3.1. Aerodynamic forces

Lift and drag coefficient polars for the aerofoil under all tested conditions of free stream turbulence are presented in figure 4. The lift coefficient from thin aerofoil theory is also plotted for comparison. The maximum solid blockage was less than $7\,\%$ , for which the errors in lift coefficients due to blockage are expected to be less than $10\,\%$ (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012a ). The reduction in lift coefficient relative to a true two-dimensional aerofoil, due to the presence of the gaps between the tips of the model and the test section walls, is estimated to be of the order of $0.1$ based on the results of Marchman (Reference Marchman1987). In the clean flow ( $\textit{Tu}=0.1\,\%$ ), the pre-stall lift is highly nonlinear with respect to the angle of attack and exceeds that predicted by thin aerofoil theory for $\alpha \leq {6}{^\circ }$ . Nonlinear lift at moderate angles of attack is typical of aerofoils operating at aerodynamically low Reynolds numbers (e.g. Ohtake, Nakae & Motohashi Reference Ohtake, Nakae and Motohashi2007; Winslow et al. Reference Winslow, Otsuka, Govindarajan and Chopra2018) and is attributed to the increased local suction that occurs due to the presence of a short LSB (Tani Reference Tani1964). Bursting of the LSB for $\alpha \gt {8}{^\circ }$ leads to an abrupt stall, decreasing lift and increasing drag.

Figure 4. (a) Lift and (b) drag coefficients. Error bars denote uncertainty ( $95\,\%$ confidence).

When the FSTI is increased to $4\,\%$ , the stall angle increases substantially and the lift at low angles of attack is relatively more linear, in agreement with the effects of moderate FSTI observed in previous studies (Kay et al. Reference Kay, Richards and Sharma2020; Hrynuk et al. Reference Hrynuk, Olson, Stutz and Jackson2024). These changes are attributed to the free stream turbulence causing expedited transition, which reduces the size of the LSB and its associated pressure plateau (e.g. Istvan et al. Reference Istvan, Kurelek and Yarusevych2018). There are no significant differences in pre-stall lift as the FSTI increases from $\textit{Tu}=4\,\%$ to $7\,\%$ , but the post-stall decrease in lift becomes less abrupt. At $\textit{Tu}=13\,\%$ , there is a substantial reduction in lift at pre-stall angles of attack, and the post-stall decrease in lift and increase in drag become notably more gradual. These changes are attributed to a continued reduction in the extent of separated flow at very high free stream turbulence intensities, which will be explored in § 3.3. The observed decrease in lift slope and more gradual stall with increasing FSTI are in agreement with the measurements of Devinant et al. (Reference Devinant, Laverne and Hureau2002). There is no significant change in aerofoil performance between integral length scales of $\varLambda _{ux}/c=1$ and $2$ at pre-stall angles of attack (compare solid and dashed lines). For $\textit{Tu}=7\,\%$ , there is a decrease in post-stall lift at the larger length scale. It is speculated that this decrease may be caused by a delay in transition due to a reduction in the amplitudes of velocity fluctuations in the free stream at the length scales to which the boundary layer is most receptive.

3.2. Overview of boundary layer transition

To investigate the mechanisms responsible for the changes in aerofoil performance at high FSTI with large integral length scales, side-view PIV measurements of the suction surface boundary layer were obtained at $\alpha ={5}{^\circ }$ , which is within ${2}{^\circ }$ of the angle of maximum lift to drag ratio for all tested cases, and at $\alpha ={12}{^\circ }$ , which represents an angle at which performance is severely degraded at low FSTI.

The boundary layer transition process on the suction surface of the aerofoil at $\alpha ={5}{^\circ }$ is illustrated using instantaneous snapshots of spanwise vorticity ( $\omega = \partial v /\partial x - \partial u /\partial y$ ) in figure 5 for $\textit{Tu}=0.1\,\%$ , $7\,\%$ and $13\,\%$ . The snapshots are separated by $0.06c/\overline {u_\infty }$ and are ordered with increasing time from top to bottom. The spanwise vorticity was computed from the circulation on the contour of the eight neighbouring velocity vectors around each PIV grid point (Reuss et al. Reference Reuss, Adrian, Landreth, French and Fansler1989) and smoothed using a $3\times 3$ spatial average. Instantaneous streamwise velocity and spanwise vorticity fields from time-resolved PIV measurements for $\textit{Tu}=0.1$ , $4\,\%$ , $7\,\%$ and $13\,\%$ at $\varLambda _{ux}/c=1$ are available in supplementary movie 1 available at https://doi.org/10.1017/jfm.2025.747.

Figure 5. Instantaneous snapshots of spanwise vorticity. Time increases from top to bottom and the snapshot separation is $0.06c/u_\infty$ .

In the clean flow (figure 5 a), due to the relatively low amplitudes of incoming disturbances, the boundary layer remains laminar over a substantial distance, leading to laminar boundary layer separation in the region of adverse pressure gradient. The separated laminar shear layer rolls up into relatively large-scale coherent vortices near $x/c=0.55$ as a consequence of the Kelvin–Helmholtz instability mechanism (e.g. Ho & Huerre Reference Ho and Huerre1984; Dovgal, Kozlov & Michalke Reference Dovgal, Kozlov and Michalke1994). The increase in wall-normal momentum transport produced by the vortices leads to reattachment and a turbulent boundary layer develops as the spanwise vortices break down into small-scale turbulence for $x/c\gt 0.6$ . This transition scenario is consistent with the features typical of short LSBs (Marxen & Henningson Reference Marxen and Henningson2011). The location of shear layer roll-up is observed to remain relatively consistent in time.

With increasing FSTI (figures 5 b and 5 c), higher amplitudes of boundary layer disturbances are generally expected to lead to earlier transition in a time-averaged sense (e.g. Simoni et al. Reference Simoni, Lengani, Ubaldi, Zunino and Dellacasagrande2017; Jaroslawski et al. Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023). The location of transition in the vorticity field snapshots can be inferred from the onset of relatively small-scale spatial variations in spanwise vorticity. Interestingly, the snapshots in figure 5 show that substantially greater variations in the location of transition occur with increasing FSTI. For $\textit{Tu}=13\,\%$ , the variations in transition location are large enough to cause the boundary layer to occasionally remain laminar across the entire PIV field of view (figure 5 c, bottom panel). It is likely that these variations are a consequence of the relatively large integral length scale of the free stream turbulence, which leads to temporal variations in effective angle of attack and incoming flow velocity. The relationship between the variations in transition location and large-scale free stream velocity fluctuations will be investigated in detail in § 3.5.

In addition to the location of transition, the FSTI also affects the development of vortical structures formed in the transition process (e.g. Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019). Whereas the roll-up vortices formed in the clean flow are relatively large and coherent, the breakdown of the laminar shear layer at $\textit{Tu}=7\,\%$ involves less organised, finer-scale structures (figure 5 b). At $\textit{Tu}=13\,\%$ , the inclined regions of increased vorticity within the boundary layer are consistent with the lift-up of low-speed streaks that eventually decay into turbulent spots (Nolan & Walsh Reference Nolan and Walsh2012). The presence of streaks is evidence that bypass transition occurs at $\textit{Tu}=13\,\%$ . The snapshots presented in figure 5 illustrate that large-scale free stream turbulence can cause substantial changes in the mean flow development and the transition dynamics relative to the clean flow.

3.3. Mean flow development

Mean streamwise velocity ( $\overline {u}$ ), standard deviation of streamwise velocity ( $\sigma _u$ ) and standard deviation of wall-normal velocity ( $\sigma _v$ ) fields for $\textit{Tu}=0.1\,\%$ , $7\,\%$ and $13\,\%$ at an integral length scale of $\varLambda _{ux}/c=1$ are presented in figures 6 and 7 for $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively. The mean streamwise velocity fields were obtained from the average correlation method, whereas the other fields were computed as the average of the instantaneous PIV vector fields. Also plotted is the mean displacement thickness, defined as $\overline {\delta ^*}=\int _{0}^{\overline {\delta }} (1-\overline {u}/\overline {u_{{e}}} ) \,{\rm d}y$ , where $\delta$ is taken as the wall-normal distance to the location of the maximum streamwise velocity ( $u_{{e}}$ ) at a given $x$ location. The inset plots in figures 6(b), 6(c), 7(b) and 7(c) show the near-wall structure of the mean flow with an enlarged $y/c$ scale. The results for $\textit{Tu}=4\,\%$ are similar to those for $\textit{Tu}=7\,\%$ and are not shown for brevity. Likewise, there are no notable differences between the mean and fluctuating flow fields for $\varLambda _{ux}/c=2$ and those shown for $\varLambda _{ux}/c=1$ , consistent with the measured lift coefficients (figure 4).

