Investigating the flow field dynamics of transonic shock buffet using particle image velocimetry

All tests were performed in the Trisonic Wind Tunnel Munich (TWM) located at Bundeswehr University Munich. This two-throat blowdown facility allows for tests in the subsonic, transonic and supersonic regimes to be carried out successfully, with a Mach number range of \(0.2<M_\infty <3.0\). Fed by two pressurized tanks with a total volume capability of 356 m\(^3\), compressed dry air is flown through the 300 mm wide, 675 mm high and 1700 mm long test section. An adjustable de Laval nozzle upstream as well as an adaptive diffuser downstream of the test section allow for \(M_\infty\) to vary throughout wind tunnel operation. Figure 1 depicts the main components of the TWM.

Ranging from \(p_0=1.2\) to 5 bar, the total pressure can also be modified by adjusting the regulator valve accordingly. The maximum operational Reynolds number per unit length is \(\hbox {Re}=8 \times 10^7\) m\(^{-1}\). As for the turbulence level of the facility, it ranges between 1.1 and 1.4% for the tested Mach number range (Scharnowski et al. 2019). The TWM is equipped with sensors in both the test section and the settling chamber in order to accurately measure the stagnation temperature, the stagnation pressure as well as the static pressure. Typical wind tunnel run times are on the order of 100 s (Accorinti et al. 2021), whereas in this study, the runs lasted approximately 20 s due to the memory capacity of the high-speed cameras. Recordings were started only after confirming that the set pressure and Mach number values were reached and stable. Additional information regarding the TWM can be found in Scharnowski et al. (2019).

The model was designed according to the OAT15A supercritical airfoil geometry. This model was selected due to its substantial investigation both experimentally (Jacquin et al. 2009; Accorinti et al. 2021) and numerically (Deck 2005; Thiery and Coustols 2005). With a chord length of \(c=150\) mm and a span of \(s=298\) mm, the model’s aspect ratio is \(AR \approx 2\). The wing’s thickness increases up to 12.3% of the chord, before decreasing to 0.5%. This specific cut-off was selected in order to avoid a very thin trailing edge which would hinder manufacturing. The wing was manufactured out of carbon fiber reinforced polymer and was painted white to reduce surface heating. A speckle pattern was later added to the wing’s suction side in order to better evaluate its deformation (Accorinti et al. 2021). A thin strip, located in the middle of the model and along its chord, was painted with chrome in order to render the surface which would be exposed to the laser reflective. Prior to its application on the model, chrome spray paint was applied to a sample and tests were performed with full laser power to confirm its reflective capability. A spanwise dot pattern, located at \(x/c=0.07\), was also applied on both sides of the model to trip the boundary layer. Spaced 6 mm apart, these dots are approximately 70 \(\upmu\)m in height and 3 mm in diameter. Figure 2 illustrates the model used in this measurement campaign. The light green shaded area represents the region exposed to the laser, which was positioned downstream of the test section and shining upstream (marked by a dark green arrow in Fig. 2).

Since the mount was not perfectly rigid, deformations occurred during a wind tunnel run which led to a difference between the set and actual angles of attack of approximately \(\alpha _\mathrm{set} - \alpha =0.7^\circ\). The values of \(\alpha\) reported herein represent the actual angles of attack during the experiment, with an approximate uncertainty of \(\pm \, 0.2^\circ\). Although present, blockage due to the wing model was not deemed significant; at the highest set angle of attack of \(\alpha = 7^\circ\), the blockage ratio, defined as the ratio between the projected solid area and the cross-sectional area of the test section, was below 4%.

