3.1. Mean flow topology

To demonstrate the mean spanwise changes in the LSB on the wing, a comparison is made with the two-dimensional airfoil. Mean surface pressure measurements on the wing and two-dimensional airfoil are presented in figure 7. The chordwise pressure distributions in figure 7(a) display a plateau downstream of the suction peak, which is characteristic of the dead-air region of the LSB (Horton Reference Horton1968). The plateau is followed by a rapid pressure recovery which indicates the reattachment of the shear layer in the aft portion of the bubble. As expected, tip effects reduce the suction developed by the wing when compared with the two-dimensional airfoil at the same geometric angle of attack of $\alpha =6^\circ$. At $\alpha =5^\circ$, the two-dimensional airfoil produces a chordwise pressure distribution that essentially matches the pressure distribution on the wing at $\alpha =6^\circ$ and $Z/c = 0.95$, implying that the effective angle of attack of the wing at $Z/c = 0.95$ is approximately $5^\circ$, as discussed in detail by Toppings et al. (Reference Toppings, Kurelek and Yarusevych2021). Figure 7(b) presents surface pressure measurements from the three rows of spanwise pressure taps for the wing at $\alpha =6^\circ$ and for the two-dimensional airfoil at the effective angle of attack of the inboard region ($\alpha =5^\circ$). The pressure distribution on the two-dimensional airfoil is essentially uniform in the spanwise direction. On the wing for $Z/c<1$, the pressure distribution also does not display significant spanwise gradients. However, for $Z/c>1$, a substantial spanwise pressure gradient develops progressively as the wingtip is approached. Figure 7(b) also indicates that the chordwise pressure gradient is reduced near the wingtip.

Figure 7.

Surface pressure distributions on the wing and airfoil. (a) Streamwise distribution and (b) spanwise distribution. Symbols in (a) mark measurements common to the streamwise and spanwise distributions. Dotted line in (b) shows the spanwise location of the measurements presented in (a).

Figure 8 presents contours of streamwise velocity on the wing at $\alpha =6^\circ$ at $z/c=1.25$ and on the two-dimensional airfoil at both the effective angle of attack of $5^\circ$ and the geometric angle of attack of $6^\circ$. Also shown is the streamwise displacement thickness $\delta ^*_x$, defined as $\delta ^*_x=\int _{0}^{\delta }(1-u/u_e){\textrm {d} y}$, where $u_e$ is the local edge velocity and $\delta$ is the boundary layer thickness. The streamwise displacement thickness was verified to closely follow the wall-normal height of the inflection point in the separated shear layer. Since the calculation of integral boundary layer parameters in three-dimensional flows requires resolving the boundary layer velocity field starting from the upstream stagnation point, planar estimations of the displacement and momentum thickness ($\delta ^*_x$ and $\theta _x$) based on a two-dimensional approximation are used here.

Figure 8.

Contours of mean streamwise and spanwise velocity on the wing and airfoil. Dashed line: streamwise displacement thickness. Panel (b) is based on data from Toppings et al. (Reference Toppings, Kurelek and Yarusevych2021)

The results in figure 8 reveal the presence of a typical LSB for all the cases shown. The extent of the mean reverse flow region (depicted by the darkest blue contour level) on the wing (figure 8a) and on the two-dimensional airfoil at the same effective angle of attack (figure 8b) show no substantial differences, confirming the close similarity between the mean LSB sufficiently far away from the wingtip and that on the airfoil at the matching effective angle. Compared with the airfoil at the same geometric angle of attack (figure 8c), the LSB location on the wing (figure 8a) is shifted downstream due to the reduction in effective angle of attack, which reduces the adverse pressure gradient and delays separation (Bastedo & Mueller Reference Bastedo and Mueller1986; Boutilier & Yarusevych Reference Boutilier and Yarusevych2012a; Herbst, Hain & Kähler Reference Herbst, Hain and Kähler2020).

Figure 9 highlights the spanwise changes in the mean velocity field towards the wingtip based on a representative subset of measurement planes. From $z/c=1.25$ to $z/c=2.05$, the streamwise velocity contours in figure 9(a) show no substantial changes in the streamwise extent of the reverse flow region. Within this spanwise region, the LSB position and length are in close agreement with those on the two-dimensional airfoil at the effective angle of attack (figure 8b). At $z/c=2.15$, the LSB moves downstream, and the reverse flow region can be seen to extend beyond the maximum downstream extent of the field of view. Closer to the wingtip, at $z/c=2.25$, the full streamwise extent of the reverse flow region is again captured within the field of view, and its length and height diminish notably. Eventually, at and beyond $z/c=2.35$, no reverse flow is detected. This is accompanied by a notable decrease in boundary layer thickness, which is due to a significant downwash induced by the wingtip vortex evidenced by the negative wall-normal component of mean velocity in figure 9(b) at $z/c=2.45$.