Figure 6. Mean streamwise velocity fields (top row), and standard deviation of streamwise (middle row) and wall-normal (bottom row) velocity fields for $\alpha ={5}{^\circ }$ . Dashed lines, $\overline{\delta^*}$ .

Figure 7. Mean streamwise velocity fields (top row), and standard deviation of streamwise (middle row) and wall-normal (bottom row) velocity fields for $\alpha ={12}{^\circ }$ . Dashed lines, $\overline{\delta^{*}}$ .

The measurements in the clean flow for $\alpha ={5}{^\circ }$ (figure 6 a) show the development of a relatively thick LSB on the suction surface of the aerofoil. The two darkest blue contours in the mean streamwise velocity plot outline the extent of mean reverse flow in the LSB, with separation at $x/c=0.25$ and reattachment at $x/c=0.58$ . Transition of the separated shear layer is evidenced by the rapid increase in velocity fluctuations that occurs near the location of maximum bubble thickness ( $x/c=0.49$ ). The ratio of the lengths of the laminar and turbulent portions of the LSB is $2.7$ , suggesting that this LSB should be classified as short (Marxen & Henningson Reference Marxen and Henningson2011).

At $\alpha ={5}{^\circ }$ and $\textit{Tu}=7\,\%$ (figure 6 b), the reverse flow region is nearly eliminated. However, the increase in streamwise velocity at $y/c\approx 0.0025$ for $0.5\lt x/c\lt 0.55$ is similar to the flow structure seen near the reattachment location in the clean flow and suggests that a thin LSB may intermittently form at $\textit{Tu}=7\,\%$ . Furthermore, the local maximum in streamwise velocity fluctuations and increase in wall-normal velocity fluctuations that occurs near $x/c=0.5$ are consistent with LSB formation, since vortex shedding from the separated shear layer is expected to cause an increase in streamwise and wall-normal velocity fluctuations near the reattachment location (Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014; Dellacasagrande et al. Reference Dellacasagrande, Barsi, Lengani, Simoni and Verdoya2020). Evidence of vortex shedding in the POD spatial modes for these conditions will be discussed in § 3.4.

At $\alpha ={5}{^\circ }$ and $\textit{Tu}=13\,\%$ (figure 6 c), the FSTI is sufficiently high that the presence of an LSB is not evident in the mean or fluctuating velocity fields. The region of increased streamwise velocity fluctuations near the wall for $x/c\leq 0.5$ without a corresponding increase in wall-normal velocity fluctuations suggests that transition at this level of FSTI is dominated by streak formation and breakdown rather than separated shear layer roll-up.

The mean flow fields presented in figure 6 show that increasing the FSTI progressively decreases the extent of separated flow at $\alpha ={5}{^\circ }$ . Because the LSB produces a low-pressure plateau in the surface pressure distribution (Horton Reference Horton1968), the decrease in lift coefficient with increasing FSTI is largely attributed to the reduction in streamwise extent of the LSB.

The reason for the substantial change in aerofoil performance at $\alpha ={12}{^\circ }$ with varying FSTI (figure 4) is evident from the mean flow fields for this angle of attack presented in figure 7. For the clean flow (figure 7 a), the reduced lift and increased drag are due to massive separation from the suction surface of the aerofoil. In this case, the amplitudes of free stream disturbances are too low for transition to occur rapidly enough to enable reattachment.

At $\textit{Tu}=7\,\%$ , the extent of separation is substantially reduced because of more rapid boundary layer transition (figure 7 b). However, a thin region of mean reverse flow was detected at the upstream end of the field of view for $\textit{Tu}=7\,\%$ and the streamwise velocity contours in the inset are consistent with those expected near the reattachment location of an LSB. Thus, a thin laminar separation bubble persists on the aerofoil for $\textit{Tu}=7\,\%$ . The presence of the LSB is consistent with the region of stronger wall-normal velocity fluctuations near the aerofoil surface for $x/c\lt 0.2$ . The persistence of mean laminar separation up to at least $\textit{Tu}=7\,\%$ is attributed to the relatively low Reynolds number and strong adverse pressure gradient for the aerofoil at $\alpha ={12}{^\circ }$ . At $\textit{Tu}=13\,\%$ , no reverse flow was detected in the PIV measurements (figure 7 c). However, the increase in streamwise velocity near the wall and plateau in $\delta ^*$ for $x/c\lt 0.4$ suggest that a thin LSB may form upstream of the field of view.

3.3.1. Mean transition

In this section, the relationship between transition and time-averaged suction surface boundary layer properties is explored. It serves to define a transition threshold for the entropy-based transition detection method that is consistent with other boundary layer transition indicators commonly applied in low-Reynolds-number flows over aerofoils. The entropy-based method is used because it is desired to characterise instantaneous variations in the transition location in § 3.4 without requiring temporal windowing for calculation of Reynolds shear stresses or integral boundary layer parameters. Figures 8(a) and 8(b) present the Reynolds shear stress ( $-\overline {u'v'}$ ) measured at $y=\overline{\delta^*}$ for $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively. The shape factors ( $H=\overline {\delta ^*}/\overline {\theta }$ , where $\overline {\theta }=\int _{0}^{\overline {\delta }}\overline {u}/\overline {u_{{e}}}(1-\overline {u}/\overline {u_{{e}}})\, {\rm d}y$ ) for $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ are plotted in figures 8(c) and 8(d), respectively.

Figure 8. (a,b) Reynolds shear stress at $y=\overline{\delta^*}$ , (c,d) shape factor and (e,f) global entropy from P-POD at (a,c,e) $\alpha ={5}{^\circ }$ and (b,d,f) $\alpha ={12}{^\circ }$ . Dashed lines, $-\overline {u'v'}=0.001\overline{u_e}^2$ ; dotted lines, maximum shape factor; shaded areas, uncertainty ( $95\,\%$ confidence).

Because a rapid increase in the magnitude of the Reynolds shear stress is indicative of transition, several transition criteria for low-Reynolds-number boundary layers have been formulated based on the Reynolds shear stress (e.g. Lang et al. Reference Lang, Rist, Wagner, Lang and Wagner2004; Ol et al. Reference Ol, McCauliffe, Hanff, Scholz and Kähler2005; Burgmann & Schröder Reference Burgmann and Schröder2008). Moreover, the streamwise location of maximum Reynolds shear stress has been shown to occur near the location of reattachment in LSBs (e.g. Lengani et al. Reference Lengani, Simoni, Ubaldi, Zunino and Bertini2017). The location where $-\overline {u'v'}$ first exceeds $0.001\overline{u_e}^2$ is indicated by the dashed lines in figures 8(a) and 8(b). This threshold for $-\overline {u'v'}$ is based on the transition threshold used by Ol et al. (Reference Ol, McCauliffe, Hanff, Scholz and Kähler2005) and Hain, Kähler & Radespiel (Reference Hain, Kähler and Radespiel2009). Other researchers have used the location of maximum shape factor to define the mean location of transition in LSBs (e.g. Michelis, Yarusevych & Kotsonis Reference Michelis, Yarusevych and Kotsonis2017; Dellacasagrande et al. Reference Dellacasagrande, Barsi, Lengani, Simoni and Verdoya2020). The locations of the most upstream local maximum of the shape factor are indicated by the dotted lines in figures 8(c) and 8(d).