Planar PIV was performed on the suction side of the OAT15A model. Upstream of the settling chamber, two PIVTEC seeding generators seeded the flow with Di-Ethyl-Hexyl-Sebacat (DEHS) tracer particles via Laskin nozzles. The DEHS tracer particles have a mean diameter of \(d_p=0.4~\upmu\)m and an approximate density of \(\rho _p=910\) kg/m\(^3\) (Paredes et al. 2012). If the fluid’s dynamic viscosity is considered to be \(\mu _f=1.655 \times 10^{-5}~\)kg/ms and its density is \(\rho _f=1.37\) kg/m\(^3\), the response time of the tracer particles is approximately (Raffel et al. 2018):which is deemed appropriate for the current study apart from a small region downstream of the shock. A dual-pulse Nd:YAG (532 nm wavelength) laser from Photonics Industries was used for all measurements. A combination of mirrors and lenses placed outside of the test section was used to form a light sheet and shine the laser tangentially along the top surface of the model in order to minimize surface reflections (Kähler et al. 2006). The illuminated region is visualized in Fig. 2. At a repetition rate of 10 kHz, each individual head of the laser system can produce an average power of 150 W, a pulse energy of 15 mJ and a pulse width of 200 ns. A 5 \(\upmu\)s pulse separation was selected in order to reach a particle image displacement of approximately 13 pixel for the freestream velocity which is considered to be well-suited for high accuracy and low uncertainty, as discussed in Scharnowski and Kähler (2020). The repetition rate during data acquisition was set to 4 kHz. A DataRay laser beam profiler was used in order to estimate the laser sheet thickness. For the laser power used during the measurement campaign, the laser sheet thickness was approximately 0.5 mm. The 50 mm thick test section windows were manufactured from Poly(methyl methacrylate), commonly known as PMMA or acrylic, for mounting purposes. Therefore, care was taken to position the camera perpendicular to the window in order to reduce the adverse effects of refraction. The overall specifications of the experiment are summarized in Table 1.

For a single run, 10,000 double images were captured using the Phantom V2640 high-speed camera, for a total of 2.5 s. Since the buffet frequency was expected to be on the order of 100 Hz (Accorinti et al. 2021), several buffet cycles were captured during this time. LaVision’s DaVis software was used to process the images upon completion of data acquisition. A sliding background was subtracted with a kernel length of \(L=6\) pixel to pre-process the images. For all processing performed, a multi-pass interrogation algorithm was used; an interrogation window of \(96 \times 96\) pixel was used for the three initial passes, followed by a refined interrogation window of \(24 \times 24\) pixel for the last three passes. A 50% window overlap was used to highlight otherwise imperceptible small-scale features (Raffel et al. 2018). A two-dimensional median filter was also applied to remove outliers in the post-processing phase.

The measurement uncertainty of the estimated velocity data is influenced by many parameters (Sciacchitano and Wieneke 2016). While the freestream velocity is believed to be accurate within 1% according to the findings in Scharnowski et al. (2019), regions of higher fluctuations are subject to larger uncertainties. The three-dimensional nature of the turbulent flow causes loss-of-pairs due to out-of-plane motion that cause a weakened correlation signal and thus higher noise for the shift vector detection. Additionally, small scale turbulent structures cause inhomogeneous motion of the traces within an interrogation window, which broadens and weakens the correlation signal. The chosen magnification and particle image displacement are a compromise between accuracy, spatial resolution and dynamic spatial range, as discussed in (Scharnowski and Kähler 2020). The uncertainty is assumed to be on the order of 0.1 pixel (corresponding to \(\approx\) 3 m/s) for regions with highly turbulent flow. However, the spatial low-pass filtering by the interrogation windows, the lack of seeding in the near wall region and distortion due to the density changes can cause other significantly enhanced errors.

As mentioned, a range of Mach numbers and angles of attack were tested to shed light on the buffet phenomenon. More specifically, the buffet frequency as well as its variation with aerodynamic parameters were investigated. Runs were first performed at a constant \(\alpha =5.8^\circ\) and varying \(M_\infty\), followed by a constant \(M_\infty =0.74\) and varying \(\alpha\) investigation. It must be noted that the \(\alpha\) variation induced a slight variation in \(M_\infty\); this was accounted for, however, by varying the diffusor cross section accordingly.