Figure 9.

Mean velocity contours in selected $x$–$y$ planes. Dashed line: streamwise displacement thickness.

The spanwise velocity contours in figure 9(c) reveal the presence of substantial spanwise flow in the LSB, reflecting the increased three-dimensionality of the flow field near the wingtip. As expected for a finite wing, the spanwise flow over the suction surface occurs in the negative $z$ direction towards the wing root. Notably, the recirculation region of the LSB near the wingtip contains substantially higher spanwise velocities than the surrounding flow, indicating that the recirculation region is influenced by the spanwise pressure gradient. At $z/c=1.25$, where the LSB on the wing is similar to that on the two-dimensional airfoil at the same effective angle of attack, the magnitude of spanwise flow does not exceed $0.1U_\infty$. However, over the range $1.95< z/c<2.15$ the magnitude of spanwise velocities in the reverse flow region exceed $0.3U_\infty$. The increased strength of the spanwise flow on this part of the span coincides with major spanwise changes in the length and height of the reverse flow region visible in figure 9(a). At $z/c=2.45$, substantial spanwise flow in the negative $z$ direction occurs across the entire field of view because of the proximity of the wingtip vortex core.

In two-dimensional flows, the locations of boundary layer separation and reattachment are traditionally defined as points where skin friction vanishes (Prandtl Reference Prandtl1904). However, in three-dimensional flows, the skin friction may be non-zero at locations of separation and reattachment (Tobak & Peake Reference Tobak and Peake1982; Délery Reference Délery2001). The topological theory of three-dimensional separation requires that separation lines be unstable manifolds of saddle points in the limiting streamline field, whereas reattachment lines be stable manifolds of saddle points (Surana, Grunberg & Haller Reference Surana, Grunberg and Haller2006). However, some studies have asserted that separation and reattachment can occur along limiting streamlines that do not connect to saddle points (Wang Reference Wang1970; Wu et al. Reference Wu, Tramel, Zhu and Yin2000), and this type of separation is commonly referred to as cross-flow separation.

Figures 10(a) and 10(b) present limiting streamlines on the two-dimensional airfoil and wing, respectively. Limiting streamlines were calculated from the velocity gradient at the surface using a third-order polynomial fit to the ten velocity vectors nearest to the surface at each $(x,z)$ location. Note that due to the higher effective angle of attack of the two-dimensional airfoil, the LSB is located farther upstream in comparison with the finite wing at the same geometric angle of attack. Although the separation line is located just outside of the field of view for the two-dimensional configuration, the results show a strongly uniform mean reattachment and nearly two-dimensional limiting streamlines (figure 10a). This is in agreement with spanwise surface pressure measurements, which show spanwise variations below $6\,\%$ of the spanwise average in the reverse flow region ($X/c=0.3$, figure 7b).

Figure 10.

Limiting streamlines. Thick solid and dashed lines: mean separation and reattachment lines, respectively; red line: $C_{fx}=0$ contour. Vertical tick marks indicate spanwise locations of side-view measurement planes. Shaded regions indicate uncertainty in separation and reattachment line estimates.

On the wing (figure 10b), the separation and reattachment lines have been heuristically identified as the attractor and repellor of nearby limiting streamlines in the separation and reattachment regions, respectively. Because the separation line of the LSB on the wing does not emanate from a saddle point, it is classified as a cross-flow separation (Wang Reference Wang1970; Tobak & Peake Reference Tobak and Peake1982). Similar to the results of numerical simulations of cross-flow separation on an ogive cylinder by Yates & Chapman (Reference Yates and Chapman1988), a well-defined onset of separation cannot be identified at the outboard end of the LSB ($2.10\leq z/c \leq 2.35$, figure 10b), where the streamlines gradually approach a common envelope. The separation and reattachment lines obtained from the limiting streamline topology are also compared with the contour of zero streamwise wall shear stress ($C_{fx}=0$) in figure 10(b). It can be seen that the contour of $C_{fx}=0$ closely follows the separation and reattachment lines across the majority of the LSB. Only in the tip region, where the limiting streamlines are oriented primarily in the spanwise direction, the zero streamwise wall shear stress contour begins to deviate from the separation line. This suggests that the two-dimensional definition of separation and reattachment based on zero streamwise skin friction can be used to identify the location and streamwise extent of an LSB on a finite wing, except for the immediate proximity of the wingtip.