At $\alpha ={5}{^\circ }$ , a rapid increase in $-\overline {u'v'}$ begins near $x/c=0.44$ for the clean flow condition (figure 8 a), which agrees well with the location of maximum shape factor for the clean flow (figure 8 d). A similar correlation between the locations of most rapid Reynolds shear stress growth and maximum shape factor is also apparent from the results at $\textit{Tu}=4\,\%$ and $7\,\%$ for $\alpha ={5}{^\circ }$ . Relative to the clean flow, the locations of maximum Reynolds shear stress and shape factor for $\textit{Tu}=4\,\%$ and $7\,\%$ are shifted upstream, indicating earlier transition. The continued presence of a maximum in the Reynolds shear stress distribution for $\textit{Tu}=4\,\%$ and $7\,\%$ provide further evidence of the presence of a thin LSB for elevated FSTI at $\alpha ={5}{^\circ }$ . Notably, there is also reasonable agreement between the Reynolds shear stress transition location criterion and the location of maximum shape factor for $\textit{Tu}=13\,\%$ , despite the absence of measurable reverse flow for this condition (figure 6 c).

At $\alpha ={12}{^\circ }$ and $\textit{Tu}=0.1\,\%$ , the flow is massively separated and transition occurs near $x/c=0.3$ , where there is a rapid increase in Reynolds shear stress (figure 8 b). When the FSTI is increased to $\textit{Tu}=4\,\%$ and $\textit{Tu}=7\,\%$ , the maximum Reynolds shear stress occurs near the upstream end of the field of view. The location of the maximum in $-\overline {u'v'}$ for these conditions suggests that the boundary layer separates and transitions upstream of the field of view and reattaches near $x/c=0.25$ , supporting the assertion made previously that the wall-normal velocity fluctuations in this region (figures 6 b and 6 c) are related to vortex shedding from an LSB. For $\alpha ={12}{^\circ }$ , the shape factor for the clean flow is not plotted since it cannot be accurately estimated when the separated shear layer is outside of the PIV field of view. Although it is within the experimental uncertainty, for the cases of elevated FSTI at $\alpha ={12}{^\circ }$ , there is a local maximum followed by a decrease in shape factor near $x/c=0.25$ that is consistent with boundary layer transition occurring near the upstream end of the PIV field of view in an LSB.

The mean transition locations determined from the Reynolds shear stress threshold and the location of the first local maximum of the shape factor are in reasonable agreement for all considered flow conditions, and indicate an upstream movement of the mean transition location with increasing FSTI and angle of attack. However, it is desired to characterise instantaneous variations in the transition location without requiring temporal windowing or time-resolved data. For this purpose, an entropy-based transition criterion is used.

Streamwise distributions of global entropy computed from the P-POD ( $S_{{g,P}}$ ) are plotted in figures 8(e) and 8( f) for the clean flow and elevated FSTI at $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively. Because noise in the PIV measurements of the thin boundary layer at the upstream end of the field of view contributes to erroneously high values of entropy in that region, the global entropy is only plotted downstream of the location of minimum global entropy. For the clean flow at both $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , there is a relatively rapid increase in global entropy near the transition locations identified from the Reynolds shear stress threshold. There is also a more rapid increase in the global entropy near the locations of transition identified from the shape factor for $\textit{Tu}=4\,\%$ and $\textit{Tu}=7\,\%$ at $\alpha ={5}{^\circ }$ . This suggests that a transition threshold can be defined based on the global entropy. We choose to define the entropy threshold for transition ( $S_{{g,t}}$ ) as the value of the global entropy at the location of maximum shape factor at $\alpha ={5}{^\circ }$ , since the location of maximum shape factor can be objectively measured for each level of FSTI. A specific threshold for each level of FSTI is used to mitigate the influence of the entropy of the free stream turbulence on boundary layer transition detection. The instantaneous transition criterion can be expressed as $S_{{s,P}}\gt S_{{g,t}}$ . For $\textit{Tu}=13\,\%$ at $\alpha ={5}{^\circ }$ , an obvious increase in the slope of the global entropy does not occur near the location of maximum shape factor. However, it will be shown that the intermittency factor distributions determined with the entropy threshold defined from the location of maximum shape factor at $\alpha ={5}{^\circ }$ are in qualitative agreement with the observed transition dynamics for each flow condition investigated. To make estimation of the transition location ( $x_{{t}}$ ) more robust to noise in the PIV measurements, we require that the instantaneous transition criterion be met over a total cumulative distance of at least $0.15c$ downstream of the location where the criterion was first met. Furthermore, $x_{{t}}$ was smoothed with a moving median filter with window length $0.14c/\overline {u_\infty }$ . By applying the transition threshold to the spatial entropy of the P-POD, turbulent flow can be identified as a function of time and streamwise location.

The mean transition locations determined from the instantaneous entropy-based transition criterion $S_{\textit{s},\textit{P}}\gt S_{\textit{g},\textit{t}}$ are compared with the location of maximum shape factor in figure 9 for $\alpha ={5}{^\circ }$ and $\varLambda _{ux}/c=1$ , where the mean transition location falls within the field of view for all levels of FSTI investigated. The standard deviations of the transition locations ( $\sigma _{x_{{t}}}$ ) are indicated by the arrows and the uncertainty in the locations of maximum shape factor are indicated by the error bars. The mean transition locations computed from the entropy-based criterion are all within the uncertainty intervals of the locations of maximum shape factor. When the FSTI is increased from $\textit{Tu}=0.1\,\%$ to $4\,\%$ , there is a significant upstream movement in the transition location, expected to result from bypass transition (e.g. Jaroslawski et al. Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023). However, the differences in mean transition locations among the cases of elevated FSTI determined using the entropy-based criterion are sensitive to the chosen transition thresholds, since the streamwise increase in entropy for these cases is more gradual (figure 8 e). It should be noted that the limited streamwise extent of the field of view is a significant source of bias in the estimates of $\overline {x_{{t}}}$ and $\sigma _{x_{{t}}}$ . Specifically, upstream excursions of $x_{{t}}$ from the field of view at elevated FSTI bias the estimates of $\overline {x_{{t}}}$ downstream of their true locations and reduce the measured $\sigma _{x_{{t}}}$ . Therefore, we refrain from drawing definitive conclusions about the change in mean transition locations between cases of elevated FSTI using the spatial entropy method. However, a more detailed exploration of the variations in the transition location in § 3.4 will show that the instantaneous transition location becomes more variable with increasing FSTI. At $\alpha ={12}{^\circ }$ , the mean transition location ( $\overline {x_{{t}}}$ ) is upstream of the PIV field of view for all cases except the clean flow, for which it is located at $\overline {x_{{t}}}/c=0.22$ . Increasing the integral length scale to $\varLambda _{ux}/c=2$ produced no substantial changes in the locations of maximum shape factor or in $\overline {x_{{t}}}$ and its standard deviation. The following discussion explores the unsteady transition dynamics responsible for the variations in transition location.

Figure 9. Locations of maximum shape factor ( $\triangle$ ) and mean transition location ( $\circ$ ) for $\alpha ={5}{^\circ }$ , $\varLambda _{ux}/c=1$ . Arrows denote standard deviation, error bars denote uncertainty ( $95\,\%$ confidence).

3.4. Transition dynamics

The effect of the adverse pressure gradient, which is stronger at higher angles of attack, on the instantaneous transition dynamics at low and high FSTIs is visualised using the contours of instantaneous spanwise vorticity for $\textit{Tu}=0.1\,\%$ and $13\,\%$ in figure 10. At both angles of attack for $\textit{Tu}=0.1\,\%$ (figures 10 a and 10 b), the separated laminar shear layer rolls up into relatively large-scale coherent vortices. At $\alpha ={5}{^\circ }$ (figure 10 a), an LSB forms, but at $\alpha ={12}{^\circ }$ (figure 10 b), the developing turbulent shear layer is unable to overcome the adverse pressure gradient and remains separated. At $\textit{Tu}=13\,\%$ (figures 10 c and 10 d), separation is suppressed due to bypass transition. The inclined structures at $x/c=0.5$ and $0.8$ in (figure 10 c) are evidence of streak formation (Nolan & Walsh Reference Nolan and Walsh2012) at the lower angle of attack. For $\textit{Tu}=13\,\%$ and $\alpha ={12}{^\circ }$ (figure 10 d), the stronger adverse pressure gradient causes earlier breakdown to small-scale turbulence within the boundary layer (e.g. Litvinenko et al. Reference Litvinenko, Chernoray, Kozlov, Grek, Löfdahl and Chun2005) and no evidence of streaks is observed in this snapshot.