As an example, Fig. 3 displays two instances of the instantaneous velocity field at \(M_\infty =0.77\) and \(\alpha =5.8^\circ\). In the top image, the shock is located at its upstream turning point, while in the bottom image, it is at its downstream turning point. For each vertical distance above the upper surface, the shock location was identified corresponding to the point where \(u=340~\)m/s. The detected shock is clearly visible in cyan, nearing the surface at \(x/c \approx 0.25\) and 0.4 in the top and bottom instances, respectively. The shock curvature is most pronounced when the shock is located most upstream; far from the surface of the wing \((y/c \approx 0.4)\), the shock resembles a normal shock, but becomes oblique as it reaches the surface of the wing. This oblique shock is a sign that the flow separates at the shock foot, which is also clearly observed as of \(x/c \approx 0.6\). In the second instance, reverse flow cannot be observed. The shock remains relatively vertical at \(x/c \approx 0.5\) for \(y/c > 0.2\), before shifting slightly upstream at \(x/c \approx 0.4\). It must be noted that the wind tunnel’s reference frame was selected in this figure, where the freestream velocity is exactly horizontal while the angle is pitched. The rotated coordinate system along the chord line of the airfoil is denoted as \(({\hat{x}}/c,{\hat{y}}/c)\) and will also be used throughout this manuscript.

Broadening the analysis, the shock position, \({\hat{x}}/c\), can be seen in Fig. 4, as a function of time for different normalized heights, \({\hat{y}}/c\). The two dotted lines correspond to the two instances highlighted in Fig. 3. Although not perfectly periodic, the different buffet cycles can clearly be seen, having an approximate period of 10 ms.

The shock position statistics are shown in Fig. 5 for a vertical distance above the upper surface equivalent to \(\approx 10\%\) of the chord length; this corresponds to \({\hat{y}}/c=0.17\). This value was chosen to easily compare results to Accorinti et al. (2022). The shock position is plotted as a function of both \(M_\infty\) (left) and \(\alpha\) (right). In order to highlight the amplitude of the shock motion, the 0.25 and 0.75 quantiles are displayed as well as the full range of all positions. In the case of a constant angle of attack (left plot), as the freestream Mach number is increased from \(M_\infty = 0.72\) to 0.74, the median shock position moves downstream. Starting from \(M_\infty =0.74\), the range of the shock motion signal becomes significantly larger, a sign that buffet onset has already taken place. Moreover, the inversion of shock motion occurs, which confirms the presence of large separation in the flow. Interestingly, the median shock position does not seem to vary between \(M_\infty =0.75\) and 0.77. In the case of a constant Mach number (right plot), strong shock oscillations are visible as of \(\alpha =5.7^\circ\). This angle is higher than the one corresponding to the inversion of shock motion (\(\alpha \approx 4.9^\circ\)). This confirms the findings in Accorinti et al. (2022), according to which the inversion of shock motion is a necessary condition for buffet onset in the case of increasing angle of attack. The results in Fig. 5 highlight differences with respect to the relevant literature in terms of buffet onset and offset. Experimentally, Jacquin et al. (2009) found that buffet onset occurred at \(\alpha =3.2^\circ\) for \(M_\infty =0.73\), whereas in the present study, it is delayed by approximately \(2^\circ\) for similar Mach numbers. Moreover, in the numerical simulations of Giannelis et al. (2018), shock buffet is quenched for \(M_\infty \ge 0.75\), while in the present work, shock oscillations can still be detected for \(M_\infty \ge 0.77\). These differences were already addressed in Accorinti et al. (2022). Potential causes, such as three-dimensional effects and wind tunnel characteristics, were suggested. In the same study, a detailed comparison of numerous literature results revealed a general sensitivity of the buffet features (frequency and amplitude of the oscillations, mean shock location as well as onset) to the numerical and experimental boundary conditions.

Taking a closer look at the temporal evolution of the shock location, Fig. 6 helps visualize the shock location and its motion for a variety of \(M_\infty\) (top) and \(\alpha\) (bottom), in the wing’s reference frame. Similar to what was displayed in Fig. 5 for one wall-normal distance, 20 time steps are visually depicted in different colors for a variety of wall-normal distances. The dashed line represents the median shock location, while the dotted lines indicate 99% of the full shock motion. Only the upstream travel of the shock was plotted for readability.

At \(t=0\) ms, the shock is positioned most downstream, while it completes half its full cycle at \(t=5\) ms; this remains true for all flow conditions. As \(M_\infty\) increases, the shock travels upstream and its amplitude increases. At \(M_\infty =0.77\), the oblique shock foot is most pronounced, spanning a larger streamwise distance. This suggests the presence of a stronger boundary layer separation. That being said, the shock motion remains larger at \(M_\infty =0.75\) than at 0.77.