The limiting streamlines on the wing in figure 10(b) reveal that the near-surface flow is largely spanwise uniform in the inboard region ($1.25 \leq z/c \leq 1.85$), despite a notable spanwise flow component within the LSB. As the wingtip is approached, the separation line remains at a nearly consistent $x/c$ location before curving upstream at $z/c\approx 2.30$, whereas the reattachment line moves downstream and exits the field of view. The limiting streamlines within the LSB display a progressively increasing spanwise orientation with increasing $z/c$, consistent with the spanwise flow observed in figure 9(c), suggesting that fluid enters into the LSB at its outboard end. For $z/c>2.25$, the limiting streamlines reorient into the positive streamwise direction due to the influence of the wingtip vortex, which suppresses boundary layer separation (Lamar Reference Lamar1974; Torres & Mueller Reference Torres and Mueller2004).

The overall limiting streamline field displays a strong similarity to the qualitative flow visualisations by Huang & Lin (Reference Huang and Lin1995), Yen & Huang (Reference Yen and Huang2009), and Ananda, Selig & Deters (Reference Ananda, Selig and Deters2018) of LSBs on finite wings. A downstream shift in LSB reattachment lines near the tips of finite wings was also reported from the flow visualisations of Marchman & Abtahi (Reference Marchman and Abtahi1985), Bastedo & Mueller (Reference Bastedo and Mueller1986) and Genç et al. (Reference Genç, Özkan, Özden, Kiriş and Yildiz2018). It is important to note that this behaviour, expected from the reduction in the effective angle of attack (Bastedo & Mueller Reference Bastedo and Mueller1986), is constrained to the relative vicinity of the wingtip ($z/c>2.15$, figure 10b). For comparison, within $1.25\leq z/c \leq 2.15$ where the effective angle of attack changes from approximately $5^\circ$ to $3^\circ$ (Toppings et al. Reference Toppings, Kurelek and Yarusevych2021), an LSB on a two-dimensional wing would move downstream by approximately $0.15c$ (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012a), i.e. approximately $75\,\%$ of the present bubble length. Thus, the spanwise variations seen in the LSB extent and position over the majority of the span are significantly lower than those expected from the changes in the effective angle of attack, and the bubble appears to settle on a configuration driven by the effective angle at the wing root, except in the proximity of the wingtip.

Figure 10(b) also reveals a distinct change in direction of the limiting streamlines at approximately $x/c=0.40$, which roughly demarcates the fore and aft portions of the LSB (i.e. the dead-air region and mean reverse flow vortex, respectively, figure 1). The limiting streamlines in the forward part of the bubble are directed more strongly in the spanwise direction, similar to the numerical simulations of Chen et al. (Reference Chen, Qin and Nowakowski2013). Because the skin friction within the LSB has a negative $z$ component, the streamwise gradient in the spanwise flow magnitude implies that the recirculating flow within the LSB follows a helical path which is similar to LSBs on nominally infinite swept wings (Horton Reference Horton1968).

Although the limiting streamline field is in qualitative agreement with previous surface flow visualisations on finite wings at low Reynolds number, resolving the attendant three-dimensional flow field is essential to reconstruct the bubble topology and understand the mechanisms by which the observed surface patterns form. Figure 11(a) presents an isosurface of zero streamwise velocity ($\bar {u}=0$) constructed from the side-view measurements that illustrates the three-dimensional shape of the LSB on the wing. The results show that the streamwise bounds of the reverse flow region closely follow the separation and reattachment lines determined from the limiting streamline topology. At the outboard end of the LSB, the reverse flow region thickness reduces, and no reverse flow is eventually detected for $z/c>2.30$. Beyond this spanwise location, downwash from the wingtip vortex causes the boundary layer to remain attached to the wing surface.

Figure 11.

(a) The $\bar {u}=0$ isosurface. Contour colours indicate normal distance from wing surface. Thin solid lines: limiting streamlines; thick solid line: separation line; thick dashed line: reattachment line. (b) Maximum height of $\bar {u}=0$ isosurface, maximum $\delta ^*_x$, and maximum height of shear layer velocity profile inflection point.

A quantitative analysis of the changes in the wall-normal extent of the LSB along the span is facilitated by figure 11(b). The spanwise variation of maximum thickness of the $\bar {u}=0$ isosurface is presented with the maximum streamwise displacement thickness. The latter is shown to closely follow the maximum height of the inflection point in the core of the shear layer. All three parameters vary gradually in the inboard region ($1.25 \leq z/c \leq 1.85$). However, a notable increase in the bubble height is observed towards $z/c\approx 2.00$, where an increase in both the maximum height of the reverse flow region and streamwise displacement thickness takes place, which agrees with the trend reported by Toppings et al. (Reference Toppings, Kurelek and Yarusevych2021). For $z/c>2.00$, a gradual decrease in LSB thickness occurs, until reverse flow is no longer observed in the tip region for $z/c>2.30$. The accompanying reduction in displacement thickness is consistent with the findings of Awasthi et al. (Reference Awasthi, Moreau and Doolan2018), who observed a substantial reduction in boundary layer thickness towards the tip of an $AR=0.5$ semispan wing.