Figure 10. Instantaneous spanwise vorticity. Grey areas are outside the PIV field of view. Thick solid line, $S_{{s,P}}$ from P-POD in turbulent regions; thin solid line, $S_{{s,P}}$ from P-POD in laminar regions; dashed line, $S_{{g,t}}$ .

The relationship between vortical structures in the transitional boundary layer and the spatial entropy computed from the P-POD at each streamwise location is demonstrated in figure 10. Relative to the laminar flow in the upstream portion of the LSB at $\alpha ={5}{^\circ }$ and $\textit{Tu}=0.1\,\%$ (figure 10 a), entropy is increased by the presence of small-scale turbulence (figure 10 d), streaks (figure 10 c) and breakdown of shear layer roll-up vortices (figure 10 a). The relationship between these flow features and the computed spatial entropy confirms that the entropy tends to increase as the boundary layer transitions from laminar to turbulent states. Qualitatively, the transition thresholds (dashed lines in figure 5) demarcate reasonably well more laminar and more turbulent flow regions in various aerofoil operating conditions.

In § 3.3, the persistence of an LSB at elevated FSTI was inferred from the time-averaged flow-fields. Another indicator of the presence of an LSB is the periodic shedding of roll-up vortices from the separated shear layer near the location of mean reattachment. These coherent vortices can be characterised from their associated spatial F-POD modes (Legrand, Nogueira & Lecuona Reference Legrand, Nogueira and Lecuona2011; Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014). Furthermore, the effect of FSTI on the most energetic coherent structures in the boundary layer can be elucidated from a comparison of the F-POD spatial modes corresponding to different flow conditions.

The relative modal energies for the F-POD ( $\lambda _k^2/\! \sum _{k}\lambda _k^2$ ), which describe the proportion of the flow’s kinetic energy that is associated with each spatial mode, are presented in figure 11. The F-POD was performed on the non-time-resolved PIV measurements because these measurements cover a larger field of view. The streamwise component of the first ten spatial modes ( $\boldsymbol{\phi }_k(\boldsymbol{x})$ ) are presented in figures 12 and 13 for angles of attack of $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively, for the clean flow and for elevated FSTI with $\varLambda _{ux}/c=1$ . The spatial modes for $\varLambda _{ux}/c=2$ (not shown for brevity) are similar to those for $\varLambda _{ux}/c=1$ and their relative energy content is also similar as shown in figure 11. Thus, a change in integral length scale from $\varLambda _{ux}/c=1$ to $2$ does not cause a substantial change in the spatial structure or relative energy content of coherent structures in the aerofoil boundary layer. Since the POD was performed without subtracting the mean flow field, the first F-POD mode, which contains over $85\,\%$ of the total energy in all cases, describes a flow field that is similar but not identical to the mean flow (e.g. Chen et al. Reference Chen, Reuss and Sick2012). The higher modes, which each contain less than $4\,\%$ of the total energy, describe spatially coherent velocity fluctuations whose spatial scale decreases with increasing mode index.

Figure 11. Relative energy of POD modes from F-POD.

Figure 12. Streamwise component of POD spatial modes for $\alpha ={5}{^\circ }$ and $\varLambda _{ux}/c=1$ .

Figure 13. Streamwise component of POD spatial modes for $\alpha ={12}{^\circ }$ .

For the clean flow at $\alpha ={5}{^\circ }$ (figure 12 a), spatial modes $3$ – $10$ display spatially periodic velocity fluctuations beginning near the location of maximum LSB thickness, which are typical of vortex shedding from an LSB (e.g. Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014; Lengani & Simoni Reference Lengani and Simoni2015). At $\textit{Tu}=4\,\%$ (figure 12 b), similar spatially periodic structures are present in the near-wall region of modes $4$ – $10$ , providing further evidence of vortex shedding from a thin LSB under these operating conditions. Mode $2$ for $\textit{Tu}=4\,\%$ describes a variation in the streamwise velocity gradient of the outer flow that is correlated to a modulation of the boundary layer. This mode is consistent with the expected response of an LSB to a change in pressure gradient. For example, if the time coefficient of this mode is positive, this mode describes a streamwise acceleration of the outer flow. The associated reduction in adverse pressure gradient would be expected to delay separation, transition and reattachment, leading to a downstream movement of the LSB, and this is consistent with the increase in streamwise velocity near the wall for $x/c\lt 0.4$ and reduction in streamwise velocity near the wall for $0.4\lt x/c\lt 0.65$ present in this spatial mode. Accounting for the reversal of sign, the similarity of the second spatial modes for the cases of $\textit{Tu}=4\,\%$ , $7\,\%$ and $13\,\%$ suggests that a thin LSB may intermittently form in all of these cases. Indeed, mode $10$ for $\textit{Tu}=7\,\%$ (figure 12 c) also displays spatially periodic structures in the boundary layer that are typical of vortex shedding from an LSB.

Mode $3$ for $\textit{Tu}=4\,\%$ describes a large-scale change in the outer flow that is correlated to a streamwise elongated region within the boundary layer, suggestive of the streaky structures formed during bypass transition (Lengani & Simoni Reference Lengani and Simoni2015). Like the second modes, the overall spatial structure of the third modes is also analogous for $\textit{Tu}=4\,\%$ , $7\,\%$ and $13\,\%$ , albeit reversed in sign. For $\textit{Tu}=7\,\%$ (figure 12 c), the fourth mode also displays a relatively large wavelength change in the outer flow that is correlated to a streamwise elongated and inclined structure within the boundary layer. For $\textit{Tu}=13\,\%$ (figure 12 d), modes $5$ and $8$ also contain similar elongated and inclined structures that are expected to form during bypass transition (Nolan & Walsh Reference Nolan and Walsh2012).

For the clean flow at $\alpha ={12}{^\circ }$ (figure 13 a), the second mode is related to oscillations in the trajectory of the separated shear layer (e.g. Fang & Wang Reference Fang and Wang2024). Similar to the clean flow at $\alpha ={5}{^\circ }$ , the spatial POD modes for $\alpha ={12}{^\circ }$ are related to periodic vortex shedding from the separated shear layer at progressively smaller scales for $4\leq k\leq 10$ . For $\textit{Tu}=4\,\%$ at $\alpha ={12}{^\circ }$ (figure 13 b), modes $6$ – $10$ exhibit streamwise periodic structures in the boundary layer that are indicative of vortex shedding from an LSB located upstream of the field of view. The second spatial modes for $\textit{Tu}=4\,\%$ , $7\,\%$ and $13\,\%$ at $\alpha ={12}{^\circ }$ are analogous and describe a large scale change in the streamwise velocity gradient of the outer flow, similar to the second modes for these levels of FSTI at $\alpha ={5}{^\circ }$ . The third modes for elevated FSTI at $\alpha ={12}{^\circ }$ describe streamwise elongated and inclined structures suggestive of streaks.

The comparison of F-POD spatial modes pertaining to different flow conditions reveals how the FSTI affects the spatial structure of the most energetic coherent structures in the boundary layer. The relatively small-scale structures associated with shear layer vortex shedding are progressively relegated to lower relative energy content and structures with relatively longer streamwise length scales gain a greater portion of the overall turbulent kinetic energy as the FSTI is increased. The modes with longer streamwise length scales contain elongated oblique structures near the aerofoil surface, consistent with the streaks that are expected to form during bypass transition. However, the F-POD modes suggest that a thin LSB may persist to at least $\textit{Tu}=7\,\%$ , for which the signature of a spatially periodic vortex train is present near the wall in mode $10$ at both angles of attack. The second F-POD modes for each of the cases of elevated FSTI describe a change in the streamwise pressure gradient. The relationship of the second modes to the streamwise pressure gradient will be used to infer changes in effective angle of attack in § 3.5.