Similarly, the shock location appears to move upstream with an increase in angle of attack. The different time steps almost collapse at \(M_\infty =0.74\) and \(\alpha =5.3^\circ\), indicating the initial stage of shock buffet. As \(\alpha\) increases, the shock curvature increases and the oblique shock foot is present for a wider \({\hat{y}}/c\) range; at \(\alpha =5.3^\circ\), the oblique shock ends around \({\hat{y}}/c=0.2\), while it ends at \({\hat{y}}/c=0.3\) when \(\alpha =6.3^\circ\). This indicates an increase in the extent of boundary layer separation.

The velocity data were further analyzed in a limited time interval (\(\Delta t=100~\)ms) to give a qualitative overview of the boundary layer evolution and the variation in wake size. In Fig. 7, the streamwise velocity, u, is plotted as a function of both time, t, and non-dimensional streamwise position, \({\hat{x}}/c\), for both pre-buffet and buffet cases. The data shown here were extracted at a fixed \({\hat{y}}/c=0.1\) and \(\alpha =5.8^\circ\). In the top image, \(M_\infty =0.72\), while it is 0.77 in the bottom figure. In both scenarios, the angle of attack is kept constant at \(5.8^\circ\) and 100 ms of data are shown, equivalent to approximately 11 buffet cycles. This allows for a much longer visualization of the temporal evolution of the shock wave compared to Fig. 6. The dashed line represents the median shock position, while the dotted lines indicate 99% of the full shock motion. The shock position ranges from \({\hat{x}}/c=0.35\) to 0.43 and from 0.28 to 0.46 in the pre-buffet and buffet cases, respectively. At \(M_\infty =0.72\), the shock buffet phenomenon has not yet reached a fully developed state as large-scale amplitudes and reverse flow downstream of the shock are absent. At \(M_\infty =0.77\), however, developed shock buffet can be observed. As expected, strong boundary layer separation is observed when the shock is most upstream, at \({\hat{x}}/c \approx 0.3\). As the shock moves downstream, the boundary layer reattaches and a subsequent cycle is initiated.

Probing further into the wake structure, the streamwise velocity in the wake can be seen in Fig. 8 as a function of both time and normalized vertical distance. The horizontal position was, in this case, set to \({\hat{x}}/c=1.3\), the angle of attack was again kept constant at \(\alpha =5.8^\circ\) and the Mach number was increased from \(M_\infty =0.72\) to 0.77 in the top and bottom figures, respectively. It must be noted that both Figs. 7 and 8 are associated with the same PIV snapshots. In the top figure, the streamwise velocity appears to range around \(\approx\) 150 m/s and does not vary significantly throughout the shock buffet cycle. However, in the buffet case, a much larger velocity range can be observed, spanning between 0 and 200 m/s.

The width of the wake also drastically increases when the shock is located most upstream. The two dotted lines in Fig. 8 represent the two instances shown in Fig. 3. The width of the separated region is larger for the first instance (shock at its upstream turning point) than for the second (shock at its downstream turning point), reaching a maximal vertical distance of \({\hat{y}}/c \approx 0.3\). Figures 7 and 8 corroborate the findings of Jacquin et al. (2009), who phase averaged the flow field’s Mach number distribution and found a maximal thickening of the wake when the shock was at its most upstream location.

A spectral analysis of the shock location was carried out in order to extract the dominant frequency of the shock motion. Figure 9 displays the spectra at various aerodynamic parameters. On the left, the spectra at \(M_\infty =0.74\), 0.75 and 0.77 are plotted at a constant \(\alpha =5.8^\circ\). On the right, \(M_\infty\) is kept constant at 0.74, while the angle of attack is varied from \(\alpha =5.3\) to \(6.3^\circ\). The normalized power spectral densities are plotted against the reduced frequency, \(k=\pi f c/u_\infty\), where f is the frequency and \(u_\infty\) is the freestream velocity, the latter of which was accurately obtained from instrumentation within the wind tunnel.