Based on the observed changes in mean flow topology along the span, three characteristic regions can be identified. The inboard region is located within $1.25\leq z/c \leq 1.85$ and characterised by nearly spanwise uniform LSB development. The tip region is confined to $z/c>2.25$, where the flow is strongly influenced by three-dimensional tip effects, and characteristic LSB features are no longer observed. Between these regions, there is an intermediate region ($1.85 < z/c \leq 2.25$), where the thickness of the LSB changes substantially and strong spanwise flow occurs within the reverse flow region.

To visualise the three-dimensional shape of the separation bubble, streamsurfaces of separation and reattachment were constructed from the side-view PIV data and are presented in figure 12. The separation surface is formed by the downstream advection of the separation line (Hultquist Reference Hultquist1992). Figure 12(a) shows that the LSB on the wing is an open bubble. Instead of forming a closed surface, the outboard portion of the separation surface is swept towards the wing root by spanwise flow, and fluid enters the LSB at its outboard open end. This is further substantiated by the trajectory of the reattachment surface in figure 12(b), which is the streamsurface that emanates from the reattachment line in reverse time (Surana et al. Reference Surana, Grunberg and Haller2006). Within the intermediate region, the reattachment surface passes over the separation surface and extends farther upstream ($2.10 \leq z/c \leq 2.25$), indicating that fluid is drawn into the LSB from above and outboard of the separation surface. Farther from the wingtip, the separation and reattachment surfaces approach the same shape, as indicated by the small segment of the reattachment surface near $z/c=1.50$ that closely follows the outside of the separation surface. The convergence of the two streamsurfaces in the inboard region of the wing suggests that the bubble topology in $x$–$y$ planes approaches that of the closed two-dimensional separation bubble depicted in figure 1 away from the wingtip.

Figure 12.

Streamsurfaces of separation (red) and reattachment (black). Thin lines: limiting streamlines; thick solid line: separation line; thick dashed line: reattachment line. Insets illustrate the relative position of separation and reattachment streamsurfaces in $x$–$y$ planes. (a) Separation surface and (b) separation and reattachment surfaces.

Because the separation and reattachment surfaces are everywhere tangent to the velocity field, there is zero mass flux across both surfaces. However, as demonstrated by the numerical simulations of Rodríguez & Theofilis (Reference Rodríguez and Theofilis2010), this does not preclude inflows or outflows from the recirculation region of the LSB in the $x$ direction. This is illustrated in the inset plots of figure 12 which present schematics of the intersection of the dividing streamsurfaces with two-dimensional $x$–$y$ planes. Near $z/c=2.10$, the reattachment surface extends upstream, passing above the separation surface, which allows fluid to enter into the mean reverse flow vortex. In contrast, near $z/c=1.90$, the separation surface extends downstream, passing above the reattachment surface, which allows fluid to exit the mean reverse flow vortex. Since the accumulated integration errors in streamsurface calculation and the limited spatial resolution of the PIV measurements make it difficult to quantify the small inflows/outflows based on the calculated separation and reattachment surfaces, a control volume analysis was employed to more precisely determine where fluid enters and exits the three-dimensional open LSB.

The planar control volume analysis was applied to the stereoscopic velocity data from the spanwise region where an LSB is fully captured within the field of view of $x$–$y$ measurement planes ($1.25\leq z/c \leq 2.10$). The definition of the planar control volume at $z/c=1.25$ is shown in figure 13(a), and the control volume boundaries at all other planes are located at the same $x/c$ and $y/c$ locations. The control volume boundary was placed around the aft portion of the LSB containing the reverse flow vortex (figure 1), where notable changes take place in the spanwise flow (figure 9b).

Figure 13.

Control volume mass balance: (a) control volume superimposed on the LSB outline at $z/c=1.25$. (b) terms (1) and (2) from (3.1).

The conservation of mass for the planar control volume in an incompressible flow can be expressed as

(3.1)

where $A$ is the area of the planar control volume bounded by $l$ and $\hat {\boldsymbol {n}}$ is the outward unit normal vector on the boundary. Term (1) represents the net in-plane outflow of fluid through the boundaries of the planar control volume due to the in-plane components ($u$ and $v$) of the mean velocity ($\bar {\boldsymbol {u}}$) at the control volume boundaries. Term (2) represents the net out-of-plane outflow of fluid due to spanwise gradients in spanwise velocity, which is equal to the derivative of the mean bound spanwise volume flow rate ($\bar {\dot {V}}$) with respect to $z$.