The observation of relatively small-scale streamwise-periodic structures consistent with vortex shedding from the separated shear layer in the F-POD modes suggest that a thin LSB may form on the aerofoil at relatively high FSTIs in large-scale free stream turbulence. Spectral analysis was performed on the wall-normal velocity fluctuations at $y=\overline{\delta^*}$ to investigate the effect of large-scale high-intensity free stream turbulence on the characteristics of shear layer rollers, namely, their frequency, wavenumber and convection speed, since the characteristics of the shed vortices affect the ability of the separated shear layer to reattach (e.g. Marxen & Henningson Reference Marxen and Henningson2011; Serna & Lazaro Reference Serna and Lazaro2015). The wall-normal velocity fluctuations are chosen for the spectral analysis because they are more strongly correlated to the passage of shear layer roll-up vortices than the streamwise velocity fluctuations (e.g. Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014). The wall-normal velocity fluctuations at $y=\overline{\delta^*}$ form a two-dimensional dataset that is a function of time and streamwise location. For the clean flow at $\alpha ={12}{^\circ }$ in the streamwise region where $y=\overline{\delta^*}$ is outside of the field of view, the wall-normal velocity fluctuation data were extracted at the upper edge of field of view. The power spectral density of the wall-normal fluctuating velocity component ( $\mathcal{P}_{fk} (v' )$ ) is plotted in figure 14 versus frequency and wavenumber. The power spectral density was estimated by averaging the squared magnitude of the two-dimensional Fourier transform obtained using windows of $128$ and $256$ samples in the temporal and streamwise dimensions, respectively, with $50\,\%$ window overlap. This windowing procedure is a generalisation of the method of Welch (Reference Welch1967) to two-dimensional data. The presented power spectral density estimates are the result of averaging over a total of $460$ windows: $2$ windows in the streamwise dimension and $230$ windows in the temporal dimension. The frequency resolution is $0.3\overline {u_\infty }/c$ , the wavenumber resolution is $30/c$ and the relative uncertainty in power spectral density is less than $8\,\%$ ( $95\,\%$ confidence). The data are normalised using the mean boundary layer edge velocity ( $\overline {u_{{e}}}$ ) at the upstream limit of the PIV field of view, which is near the location of boundary layer separation for those cases where there is an LSB or massively separated flow (figures 6 and 7).

Figure 14. Frequency-wavenumber power spectral density of wall-normal velocity fluctuations at $y=\overline{\delta^*}$ for clean flow and elevated FSTI with $\varLambda _{ux}/c=1$ at (a,c,e,g) $\alpha ={5}{^\circ }$ and (b,d,f,h) $\alpha ={12}{^\circ }$ .

At $\alpha ={5}{^\circ }$ (figure 14 a), vortex shedding from the LSB in the clean flow produces a peak at $\textit{fc}/\overline {u_{{e}}}=6.6$ ( $\textit{fc}/\overline {u_\infty }=9.8$ ) and $kc=110$ , which is consistent with the fundamental frequencies and wavenumbers for vortex shedding from LSBs at similar chord Reynolds numbers (Toppings & Yarusevych Reference Toppings and Yarusevych2023, Reference Toppings and Yarusevych2024). The estimated wavenumber corresponds to a streamwise wavelength of $0.06c$ , which is in agreement with the streamwise wavelength of the stuctures near the location of maximum LSB height ( $x/c=0.49$ ) in POD modes $6$ – $10$ for the clean flow (figure 12 a). Relative to the clean flow, there is a reduction in the frequency, wavenumber and amplitude of the fundamental vortex shedding peak for the cases of elevated FSTI. The reduction in frequency and wavenumber of the vortex shedding peak is attributed to the reduction in the distance of the separated shear layer from the aerofoil surface, which is expected to reduce the frequency and wavenumber of the most amplified disturbances (e.g. Dovgal et al. Reference Dovgal, Kozlov and Michalke1994). The reduction in amplitude of the vortex shedding peak is attributed to bypass transition, which leads to more rapid breakdown of the spanwise roll-up vortices into three-dimensional turbulence (Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019). Furthermore, the streaks formed due to free stream turbulence are expected to produce larger streamwise velocity fluctuations than wall-normal velocity fluctuations (e.g. Nolan & Walsh Reference Nolan and Walsh2012), consistent with the fluctuating velocity statistics presented in figures 6 and 7. However, the discernible spectral peak of the wall-normal velocity fluctuations for $\textit{Tu}=7\,\%$ at $\textit{fc}/\overline{u_e}=5.7$ and $kc=84$ (figure 14 e) provides additional evidence that an LSB with periodic vortex shedding may persist at this FSTI.

At $\alpha ={12}{^\circ }$ , there are relatively stronger and more broadband wall-normal velocity fluctuations in the clean flow (figure 14 b) because the greater separation angle reduces the stability of the separated shear layer (e.g. Dovgal et al. Reference Dovgal, Kozlov and Michalke1994). Similar to the results from the lower angle of attack, increasing the FSTI at $\alpha ={12}{^\circ }$ also decreases the amplitudes of wall-normal velocity fluctuations. This is consistent with the more rapid breakdown of the roll-up vortices into three-dimensional turbulence at elevated FSTI observed in previous studies where bypass transition mechanisms become dominant (e.g. Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019; Aniffa et al. Reference Aniffa, Caesar, Dabaria and Mandal2023; Jaroslawski et al. Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023). The absence of a distinct spectral peak for the cases of elevated FSTI at $\alpha ={12}{^\circ }$ is attributed to the upstream movement of the vortex roll-up location relative to the PIV field of view at this angle of attack, with the consequence of earlier vortex breakdown and reduced coherence of wall-normal velocity fluctuations in the field of view.

In all of the spectra presented in figure 14, the amplitude of wall-normal velocity fluctuations is greatest along a convective ridge (e.g. Abraham & Keith Reference Abraham and Keith1996). The convective speed of these fluctuations, which is approximately equal to the vortex convection speed for cases where coherent vortex shedding occurs, was estimated from a linear fit to the highest amplitude wavenumber for each frequency. The uncertainty in convection speeds is estimated to be $0.1\overline{u_e}$ from the variations in convection speeds obtained with various window widths for the power spectral density estimation. At $\alpha ={5}{^\circ }$ , the convective speed is within $0.4\leq \overline{u_e}\leq 0.5$ , consistent with the convective speeds expected for shear layer roll-up vortices in LSBs (Yarusevych, Sullivan & Kawall Reference Yarusevych, Sullivan and Kawall2009). At $\alpha ={12}{^\circ }$ , the convective speed remains at $0.5 \overline{u_e}$ for both the clean flow condition, where the flow is massively separated, and for the attached flow conditions at elevated FSTI. Although the frequency content of wall-normal velocity fluctuations in the boundary layer becomes more broadband with increasing FSTI, the results of the frequency-wavenumber analysis suggest that the angle of attack and FSTI have a negligible effect on the convection speed of coherent structures formed in the transition process when normalised by the boundary layer edge velocity. The measured convection speeds are in agreement with the universal roll-up frequency scaling proposed by Yarusevych et al. (Reference Yarusevych, Sullivan and Kawall2009).

To illustrate the temporal variations in transition location that occur at elevated FSTI, the spatial entropy from the P-POD is plotted versus time and streamwise location in figures 15 and 16 for $\textit{Tu}=0.1\,\%$ , $7\,\%$ and $13\,\%$ at $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively. The entropy contours in these figures were obtained from time-resolved PIV measurements. All spatial entropy values below the transition threshold ( $S_{{s,P}}/S_{{g,t}}\lt 1$ ) are covered by the lowest contour level. The overall entropy of the entire velocity field at each moment in time is quantified by the spatial entropy computed from the F-POD ( $S_{{s,F}}$ ), plotted as magenta lines in figures 15 and 16. Histograms of the instantaneous location of $x_{{t}}$ from non-time-resolved data over a wider field of view are provided in the right column for each level of FSTI. The probabilities in the histograms are defined as the fraction of the time that the transition location is within each bin of width $0.025c$ .