An increase in either \(M_\infty\) or \(\alpha\) induces a positive shift in reduced frequency. As \(M_\infty\) increases from 0.74 to 0.77, the dominant value of k increases from 0.20 to 0.22. Similarly, k varies from 0.14 to 0.25 as \(\alpha\) increases from 5.3 to \(6.3^\circ\). Although both trends corroborate the findings of Accorinti et al. (2022), the exact values of the dominant frequency slightly differ. This might be due to the slightly different rigidity between models used. The buffet frequency found for \(M_\infty =0.74\) and \(\alpha =5.8^\circ\) is very similar to those found in Jacquin et al. (2009) and Giannelis et al. (2018) for \(M_\infty =0.73\) and \(\alpha =3.5^\circ\), which are 0.21 and 0.22, respectively.

In order to confirm the validity of the results herein, simultaneous pressure measurements were obtained from XCQ-062 pressure transducers located on the upper surface of the wing model and mounted along the chord direction. Their acquisition rate was set to 100 kHz. Figure 10 displays the normalized power spectral density of the pressure signal, \(\phi _p\), as a function of k, for two distinct pressure taps; the first pressure tap was located at \({\hat{x}}/c=0.42\) (top plots), while the second was further downstream at \({\hat{x}}/c=0.57\) (bottom plots). Since the shock hovered around the first pressure tap under certain flow configurations, it was deemed useful to include data from a second pressure tap as a comparative measure. This second tap captured the frequency content of the boundary layer more and hence recorded lower power spectral density. For each pressure tap, the variation in both the freestream Mach number (left plots) and the angle of attack (right plots) was investigated, while maintaining the other variable constant. Focusing on the left plots, as \(M_\infty\) increases, the value of k corresponding to a peak in \(\phi _p\) also increases. Focusing particularly on the data obtained at \({\hat{x}}/c=0.42\), the peak is smaller for the \(M_\infty =0.77\) case than for the other two cases. Recalling Fig. 5, given that the shock is rarely positioned at \(x/c=0.42\) although the shock motion remains large, this might lead to a reduced signal at \(M_\infty =0.77\). However, at the same Mach number but slightly downstream at \(x/c=0.57\), there does not appear to be a pronounced peak at all. Three phenomena could cause this flattening, notably the strong boundary layer separation dampening the peak, the buffet phenomenon moving towards its offset and a stronger fluid structure interaction with heave. The power spectral density peak also seems to shift towards greater values of k as \(\alpha\) increases. This agrees well with the results shown in Fig. 9 as well as the findings of McDevitt and Okuno (1985). For instance, at \({\hat{x}}/c=0.42\) and \(\alpha =5.8^\circ\), the dominant reduced frequency increases from \(k=0.19\) at \(M_\infty =0.74\) to \(k=0.23\) at \(M_\infty =0.77\), while it ranged from \(k=0.20\) to 0.22 in Fig. 9. Similarly, a positive shift occurs with an increase in \(\alpha\); again at \({\hat{x}}/c=0.42\), the dominant reduced frequency increases from \(k=0.17\) to 0.24 as \(\alpha\) increases from 5.3 to \(6.3^\circ\), whereas it varied from \(k=0.14\) to 0.25 in Fig. 9. The same peaks are found with the pressure tap slightly downstream, at \({\hat{x}}/c=0.57\).

Looking closely at the power spectral density obtained via the pressure sensor at \({\hat{x}}/c=0.42\), two smaller peaks can be observed at \(k \approx 1.8\) and 3.3 (or, equivalently, 920 and 1690 Hz). The peak at \(k \approx 1.8\) can also be seen in the PIV data, for the case of \(M_\infty =0.74\) and \(\alpha =5.3^\circ\). The origin of the peaks at higher frequency than the fundamental buffet frequency could be upstream travelling waves (UTW), although this hypothesis remains to be further investigated. D’Aguanno et al. (2021) observed UTW at a tenfold higher frequency than the buffet frequency (in their case 2000 Hz, or equivalently, \(k=2.76\)), which compares well with both the findings of Hartmann et al. (2013) and the current study. The slightly smaller peak at \(k \approx 3.3\) could potentially be a higher harmonic of the peak indicative of UTW. It must be noted that both peaks are approximately two orders of magnitude smaller than the one indicative of shock buffet.