Figure 13(b) presents the values of terms (1) and (2) for each of the planar control volumes. The observed changes in the spanwise gradient of the spanwise volume flow rate, and consequently term (2) in (3.1), necessitate changes in the net in-plane outflow (term (1), (3.1)) through the control volume boundaries. It can be seen that, at a given spanwise location, the two terms add to zero to within the experimental uncertainty. Of particular interest, however, are the signs of the respective terms, distinguishing between inflow and outflow.

The last outboard plane that captures the full extent of the LSB is located at $z/c=2.10$, where the LSB topology experiences notable changes due to the proximity of the wingtip, including the reduction of the bubble height and downstream movement of the reattachment location (figure 11). In figure 13(b), term (1) is negative at this location, reflecting the inflow into the open end of the LSB. As the gradient of the spanwise volume flow rate inside the LSB diminishes inboard, in-plane outflow (i.e. positive term (1)), is eventually attained, peaking in the intermediate region and then gradually decreasing towards the expected zero value for the nearly two-dimensional flow in the inboard region. The spanwise region of maximum in-plane outflow from the planar control volume matches that of maximum LSB thickness in figure 11(b). Thus, the increase in LSB thickness in the intermediate region is attributed to changes in the mass exchange balance between the LSB and the surrounding fluid driven by the spanwise flow across the open end of the LSB. Consequently, the strong three-dimensional flow induced by the wingtip vortex not only causes open LSB formation and attached flow near the tip, but also induces notable spanwise changes to the LSB position, length, and height in the intermediate region.

3.2. Laminar to turbulent transition

Due to the critical role of shear layer transition in determining the overall mean LSB topology, the observed spanwise variations in the mean flow of the three-dimensional LSB on the wing are expected to be linked to changes in the transition process. The evolution of velocity perturbations in the shear layer is explored in figure 14, which presents contours of r.m.s. velocity fluctuations. For the inboard region (e.g. at $z/c=1.25$), the results reveal the progressive amplification of streamwise velocity fluctuations along the core of the separated shear layer, which is most prominent around the maximum height location. The attendant flow transition leads to mean flow reattachment, where saturation of velocity fluctuations is observed, agreeing with previous studies (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012b; Marxen et al. Reference Marxen, Lang and Rist2013). Near the $x/c$ location of maximum bubble height, identifiable from the streamwise displacement thickness plots (dashed lines), the streamwise velocity fluctuation contours (figure 14a) exhibit three peaks, with the middle one aligned along the core of the shear layer, a structure expected from other experiments in separated shear flows (Dovgal, Kozlov & Michalke Reference Dovgal, Kozlov and Michalke1994; Häggmark, Bakchinov & Alfredsson Reference Häggmark, Bakchinov and Alfredsson2000) and linear stability predictions (Michalke Reference Michalke1991; Marxen et al. Reference Marxen, Kotapati, Mittal and Zaki2015). The wall-normal velocity fluctuations (figure 14b) for the inboard planes display a single peak along the trajectory of the separated shear layer, consistent with the formation of shear layer vortices. The observed trends persist in all measurement planes for $z/c\lesssim 2.00$ and are similar to those expected for a two-dimensional LSB (Lengani et al. Reference Lengani, Simoni, Ubaldi, Zunino and Bertini2017b). The spanwise velocity fluctuations captured in the present investigation (figure 14c) offer a novel insight into the progression of the last stages of the transition process, where onset of spanwise fluctuations is expected. It can be seen that similar levels of spanwise velocity fluctuations are achieved farther downstream compared with either streamwise or wall-normal components. As will be discussed later, these arise from spanwise deformations and subsequent breakdown of dominant shear layer vortices in the aft portion of the LSB.

Figure 14.

The r.m.s. velocity contours. Dashed line: streamwise displacement thickness. Masked flow area within $0.2< x/c<0.3$ pertains to region of relatively high random errors near the wall.