Figure 15. Left column shows contours of spatial entropy from P-POD at $\alpha ={5}{^\circ }$ . Grey areas are outside the PIV field of view. Magenta lines, spatial entropy from F-POD. Right column shows histograms of $x_{{t}}$ . Solid lines, $\overline {x_{{t}}}$ ; dashed lines, $\overline {x_{{t}}}\pm \sigma _{x_{{t}}}$ .

Figure 16. Left column shows contours of spatial entropy from P-POD at $\alpha ={12}{^\circ }$ . Grey areas are outside the PIV field of view. Magenta lines, spatial entropy from F-POD. Right column shows histograms of $x_{{t}}$ . Solid lines, $\overline {x_{{t}}}$ ; dashed lines, $\overline {x_{{t}}}\pm \sigma _{x_{{t}}}$ .

In the clean flow at $\alpha ={5}{^\circ }$ (figure 15 a), there is a rapid increase in entropy at $x/c\approx 0.5$ , corresponding to the onset of transition near the location of maximum LSB thickness (figure 6 a). For this case, the transition location determined from the entropy-based criterion remains relatively constant, with a mean of $\overline {x_{{t}}}/c=0.48$ and a standard deviation of $\sigma _{x_{{t}}}/c=0.02$ , as shown in the histogram. The boundary layer consistently remains turbulent downstream of the mean reattachment location of the LSB ( $x/c=0.6$ figure 6 a). Regions of the boundary layer with higher spatial disorder indicative of turbulent flow appear as oblique ridges of higher entropy. The slope of the ridges in the plot is proportional to their convective speed. The convective speed of the high-entropy ridges is approximately $0.4\overline {u_{{e}}}$ ( $0.7\overline {u_\infty }$ ), which is consistent with the convective speed of the shear layer roll-up vortices estimated from the frequency wavenumber spectrum in figure 14. The consistency of the estimated convective speeds of the wall-normal velocity fluctuations and the spatial entropy fluctuations holds for varying flow conditions, confirming the suitability of spatial entropy as a spatio-temporally localised indicator of velocity field disturbances.

For $\alpha ={5}{^\circ }$ with elevated FSTI (figures 15 b and 15 d), the mean streamwise location of transition is relatively unaffected by FSTI beyond $\textit{Tu}=4\,\%$ , but becomes more variable in time as the FSTI is increased. For example, for $\textit{Tu}=13\,\%$ (figure 15 d) at $t\overline {u_\infty }/c=9.5$ , transition occurs upstream of the PIV field of view and at $t\overline {u_\infty }/c=11.8$ , laminar flow persists across the entire streamwise extent of the field of view. The mean measured transition locations for $\textit{Tu}=4\,\%$ , $7\,\%$ and $13\,\%$ are $\overline {x_t}/c=0.25$ , $0.28$ and $0.30$ , respectively, and the standard deviations are $\sigma _{x_t}/c=0.07$ , $0.08$ and $0.09$ , respectively.

The overall level of disorder in the flow over the entire field of view can be quantified by the spatial entropy obtained from the F-POD (magenta lines in figures 15 and 16). When the transition location moves upstream, the spatial entropy from the F-POD tends to increase. The spatial entropy from the F-POD will be used in § 3.5 to correlate overall changes in the disorder of the flow over the suction surface of the aerofoil to free stream velocity fluctuations upstream of the aerofoil.

Similar to the foregoing results for the clean flow at $\alpha ={5}{^\circ }$ , the transition location also remains relatively consistent for the clean flow at $\alpha ={12}{^\circ }$ (figure 16). However, the separated shear layer transitions further upstream at the higher angle of attack because the more adverse pressure gradient causes earlier separation, which destabilises the shear layer (e.g. Diwan & Ramesh Reference Diwan and Ramesh2009). This is consistent with the earlier shear layer roll-up in figure 10(b). For $\textit{Tu}=4\,\%$ at $\alpha ={12}{^\circ }$ , transition frequently occurs sufficiently upstream to prevent massive laminar separation from stalling the aerofoil (figure 16 b). For all levels of elevated FSTI at $\alpha ={12}{^\circ }$ (figure 16 b–d), the mean computed transition location is near the upstream limit of the field of view. Therefore, the standard deviations of the transition locations (dashed lines in figure 16) are likely underestimated. For the cases of $\textit{Tu}=4\,\%$ and $7\,\%$ (figures 16 b and 16 c), the computed transition location is upstream of $x/c=0.175$ for more than $98\,\%$ of the measurement period. Intermittent periods of reduced spatial entropy indicative of laminar flow still occur in the attached boundary layer at $\alpha ={12}{^\circ }$ , notably at $\textit{Tu}=13\,\%$ (figure 16 d). However, the spatial and temporal extent of these laminar periods are substantially reduced compared with those for elevated FSTI at $\alpha ={5}{^\circ }$ . For $\textit{Tu}=13\,\%$ at $\alpha ={12}{^\circ }$ , there are also notable intermittent periods of substantially increased spatial entropy. It will be shown later that these periods are related to intermittent massive separation of the suction surface boundary layer.

Using the spatial entropy from the P-POD ( $S_{{s,P}}$ ) and the transition thresholds ( $S_{{g,t}}$ ), the intermittency factor ( $\gamma$ ) can be calculated. The intermittency factor is defined as the fraction of time that $S_{{s,P}}\gt S_{{g,t}}$ and thus indicates the probability that the flow is turbulent at a given streamwise location (e.g. Hedley & Keffer Reference Hedley and Keffer1974). The intermittency factor is plotted in figure 17 for locations downstream of the minimum global entropy. The locations of maximum shape factor at $\alpha ={5}{^\circ }$ used to define the transition thresholds are indicated by the dashed lines in figure 17(a). For all levels of FSTI at $\alpha ={5}{^\circ }$ , $\gamma \approx 0.25$ at the location of maximum shape factor. In the clean flow at both $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , the intermittency factor rapidly increases from $\gamma \approx 0$ to $\gamma \approx 1$ over a distance of approximately $0.15c$ . The reduction in intermittency factor for elevated FSTI relative to the clean flow for $x/c\gt 0.5$ at $\alpha ={5}{^\circ }$ is consistent with the increased variability in transition location and intermittent periods of laminar flow for elevated FSTI in figure 15. For $x/c\lt 0.8$ at $\alpha ={12}{^\circ }$ (figure 17 b), there is a decrease in intermittency factor with increasing FSTI, consistent with the larger variations in spatial entropy with increasing FSTI in figure 16. Overall, as the FSTI is increased, the maximum rate of change of the intermittency factor becomes more gradual, indicating a higher probability of laminar flow persisting downstream of the mean transition location.

Figure 17. Intermittency factor for clean flow and elevated FSTI with $\varLambda _{ux}/c=1$ . Dashed lines, location of maximum shape factor. Shaded areas indicate uncertainty ( $95\,\%$ confidence).

To characterise the frequencies associated with the large variations in transition location observed for $\alpha ={5}{^\circ }$ , the power spectral density of the spatial entropy fluctuations from the P-POD ( $\mathcal{P}_f (S'_{{s,P}} )$ ) performed on the non-time-resolved PIV measurements for $\alpha ={5}{^\circ }$ are presented in figure 18(a, b) for $\varLambda _{ux}/c=1$ and $2$ , respectively. The spectra presented in figure 18 were obtained after averaging the spectra over $0.4\lt x/c\lt 0.5$ , where relatively large amplitude and low-frequency variations in spatial entropy occur for all levels of FSTI tested (figure 15). The clean flow with $\varLambda _{ux}/c=0.2$ is shown in both plots for reference. The spectra were obtained using Welch’s method with a Hamming window of length 128 samples and $50\,\%$ window overlap, yielding a frequency resolution of ${0.007}\,\textrm {Hz}$ .

Figure 18. Power spectral density of spatial entropy fluctuations from P-POD averaged over $0.4\lt x/c\lt 0.5$ for $\alpha ={5}{^\circ }$ .