Additionally, the power spectral density was computed for each streamwise velocity fluctuation obtained via PIV as a comparative measure to the two previous spectral analyses. A field average was later computed, from which the dominant frequency of \(f_{\text {dom}}=98~\)Hz or, equivalently, \(k_{\text {dom}}=0.19\), was extracted for the case where \(\alpha =5.8^\circ\) while \(M_\infty =0.74\). This frequency closely matches the results obtained previously via shock location and surface pressure measurements. Figure 11 highlights the regions exhibiting this dominant frequency; the normalized power spectral density at \(k_{\text {dom}}\) is plotted within the field of view. Interestingly, this dominant frequency appears mostly at the shock and the wake regions, showcasing a hundredfold increase in energy compared to the remaining flow field. This result closely matches that of Crouch et al. (2007).

According to the model derived by Lee (2001), shock buffet is sustained by a feedback mechanism between downstream and upstream travelling waves. The velocity of the downstream travelling waves was estimated in Lee (2001), Xiao et al. (2006), Deck (2005), Jacquin et al. (2009), Garnier and Deck (2010) as well as Hartmann et al. (2013) by correlating the pressure signals on the upper surface. In Hartmann et al. (2013), the convection velocity was also computed by correlating the absolute velocity fluctuations at various streamwise positions along a time-averaged streamline above the region of separated flow, yielding a good agreement with the one obtained via pressure sensors. Determining the convective velocity at which the downstream travelling waves travel is an important part of the validation of Lee’s model.

Due to the available highly resolved field information obtained via PIV, a similar analysis was performed herein, where the streamwise velocity correlation, \(R_{uu}\), was computed at various streamwise and wall-normal positions close to the model’s upper surface. \(R_{uu}\) is defined as follows:where \(u_n'(t)\) and \(u_m'(t+\tau )\) represent streamwise velocity fluctuation signals at positions \((x_n/c,y_n/c)\) and \((x_m/c,y_m/c)\) respectively, the second data set having been shifted by \(\tau\). Their product is then averaged over all time steps, t. As for \(\sqrt{\overline{u_n'(t)^2}}\) and \(\sqrt{\overline{u_m'(t)^2}}\), they represent the standard deviation of both fluctuating velocity signals.

The correlation of the streamwise velocity was computed for different values of \(({\hat{x}}_n/c,{\hat{y}}_n/c)\), ranging from \(0.6 \le {\hat{x}}_n/c \le 1.1\) and \(0.10 \le {\hat{y}}_n/c \le 0.20\), in order to examine how flow structures convect within and slightly above the wake.

Color-coded between \(-1\) and 1, Fig. 12 displays \(R_{uu}\) for three consecutive time steps, where \(\tau = -0.25\) and \(\tau =0.25~\)ms are both associated with a negative and positive shift of one snapshot, respectively. The x-shaped marker represents the \(({\hat{x}}_n/c,{\hat{y}}_n/c)\) position, while the circular marker represents the maximum \(R_{uu}\) value along \({\hat{y}}_n/c\) for a given \(\tau\). That being said, the x-shaped marker position remains unchanged as \(\tau\) varies, while the circular marker detects the streamwise position which is most correlated to the origin for a specific \(\tau\) and therefore varies in time. Detecting this maximum enables the computation of the convective velocity, \(u_c\), defined as:

Since data with two non-zero values of \(\tau\) were computed, an average velocity was extracted:Figure 13 is a compilation of the average convective velocity for different values of \({\hat{x}}_n/c\) and \({\hat{y}}_n/c\), where three compelling aspects can be highlighted. Firstly, the convection velocity exhibits an increase as it moves downstream. Focusing on the pink data points representing \({\hat{y}}_n/c=0.19\), \(u_{c,\mathrm{avg}}=79~\)m/s (\(\approx 0.33u_\infty\)) at \({\hat{x}}_n/c=0.6\), while it is rather \(\approx 150~\)m/s (\(\approx 0.63u_\infty\)) at \({\hat{x}}_n/c=1.1\). A similar trend can be recognized for the other wall-normal positions. A streamwise increase of the downstream convection velocity was also reported in Lee (2001) for some aerodynamic cases. However, in Xiao et al. (2006), Deck (2005), Jacquin et al. (2009), Garnier and Deck (2010) as well as Hartmann et al. (2013), constant downstream convection velocities were found: 0.12\(u_\infty\), 0.05\(u_\infty\), 0.07\(u_\infty\), 0.06\(u_\infty\) and 0.08\(u_\infty\) respectively.