Significant changes in the development of shear layer velocity fluctuations take place in the intermediate region for $z/c>2.00$ (figure 14). Most notably, a progressive downstream shift of the transition process can be seen between $z/c=2.05$ and $2.25$, and no significant velocity fluctuations are observed within the field of view in the tip region at $z/c=2.25$ and $2.35$. Figure 15(a) presents a convenient comparison basis for the assessment of spanwise variations in turbulent kinetic energy (TKE) at the core of the shear layer superimposed on the mean separation and reattachment lines. Also presented is the streamwise location of maximum shape factor $H_x=\delta ^*_x/\theta _x$, which has been shown to be a robust criterion for defining a mean transition location in two-dimensional LSBs (Kurelek, Kotsonis & Yarusevych Reference Kurelek, Kotsonis and Yarusevych2018). A rapid growth in turbulent kinetic energy occurs in the aft portion of the LSB for $z/c\lesssim 2.10$. Within the same spanwise region, the $x/c$ location of maximum shape factor follows the shape of the TKE contours and the reattachment line, indicating that the spanwise variation seen in the reattachment location is related to the variation in the transition process. In contrast, for $z/c\gtrsim 2.15$, the location of maximum shape factor moves upstream, while no significant TKE levels are reached within the field of view. This signifies that estimation of the transition location using the two-dimensional definition of the shape factor is no longer valid near the wingtip, and the flow behaviour is largely governed by the three-dimensional wingtip effects. In contrast, within the inboard and intermediate regions of the span, the formation of the LSB is largely driven by the shear layer transition, similar to the two-dimensional LSB formation.

Figure 15.

(a) Contours of TKE at $y=\delta ^*_x$. Solid line: separation line; dashed line: reattachment line; diamonds: location of maximum streamwise shape factor. Thin lines indicate uncertainty in separation and reattachment lines. (b) Maximum shape factor and minimum streamwise velocity.

Added insight into local flow stability can be gained from the results in figure 15(b), which presents the spanwise variation in maximum shape factor and minimum streamwise velocity. In the inboard region, both the shape factor and minimum streamwise velocity remain unchanged within the experimental uncertainty. However, in the intermediate region ($1.85< z/c\leq 2.25$) changes in both parameters take place, and significant extrema occur around $z/c=2.10$. Here, the minimum streamwise velocity reaches $-0.25U_\infty$, which exceeds the threshold of $-0.15U_\infty$ to $-0.2U_\infty$ for absolute instability in LSBs reported by Alam & Sandham (Reference Alam and Sandham2000) from direct numerical simulations. Although the increase of shape factor and reverse flow velocity with increasing $z/c$ from $z/c=1.95$ to $z/c=2.10$ would be expected to increase the instability of the separated shear layer (Dovgal et al. Reference Dovgal, Kozlov and Michalke1994), a downstream movement in the location of maximum shape factor and delay in TKE production occurs. Considering that these changes coincide with the region of substantial spanwise flow within the LSB (figure 9c), it implies that the transition process becomes more strongly influenced by three-dimensional effects closer to the wingtip. In the tip region ($z/c\geq 2.25$) where transition is delayed, a continued decrease in maximum shape factor and increase in minimum streamwise velocity occurs with increasing $z/c$. Concurrently, the thickness of the reverse flow region is reduced before separation ceases entirely near the wingtip (figure 11b), consistent with the reduction of the adverse pressure gradient (figure 7b).

To further explore the spanwise variations in the transition process, the growth of wall-normal velocity fluctuations, which are strongly associated with the formation of spanwise vortices that lead to transition (Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014), is explored in figure 16. Wall-normal r.m.s. velocity fluctuations at $y=\delta ^*_x$ are compared at representative spanwise locations in the inboard ($z/c=1.25$) and intermediate ($z/c=1.95$, $2.15$ and $2.20$) regions in figure 16(a). Only $v'_{rms}$ values exceeding the uncertainty from correlation statistics with a $95\,\%$ confidence level are shown. Also plotted for reference is the growth of $v'_{rms}$ from the two-dimensional airfoil at the same effective angle of attack as the inboard region of the wing ($5^\circ$), based on data from Toppings et al. (Reference Toppings, Kurelek and Yarusevych2021). Maximum streamwise growth rates of r.m.s. wall-normal velocity fluctuations ($\sigma _{v'max}=\ln (\Delta v'_{rms}/U_\infty )/(\Delta x/c)$, e.g. Istvan & Yarusevych Reference Istvan and Yarusevych2018) are plotted in figure 16(b). The markers in figure 16(a) indicate the regions of approximately constant exponential growth that were used to estimate the maximum growth rates for each respective $v'_{rms}$ curve.

Figure 16.

(a) Streamwise growth of $v'_{rms}$ at $y=\delta ^*_x$ on the wing at $\alpha =6^\circ$ compared with the airfoil at the effective angle of attack of the midspan. Shaded regions indicate uncertainty. Markers indicate interval used to calculate $\sigma _{v'max}$ (b) Maximum streamwise growth rates of $v'_{rms}$. Dashed line: airfoil at effective angle of attack of midspan.