In the clean flow, the transition location is relatively steady and there are no substantial peaks in the spatial entropy spectrum. For increasing FSTI, there is an increase in the amplitudes of spatial entropy fluctuations, consistent with the low-frequency variability in the transition location observed for high FSTI (figure 15). For $\textit{Tu}=4\,\%$ and $7\,\%$ , increasing the integral length scale from $\varLambda _{ux}/c=1$ to $2$ has the effect of shifting the largest amplitude fluctuations to lower frequencies. For $\varLambda _{ux}/c=1$ , there are prominent peaks at $\textit{fc}/u_\infty =0.06$ and $0.08$ for $\textit{Tu}=4\,\%$ and $7\,\%$ , respectively (figure 18 a). These peaks are consistent with the frequencies of the largest amplitude free stream velocity fluctuations for these conditions (cf. figure 3). For $\varLambda _{ux}/c=2$ , the frequencies of the largest amplitude spatial entropy fluctuations are reduced to $\textit{fc}/u_\infty =0.03$ for both $\textit{Tu}=4\,\%$ and $7\,\%$ (figure 18 b), consistent with the reduction in frequency of the largest amplitude free stream velocity fluctuations. There is also a reduction in the frequency of the largest amplitude spatial entropy fluctuations for the case of $\textit{Tu}=16\,\%$ , $\varLambda _{ux}=2$ relative to $\textit{Tu}=13\,\%$ , $\varLambda _{ux}=1$ . Considering that the change of integral length scale had a negligible influence on the mean flow, these results suggest that variations in $\varLambda _{ux}$ , when $\varLambda _{ux}\sim c$ , do not directly affect transition through the receptivity process. Instead, when $\varLambda _{ux}\sim c$ , changes in $\varLambda _{ux}$ influence the frequency at which variations of the global flow development occur about the mean.

Figure 19. Instantaneous streamwise velocity measurements for the (a,b) $1$ st percentile of $S_{{s,F}}$ and (c,d) $99$ th percentile of $S_{{s,F}}$ from the F-POD observed at (a,c) $\alpha ={5}{^\circ }$ and (b,d) $\alpha ={12}{^\circ }$ for $\textit{Tu}=13\,\%$ and $\varLambda _{ux}/c=1$ .

The relationship between the flow field development and the spatial entropy computed from the F-POD (lines in figure 15, 16) is demonstrated in figure 19, which presents instantaneous contours of streamwise velocity for the PIV snapshots of the $1$ st percentile of $S_{{s,F}}$ (figure 19 a,b) and the $99$ th percentile (figure 19 c,d) of $S_{{s,F}}$ at $\textit{Tu}=13\,\%$ . Recall, the value of $S_{{s,F}}$ computed from the F-POD quantifies the spatial disorder in the flow over the entire field of view. In the PIV snapshots of the the $1$ st percentile of $S_{{s,F}}$ , the boundary layer remains thin and laminar over a substantial portion of the field of view. For the snapshots of the $99$ th percentile of $S_{{s,F}}$ , the boundary layer is relatively thicker and at $\alpha ={12}{^\circ }$ , reverse flow covers a substantial portion of the field of view. Figure 19(d) confirms that the periods of relatively higher spatial entropy for $\alpha ={12}{^\circ }$ and $\textit{Tu}=13\,\%$ in figure 16(d) are associated with intermittent massive separation from the aerofoil surface. It is likely that these periods of massive separation are partly responsible for the measured reduction in time-averaged lift coefficient for $\textit{Tu}=13\,\%$ and $16\,\%$ relative to $\textit{Tu}=4\,\%$ and $7\,\%$ at $\alpha ={12}{^\circ }$ (figure 4).

3.5. Correlation of oncoming disturbances to boundary layer development

Measurements from the hotwire anemometer placed $10c$ upstream of the aerofoil and synchronised with the PIV system are used to explore the relationship of the oncoming velocity magnitude to the flow over the suction surface of the aerofoil. The correlation coefficient between the velocity magnitude measured by the hotwire and the mean streamwise velocity averaged over the PIV field of view ( $\langle u\rangle$ ) is plotted in figure 20(a,b) versus time lag for $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ , respectively. Data were used from the PIV measurements at ${100}\,\textrm {Hz}$ for better statistical convergence at the expense of temporal resolution. The results show a positive peak at $t\overline {u_\infty }/c\approx 10$ for all levels of elevated FSTI at both $\alpha ={5}{^\circ }$ and ${12}{^\circ }$ . Since the hotwire was positioned a distance of $10c$ upstream of the quarter-chord of the aerofoil, this peak corresponds to disturbances in the oncoming flow that convect at the free stream velocity and are sufficiently large to maintain their temporal coherence over that distance. The negative correlation peaks that occur at $tu_\infty /c\approx 5$ and $15$ are attributed to inevitable opposite-sign changes of incoming flow velocity preceding and following that responsible for the dominant correlation peak. For example, due to the requirement for continuity, a significant, large-scale increase in free stream velocity must be preceded or followed by a decrease in free stream velocity to maintain the overall average flow rate through the test section.

Figure 20. (a, b) Correlation coefficient between upstream hotwire anemometer velocity and spatially averaged streamwise velocity from side-view PIV and (c,d) transition location from P-POD. Values outside the dashed lines are significant at a $95\,\%$ confidence level.

Because of the closed test section, the active grid also produces pressure waves that likely affect the boundary layer development on the aerofoil (e.g. Collins & Zelenevitz Reference Collins and Zelenevitz1975). The time lag for a pressure wave to propagate from the hotwire to the aerofoil is $t\overline {u_\infty }/c\approx 0.003$ . Although the influence of such pressure waves on the aerofoil boundary layer cannot be discounted, the convective correlation peaks at $t\overline {u_\infty }/c\approx 10$ are substantially larger in magnitude than the correlations for $t\overline {u_\infty }/c\approx 0.003$ . This suggests that convective vortical structures in the oncoming flow are primarily responsible for the variations in boundary layer development on the suction surface of the aerofoil. In the clean flow, there are no significant correlations between the oncoming flow velocity magnitude and the spatially averaged velocity magnitude of the PIV measurements due to the order of magnitude lower FSTI.

The correlation coefficients between the upstream velocity magnitude and the transition location determined using the entropy-based method from the P-POD (§ 3.3.1) are presented in figures 20(c) and 20(d). Note that, for the case of $\textit{Tu}=4\,\%$ at $\alpha ={12}{^\circ }$ , no correlation coefficient could be computed because the transition location was always upstream of the side-view PIV field of view. The correlations are largely insignificant for all time lags at a $95\,\%$ confidence level. Furthermore, no statistically significant correlations between the oncoming velocity magnitude and the turbulent kinetic energy within the boundary layer were found when the velocity fluctuations were normalised by the spatially averaged velocity within the boundary layer to account for the expected increase in turbulent kinetic energy with increasing free stream velocity. Thus, it is unlikely that large-scale fluctuations in effective oncoming velocity magnitude are responsible for the intermittent periods of expedited and delayed transition observed at high FSTI.

In addition to fluctuations in the free stream velocity magnitude, large-scale free stream turbulence also leads to fluctuations in effective angle of attack, which may influence the boundary layer transition process on the aerofoil (e.g. Herbst et al. Reference Herbst, Kähler and Hain2018; Kay et al. Reference Kay, Richards and Sharma2020). The variations in effective angle of attack due to fluctuations in free stream velocity direction were characterised from estimated variations in the suction surface pressure coefficients. The estimations were performed for $\alpha ={5}{^\circ }$ , where the boundary layer remains relatively thin and massive separation is unlikely to occur. The instantaneous surface pressure coefficient at a given $x$ location was estimated from the local instantaneous boundary layer edge velocity as $C_p=1- (u_{{e}}/\overline {u_\infty } )^2$ , assuming that the wall-normal pressure gradient may be neglected within the boundary layer (e.g. Diwan & Ramesh Reference Diwan and Ramesh2012). The free stream velocity used for calculating the pressure coefficients was the mean free stream velocity ( $\overline{u_\infty}$ ) obtained from a calibration of the pressure drop across the wind tunnel contraction upstream of the active grid.