Secondly, the values of convection velocity measured in the mentioned literature are significantly smaller than the ones in Fig. 13. Nonetheless, it should be recalled that in the present work, the convection velocity was not estimated at the model’s surface but at a certain distance from it (between \({\hat{y}}_n/c=0.1\) and \({\hat{y}}_n/c=0.2\)). Therefore, the so-computed velocity could correspond to vortical structures travelling within the mixing layer. In Jacquin et al. (2009), it was stressed that if one associated the downstream travelling waves of Lee’s model to Kelvin-Helmholtz type structures in the mixing layer instead of to perturbations at the surface, a value on the order of 0.5\(u_\infty\) should be expected for the convection velocity. Neither in the present work nor in Xiao et al. (2006), Deck (2005), Jacquin et al. (2009) or in Garnier and Deck (2010) was the estimated shedding frequency of the vortical structures associated with the computed convection velocities. Therefore, no proof was provided that these very downstream travelling structures are responsible for the formation of upstream travelling waves. Only in Hartmann et al. (2013) was a model proposed, which connects downstream travelling vortical structures detected via PIV to the generation of acoustic waves. The characteristic length of the vortices was extracted from temporal snapshots. The velocity of the vortices was not directly measured. Instead, the hypothesis was made that they travel at the convection velocity (0.08\(u_\infty\)) estimated via cross-correlation of the pressure signal at the surface. The shedding frequency of the vortices, computed by dividing their velocity by their characteristic length, agreed well with the one of the UTW. Based on the equality between these two shedding frequencies, Hartmann et al. (2013) gave a detailed description of the physical mechanism of shock buffet. The variation in strength of the vortical structures passing by the trailing edge and the pulsation of the separated boundary layer modulated the strength of acoustic UTW. The variation in intensity of the acoustic waves sustained the shock oscillation and completed the feedback. However, the vortices detected via PIV lay at a distance from the upper surface that is comparable to the ones investigated in the present study. Therefore, it cannot be excluded that their convection velocity should be considered higher. This would result in a mismatch between the shedding frequencies of the downstream travelling vortices and of the upstream travelling waves and would weaken the model proposed by Hartmann et al. (2013). This mismatch was found in D’Aguanno et al. (2021), where downstream travelling vortical structures were detected in the shear layer via autocorrelation of POD-reconstructed PIV snapshots. Their characteristic length was also extracted from time snapshots and their velocity was assumed to be equal to the average of the streamwise velocity in the shear layer region where the correlation was performed (\(\approx 0.5u_\infty\)). The resulting shedding frequency of the vortices was approximately three times higher than the one computed for the UTW. It should be stressed that both in D’Aguanno et al. (2021) and Hartmann et al. (2013) was an assumption made for the velocity of the vortices, which led to two remarkably different results. Further research is needed to understand which of the downstream travelling structures are actually responsible for the formation of the upstream travelling waves at the trailing edge.

Finally, a general sensitivity of the convection velocity to the wall-normal distance can also be observed in Fig. 13. Although the trend of the convection velocity appears to converge irrespective of \({\hat{y}}_n/c\) for a given \({\hat{x}}_n/c > 0.7\), differences on the order of 10 m/s can be spotted at several streamwise positions. A closer look was taken to understand the spread around \({\hat{x}}_n/c=0.6\). The lower values of \(u_{c,\mathrm{avg}}\) occur at \({\hat{y}}_n/c=0.16\) and 0.17, which correspond to the wake-freestream interface, where a distinct correlation peak is not present. Additionally, the sudden drops in \(u_{c,\mathrm{avg}}\), which sometimes occur for \({\hat{y}}_n/c \ge 0.17\), are a result of the wake-freestream interface’s motion. An upward shift in the interface would lead to a sudden uncorrelation between the origin and the structures downstream. Uncertainties in estimating the location of maximal \(R_{uu}\) might also lead to the discontinuous increase in \(u_{c,\mathrm{avg}}\). The correlating flow structures are of different size and shape and, consequently, the correlation shows a rather broad and weak peak. This makes it difficult to estimate its maximum location accurately. The uncertainty of the maximum locations for \(\tau =\pm \, 0.25\) ms is estimated to be on the order of \(5\%\) of the chord length. This leads to the spread of the estimated convection velocity shown in Fig. 13.