against the normalized height. The graphs compare experimental data with linear stability theory (LST) results for frequencies between 350 and 370 Hz. The top row (a-d) and bottom row (e-h) correspond to different chordwise positions (x/c) of 0.550, 0.575, 0.625, and 0.650. The experimental data is represented by circles, while the LST results are shown as dashed lines. The graphs illustrate the growth and damping of the 360 Hz mode at various positions along the chord.

Figure 16.

Growth of the instability modes represented by the maximum streamwise fluctuating velocity, $u'_{max}$ , of each mode along the streamwise direction. The solid red lines represent the start and end of the LSB for the case of ${\textit{Re}}=2.0\times 10^5$ and only the beginning of the LSB for ${\textit{Re}}=2.0\times 10^5$ .

Figure 16. Long description

Two line graphs depict the growth of instability modes along the streamwise direction for different Reynolds numbers. The x-axis represents the streamwise direction, while the y-axis represents the maximum streamwise fluctuating velocity on a logarithmic scale. The graphs show data for two different Reynolds numbers, 2.0 times 10 to the power of 5 and 2.5 times 10 to the power of 5. Each graph includes multiple data series represented by different symbols and colors, indicating various frequency ranges. The solid red lines mark the start and end of the laminar separation bubble for the case of 2.0 times 10 to the power of 5 and only the beginning for the case of 2.5 times 10 to the power of 5. The data points show how the instability modes grow or diminish along the streamwise direction, with some modes reaching higher velocities than others. The graphs illustrate the behavior of these modes under different conditions, providing insights into their development and interaction within the boundary layer.

On the other hand, at ${\textit{Re}}=2.5\times 10^5$ , from figure 13, the LST shows a growth of the 360 Hz mode linked with $D_5$ , right from the neutral point, with no damping as for the case of ${\textit{Re}}=2.0\times 10^5$ . It is important to note that this mode was purely a duct mode and there was no lock-in with the structural mode observed from figure 5, especially at ${\textit{Re}}\gt 2.0\times 10^5$ . When the experimental amplitude of this mode is adjusted to match the LST at $x/c\gt 0.55$ , similar trend is recovered. However, for the mode linked to $D_2$ , at ${\textit{Re}}=2.5\times 10^5$ it was more challenging to match the experimental amplitudes with the N-factor from the LST. From figure 13 the best match in order to extrapolate the trend past the LSB shows that the amplitude of the 210 Hz mode linked to $D_2$ at $x/c\lt 0.65$ is significantly lower than the 360 Hz mode linked to $D_3$ , which is not the case when looking at the spectra in figure 11 and is clearer from figure 16. This is the issue linked with arbitrarily matching the N-factor in the absence of measurement of the initial amplitude, $A_0$ , as mentioned earlier. In figure 15(e–h) the 360 Hz mode linked with $D_5$ has the classical TS mode shape and once again the contribution of the duct mode on the TS mode is highlighted. Unlike at ${\textit{Re}}=2.0\times 10^5$ the LST now shows that the 210 Hz mode, linked to $D_2$ is damped until $x/c\approx 0.4$ , but during the experiment the rapid growth was further delayed until $x/c\gt 0.6$ , coinciding with the location of the LSB. From figure 14(e–h) the peak amplitude in the 210 Hz mode $(D_2)$ is almost evenly distributed between the outer and inner mode, so this showed the presence of a KH mode as reported by Marxen et al. (Reference Marxen, Lang and Rist2012). Even if the computation cannot be extended to separation and beyond, the KH mode operated even upstream as confirmed by figure 14(e–h), where a good agreement can be observed between the LST and the experiment. In this case, both KH at 210 Hz and TS mode at 360 Hz, were operational upstream of the bubble. However, the KH mode grew more rapidly as shown in figure 16.

In summary, the impact of the duct modes is seen to be stronger on the TS modes where at ${\textit{Re}}=2.0\times 10^5$ it is driven by the $D_2$ and by $D_5$ at ${\textit{Re}}=2.5\times 10^5$ . This can be confirmed by the fact that the KH mode, when not dominated by the TS mode, appears to grow mainly in the vicinity and within the LSB. Nonetheless, in theory at ${\textit{Re}}=2.0\times 10^5$ the initial amplitude and growth rate of the perturbation is supposed to be lower between the two cases analysed here, but this was not the case. As reported in § 5.3 the earlier turbulent reattachment was through the nonlinear growth and destabilisation of TS modes that are significantly boosted upstream. This could be due to enhanced growth from acoustic forcing or model vibration, or a combination of both. From the accelerometer spectrogram in figure 5, this Reynolds number corresponds to the end of the tonal noise regime, so their contribution to acoustic forcing will be negligible; however, there was still a strong lock-in between structural and duct mode, $S_2/D_2$ , where a non-negligible level of vibration can be observed. As mentioned earlier in § 3 the root cause of this vibration cannot be isolated from the experiment, but if triggered by the unsteady loading due to the oscillation of the LSB, it could be regarded as a form of buffeting. To the authors’ knowledge this would thus constitute a first in terms of experimental demonstration of the impact of a three-way aero–vibroacoustic coupling on boundary layer instability.

5.3. Artificial forcing and destabilisation

The linear stability analysis in § 5.2 confirmed that the modes identified from the spectra in figure 11 are instabilities resulting from the response of the boundary layer to the perturbations generated by the aero–vibroacoustic interactions and lock-in between the duct and structural mode, $D_2/S_2$ . As seen in § 3, the 210 and 360 Hz modes emerging from the interaction with the aerofoil can excite boundary layer instability modes upstream of the LSB. However, the LST only shows rapid linear amplification of primary modes and cannot provide any detail on the destabilisation process. In the absence of spatiotemporal and 3-D measurements the exact source of the secondary interactions and nonlinear growth cannot be deduced from this experiment. Nevertheless, the development of the superharmonics in the spectra would point to the potential presence of Klebanoff modes. Here the aim was to check which mode was driving the breakdown process and to provide further insight into the mechanism linked to the shrinkage of the bubble at ${\textit{Re}}=2.0\times 10^5$ .