The mean surface pressure coefficient distributions for $\alpha ={5}{^\circ }$ are plotted in figure 21, along with inviscid surface pressure distributions for angles of attack of $\alpha ={0}{^\circ }$ , ${5}{^\circ }$ and ${10}{^\circ }$ computed using the XFOIL software (Drela Reference Drela and Thomas1989). The mean surface pressure coefficients for the clean flow with $\varLambda _{ux}/c=0.2$ are plotted in both figures 21(a) and 21(b) for reference. The maximum uncertainty in the estimated mean pressure coefficients is less than $0.2$ . In the clean flow, the $C_{\overline{p}}$ distribution initially follows the inviscid distribution at the geometric angle of attack of $\alpha ={5}{^\circ }$ , before diverging near the boundary layer separation point and plateauing. A relatively rapid pressure recovery occurs between the location of maximum LSB thickness and the mean reattachment point, consistent with the surface pressure distribution expected of a short LSB (Horton Reference Horton1968). For elevated FSTI, the pressure plateau is suppressed and the pressure distributions follow that for the inviscid flow at $\alpha ={5}{^\circ }$ more closely, consistent with the thinning of the LSB. However, there is still an increase in suction relative to the inviscid flow for $\textit{Tu}=4\,\%$ and $7\,\%$ , suggesting that an LSB persists.

Figure 21. Estimated mean surface pressure distributions at $\alpha ={5}{^\circ }$ . Shaded areas denote uncertainty ( $95\,\%$ confidence). Dotted lines, inviscid surface pressure distributions from XFOIL (Drela Reference Drela and Thomas1989).

Previously, it was shown that the second POD mode for the cases of elevated FSTI was related to the streamwise pressure gradient. Since increases in effective angle of attack are expected to lead to a more adverse streamwise pressure gradient, variations in the effective angle of attack at elevated FSTI were characterised using conditional averaging of the estimated pressure coefficient distributions based on the temporal coefficient of the second POD mode. The surface pressure distributions of the conditional averages of snapshots for $a_2$ falling below or above the mean value by more than one standard deviation are plotted in figure 22 as dashed and dash-dotted lines, respectively, for the cases of elevated FSTI. Depending on the sign of the second spatial POD mode (figure 12), the $C_{\overline{p}}$ distributions conditioned on $a_2\lt \overline {a_2}-\sigma _{a_2}$ and $a_2\gt \overline {a_2}+\sigma _{a_2}$ show either a decrease or increase in pressure gradient consistent with a change in effective angle of attack. For $\textit{Tu}=13\,\%$ (figure 22 a) and $\textit{Tu}=16\,\%$ (figure 22 b), the conditional pressure distributions begin to approach the inviscid pressure distributions for $\alpha ={10}{^\circ }$ and $\alpha ={0}{^\circ }$ . The pressure gradients of the conditional averages for $\textit{Tu}=16\,\%$ are consistent with changes in effective angle of attack of nearly $\pm {5}{^\circ }$ relative to the aerofoil’s fixed geometric angle of attack of $\alpha ={5}{^\circ }$ . This range of effective angle of attack variation is in reasonable agreement with the results of Kay et al. (Reference Kay, Richards and Sharma2020), who reported standard deviations of the effective angle of attack of ${2.4}{^\circ }$ and ${6.9}{^\circ }$ for $\textit{Tu}=5\,\%$ and $15\,\%$ , respectively. For $\textit{Tu}=4\,\%$ and $7\,\%$ , the variations in effective angle of attack in the present study are more difficult to estimate from a comparison with inviscid surface pressure distributions due to the increase in suction caused by the relatively larger LSB forming at these FSTIs. However, the conditional surface pressure distributions from these cases are consistent with the expected increase in effective angle of attack variation with increasing FSTI. Note that the differences in the surface pressure distributions for $\varLambda _{ux}/c=1$ and $2$ at constant FSTI ( $\textit{Tu}=4\,\%$ and $7\,\%$ ) are within the experimental uncertainty. This finding implies changes in $\varLambda _{ux}$ , when $\varLambda _{ux}$ remains of the order of the aerofoil chord, do not substantially affect the effective angle of attack variations experienced by the aerofoil.

Figure 22. Estimated conditional surface pressure distributions at $\alpha ={5}{^\circ }$ . Dashed lines, conditional average of $a_2\lt \overline {a_2}-\sigma _{a_2}$ ; dash-dotted lines, conditional average of $a_2\gt \overline {a_2}+\sigma _{a_2}$ ; dotted lines, inviscid surface pressure distributions from XFOIL (Drela Reference Drela and Thomas1989).

Figure 23. Correlation between the temporal coefficient of the second mode $a_2$ and spatial entropy $S_{{s,F}}$ from F-POD. Red lines show best linear fit.

Figure 22 demonstrates that substantial changes in pressure gradient occur at high FSTI that are likely associated with changes in effective angle of attack. When the second POD mode for the cases of elevated FSTI is related to a large-scale change in the edge velocity gradient (cf. figures 12 and 13) and therefore to a large-scale change in surface pressure distribution, the temporal coefficient of the second POD mode ( $a_2$ ) is expected to be correlated to the spatial entropy from the F-POD. This is because stronger adverse pressure gradients lead to earlier boundary layer transition (Dellacasagrande et al. Reference Dellacasagrande, Barsi, Lengani, Simoni and Verdoya2020), causing an increase in disorder within the flow field. The relationship between $a_2$ and the spatial entropy from the F-POD is shown in figure 23 for the cases of elevated FSTI. If the degree of disorder within the flow field was independent of the second POD mode, the distribution of the data points in each scatter plot would have reflective symmetry about $a_2=0$ . In all cases, the scatter plots are visibly asymmetric about $a_2=0$ , and the correlations between $a_2$ and $S_{{s,P}}$ are significant at a $95\,\%$ confidence level as verified by statistical $P$ -value evaluation. The sign of the correlation coefficients ( $\rho _{a_2 S_{{s,F}}}$ ) vary according to the sign of the spatial modes, but are all consistent with increasing spatial entropy with increasing adverse pressure gradient when accounting for the sign of the spatial modes (cf. figure 12 for $\varLambda _{ux}/c=1$ ). Weaker but still statistically significant correlations were found directly between $a_2$ and $x_{{t}}$ , consistent with the tendency for earlier transition during periods of stronger adverse pressure gradient. Thus, the observed correlations between $a_2$ and the spatial entropy suggest that the variations in effective angle of attack caused by large-scale free stream turbulence are linked to the relatively large variations in transition location and boundary layer thickness observed at high FSTI.

The prevalent effect of angle of attack fluctuations compared with that of the velocity magnitude is consistent with the much higher sensitivity of the transition process to comparable relative changes in angle of attack than chord Reynolds number for quasi-steady conditions (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012b ; Park, Shim & Lee Reference Park, Shim and Lee2020). For example, the data of Boutilier & Yarusevych (Reference Boutilier and Yarusevych2012b ) indicate that for a NACA 0018 aerofoil operating at ${\textit{Re}}_c=1\times 10^5$ and $\alpha ={5}{^\circ }$ , a $50\,\%$ increase in ${\textit{Re}}_c$ would cause a $0.05c$ upstream movement in transition location, compared with a $0.1c$ upstream movement for a $50\,\%$ increase in angle of attack. Furthermore, the inferred relative fluctuations in angle of attack at high FSTI are substantially higher than those in chord Reynolds number. For example, at $\textit{Tu}=16\,\%$ , $\varLambda _{ux}/c=2$ , the relative fluctuations in effective chord Reynolds number are of the order of $10\,\%$ . In contrast, an order of $0.1\overline {u_\infty }$ fluctuations in vertical velocity component correspond to approximately $\arctan { (0.1 )}\approx {6}{^\circ }$ fluctuations in the angle of attack (i.e. ${\sim} 100\,\%$ of the geometric angle of attack). Thus, the relative fluctuations in angle of attack are an order of magnitude higher than those in Reynolds number for this test case.