For $z/c\leq 2.10$, the initial growth rates of $v'_{rms}$ are similar to the two-dimensional airfoil at the effective angle of attack, and the velocity fluctuations saturate at a similar level. However, closer to the wingtip, the maximum growth rate of $v'_{rms}$ is reduced substantially. The notable reduction in the growth rate of $v'_{rms}$ at the outboard end of the intermediate region (figure 16b) is consistent with the delay in transition observed on this part of the span in figures 14 and 15(a). This delay in transition is in agreement with the results of Smith & Ventikos (Reference Smith and Ventikos2021), who observed a reduction in the growth of Tollmien–Schlichting waves near the wingtip in an attached boundary layer. The results confirm that the flow development in the tip region is largely governed by the wingtip vortex induced flow, which eliminates separation and terminates LSB, while delaying transition significantly.

3.3. Vortex dynamics

Shear layer vortex shedding is known to play a key role in the LSB dynamics (Watmuff Reference Watmuff1999; Marxen et al. Reference Marxen, Lang and Rist2013). The development of shear layer vortices in the investigated LSB is illustrated in figure 17, which presents an instantaneous streamwise velocity fluctuation field ($u'$) from the top-view PIV configuration and insets of instantaneous spanwise vorticity and $\lambda _2$ criterion (Jeong & Hussain Reference Jeong and Hussain1995) contours at selected spanwise positions from uncorrelated side-view PIV measurements. In the top-view snapshot, the spanwise structure of the rollers can be observed. Because the light sheet was positioned to intersect the tops of the roll-up vortices, the vortices appear as spanwise bands of positive streamwise velocity fluctuations. Notable periodic velocity fluctuations can be seen just upstream of and at the mean reattachment line. They display a strong degree of spanwise coherence along the inboard portion of the wing, comparable to observations on two-dimensional geometries at similar turbulence levels (Nati et al. Reference Nati, de Kat, Scarano and van Oudheusden2015; Kurelek, Lambert & Yarusevych Reference Kurelek, Lambert and Yarusevych2016; Istvan & Yarusevych Reference Istvan and Yarusevych2018; Toppings et al. Reference Toppings, Kurelek and Yarusevych2021). However, as the wingtip is approached, the vortices begin to develop progressively more pronounced spanwise undulations. Eventually, no clear signs of periodic shedding are observed in the tip region ($Z/c> 2.25$). Across the portion of the span where shedding is seen, the spanwise structures rapidly lose coherence and break down downstream of mean reattachment, with signs of vortex breakdown appearing earlier upstream in the intermediate region ($1.85< Z/c\leq 2.25$).

Figure 17.

Instantaneous flow field snapshots illustrating vortex shedding in LSB. The central image presents streamwise velocity fluctuation contours from top-view measurements. Insets present contours of spanwise vorticity and $\lambda _2 c^2/U_\infty^2 =-1500$ from uncorrelated side-view PIV measurements at selected $x$–$y$ planes along the span. Dashed line: reattachment line.

The presence of periodic shedding of shear layer structures across the majority of the span is confirmed by the insets from side-view measurements. Note that the streamwise extent of the side-view field of view is smaller than that of the top-view measurements. The insets show that the formation of vortices occurs in the aft portion of the LSB, and the roll-up location remains similar across the span for $z/c<2.00$. With increasing $z/c$ in the inboard region, the distance between the vortex trajectory and the surface also increases, and this trend continues in the intermediate region up to $z/c=2.00$, following the increase in the mean height of the separation bubble (figure 11b). Closer to the wingtip, at $z/c=2.15$, vortices can still be identified, but they form farther downstream and closer to the wing surface. The strength of the roll-up vortices in the field of view rapidly diminishes between $z/c=2.15$ and $2.20$, and no roll-up can be seen at $z/c=2.25$. This is consistent with the less rapid growth of wall-normal velocity fluctuations at these spanwise locations in figure 16(b), and is in agreement with linear stability theory, which predicts a decrease in disturbance growth rates as the inflection point of the shear layer moves closer to the wall (Michalke Reference Michalke1991). The absence of shear layer roll up at and beyond $z/c=2.25$ is substantiated by the top-view results that cover a larger streamwise extent of the flow, indicating that shedding is suppressed and that the flow remains laminar near the wingtip. Despite the absence of shear layer roll-up in the tip region, the separated shear layer still curves towards the wing surface. However, mean reattachment is not driven by the increase in wall-normal momentum transfer caused by shear layer roll-up vortices but by the rapid decrease in the adverse pressure gradient and downwash induced by the wingtip vortex, which eventually suppress separation closer to the wingtip.