From figure 16, during the baseline natural forcing, the 210 Hz mode, associated with $D_2$ was the most dominant in both Reynolds number cases as it attained the threshold of $1\,\%$ of freestream velocity fluctuation, where nonlinear interactions can be expected. At ${\textit{Re}}=2.0\times 10^5$ , it took the form of a TS wave that underwent very rapid growth and destabilisation, as the primary mode attained the amplitude threshold at $x/c\approx 0.65$ and the first harmonic started growing rapidly at $x/c\approx 0.55$ before even encountering the LSB. Both the primary mode and the first harmonic at 410 Hz saturated at $x/c\approx 0.75$ . From figure 6 saturation of the $u_{rms}$ started at a similar position and from the oil droplet visualisation turbulent reattachment occurred at $x/c\approx 0.83$ . Saturation of the $u_{rms}$ is a better representation of breakdown as it represents the integration of the amplitudes within a larger frequency band and therefore incorporates the superharmonics, which will amplify during the secondary nonlinear growth. Rapid growth of the harmonic upstream of the LSB further corroborates the idea that destabilisation had already taken place upstream due to rapid growth of the TS mode, boosted by the lock-in during the aero–vibroacoustic interaction.

At ${\textit{Re}}=2.5\times 10^5$ , the 210 Hz mode linked with $D_2$ was mainly damped upstream and underwent more rapid growth at $x/c\approx 0.6$ , ahead of the bubble and experienced a jump in amplitude at $x/c\gt 0.65$ , within the LSB. The mode attained the threshold slightly later, at $x/c\approx 0.7$ and saturation was imminent around $x/c\approx 0.8$ . Breakdown and turbulent reattachment would take place around $x/c\approx 0.87$ , based on the saturation of the $u_{rms}$ in figure 6 and the oil droplet visualisation in figure 7(b). At this Reynolds number, the LST in figure 13 showed that the 360 Hz mode linked with $D_5$ was quite dominant, however, the spectra showed that this mode was slightly less energetic and, as confirmed from the amplitude growth in figure 16, it did not surpass the 210 Hz mode and did not attain the threshold of $1\,\%$ . The LST showed that it was a TS mode and was also corroborated by the experiment in figure 15(e–h). But, unlike the TS mode linked to $D_2$ at ${\textit{Re}}=2.0\times 10^5$ it did not amplify significantly upstream of the LSB to undergo destabilisation and breakdown. The first harmonic of this mode underwent rapid growth at $x/c\gt 0.675$ , which corresponds to the start of the LSB. This is due to the highly destabilising nature of the LSB and as shown by Xu et al. (Reference Xu, Mughal, Gowree, Atkin and Sherwin2017), the LSB tends to distort the TS wave and in the presence of a 3-D bubble this would lead to rapid formation of $\lambda$ -structures and breakdown. But here, the primary mode had not reached the $1\,\%$ threshold within the experimental domain, so this shows that it had not experienced critical destabilisation so as to undergo rapid breakdown and thus extending the length of the LSB. In the absence of 3-D spatiotemporal measurements, the interaction of the TS modes with the LSB cannot be quantified. However, since this is a nominally 2-D LSB the distortion will be less severe, unlike those presented by Xu et al. (Reference Xu, Mughal, Gowree, Atkin and Sherwin2017) and thus allowing the bubble to extend farther downstream.

Once again, the artificial excitation could be employed to confirm the scenario observed during natural forcing. At ${\textit{Re}}=2.0\times 10^5$ artificial excitation did not affect the transition scenario, as similar amplitudes were attained by the fundamental and harmonic modes with and without forcing in figure 16. However, the initial amplitude of the 360 Hz mode, which was also a TS mode at this Reynolds number, was boosted. The amplitude of the harmonic of this mode started growing rapidly while developing over the LSB, but since the primary mode did not attain the threshold of $1\,\%$ , it was unlikely to contribute to breakdown. On the other hand, excitation at ${\textit{Re}}=2.5\times 10^5$ boosted the initial amplitude of the 210 Hz KH mode; however, no significant difference in growth rate was observed with the naturally excited baseline case. The artificial excitation appears to have a stronger effect on the TS mode at 360 Hz, which was significantly boosted upstream of the bubble and also reached the threshold of $1\,\%$ at a similar position as both the naturally and artificially forced KH modes. This also led to forcing of the harmonics of the 360 Hz that grew more rapidly within the LSB. The harmonic of the 210 Hz mode attained earlier saturation, at lower amplitudes compared with the naturally excited case and attaining similar amplitudes as the harmonic of 210 Hz mode under artificial excitation. In this case, the TS and the KH will be expected to undergo earlier secondary interactions, but despite being significantly boosted, the TS mode was still not able to overtake the instability process upstream of the bubble, as is in the case of ${\textit{Re}}=2.0\times 10^5$ , and turbulent reattachment was at similar location to the naturally excited case. Here, the amplitude of the artificial forcing could not be further increased due to physical limitation of the excitor, in order to boost the 360 Hz mode at ${\textit{Re}}=2.5\times 10^5$ and to check it could overtake the KH mode like in the case of ${\textit{Re}}=2.0\times 10^5$ . Nevertheless, the artificial excitation confirmed that the unstable modes predicted by LST can be manipulated by aero–vibroacoustic interaction to modify the route to transition.