To explore the interaction of spanwise flow with the shear layer rollers, figure 18 presents uncorrelated phase-averaged spanwise vorticity ($\tilde {\omega _z}$), spanwise velocity ($\tilde {w}$) and $\lambda _2$ contours for selected $x$–$y$ planes. The full phase-averaged cycle sequences from the same planes are available in supplementary movies 1–4. In the inboard region ($z/c=1.25$, figure 18a), no significant spanwise velocity fluctuations are associated with the shear layer rollers. In the intermediate region, at $z/c=2.00$ (figure 18b), shed vortices entrain high spanwise momentum fluid concentrated within the LSB and advect it downstream. A similar interaction is seen at $z/c=2.15$ (figure 18c). Eventually, sufficiently close to the wingtip, shedding is suppressed ($z/c=2.25$, figure 18d), and only minor perturbations can be seen in the spanwise velocity contours. The results in figure 18 show that the interaction between shear layer vortices and spanwise flow leads to the entrainment of spanwise momentum into the vortex cores in the intermediate region. This may be the underlying cause of the stronger spanwise undulations and earlier vortex breakdown in the intermediate region (figure 17), since axial flow is known to destabilise vortex filaments (Lucca-Negro & O'Doherty Reference Lucca-Negro and O'Doherty2001).

Figure 18.

Phase-averaged $\tilde {\omega _z}$ and $\tilde {w}$ contours at selected side-view measurement planes. Black lines are contours of $\tilde {\lambda _2}c^2/U_\infty ^2=-1500$. Full cycle sequence is available in supplementary movies 1–4 at https://doi.org/10.1017/jfm.2021.881.

The phase-averaged streamwise velocity results from the top view configuration in figure 19 confirm the strong two-dimensionality of the roll-up vortices in the inboard region. Closer to the wingtip, greater spanwise variations in the vortex cores take place within $1.75< Z/c<2.10$, and the associated changes in the spanwise vortex shedding are best seen in supplementary movie 5. The eventual termination of the shedding process closer to the wingtip is also reflected in figure 19. In the range $1.85< Z/c<2.10$, where earlier breakdown of coherent fluctuations is seen in figure 17, the phase-averaged roll-up vortices in figure 19 do not persist as far downstream into the redeveloping turbulent boundary layer as they do in the inboard region, providing further evidence that the entrainment of spanwise momentum into the vortex cores in the intermediate region leads to earlier vortex breakdown.

Figure 19.

Phase-averaged $\tilde {u}$ contours from the top-view configuration. Dashed line: reattachment line. Full cycle sequence is available in supplementary movie 5.

To analyse spanwise changes in the vortex dynamics in more detail, POD was conducted on the measurements of the three-component velocity fluctuation vector $\boldsymbol {u'}$, and the in-plane velocity fluctuation vector $(u',w')$, from the side- and top-view configurations, respectively. The relative POD modal energies of the first twenty POD modes are presented in figures 20(a) and 20(b), for the side- and top-view configurations, respectively. Modes $1$ and $2$ of $\boldsymbol {u'}$ contain a notably greater fraction of the total energy than the higher modes and are paired for $z/c\leq 2.15$, which was confirmed by the examination of the corresponding spatial modes. Pairing also occurs between the first two POD modes of $(u',w')$ from the top view. The side-view results are consistent with the prominent shedding of shear layer vortices in the same spanwise region seen in figure 17. At $z/c>2.15$, the relative energy content of the first two modes of $\boldsymbol {u'}$ is reduced significantly, consistent with the suppression of shear layer shedding near the wingtip.

Figure 20.

Relative modal energy distribution of the first $20$ POD modes of (a) $\boldsymbol {u'}$ from side-view PIV and (b) $(u',w')$ from top-view PIV.

Since the level of TKE varies substantially in the spanwise direction in the aft portion of the LSB (figure 15a), it is instructive to compare the absolute energy content of the side-view POD modes. Figure 21 plots the spatially averaged TKE ($\langle \text {TKE}\rangle$) of the first two modes for each measurement plane, which can be used to compare absolute modal energy content between different spanwise locations. A relative increase in the energy content of the first two modes is seen at $z/c=2.00$, where formation of stronger vortices was observed due to the increase in the height of the mean LSB. However, similar to the distribution of relative modal energy, the absolute modal energy of the dominant mode pair is greatly reduced in the tip region, where the shear layer vortices weaken and shedding eventually ceases. Although a peak in relative modal energy of POD modes $1$ and $2$ of $\boldsymbol {u'}$ occurs at $z/c=2.15$ (figure 20a), figure 21 shows that the absolute energy contained in these modes at $z/c=2.15$ is substantially less than that at the neighbouring inboard planes ($z/c\leq 2.10$). This is attributed to the delayed shear layer roll-up (figures 17d and 18c) and the associated reduction in the energy of random turbulent fluctuations within the field of view. Thus, although weaker velocity fluctuations are produced by the vortices here, the relative energy of the associated modes increases since the structures do not break down within the field of view.

Figure 21.

Spatially averaged TKE of the first two POD modes of $\boldsymbol {u'}$ at each side-view plane.

The most energetic spatial POD mode ($\boldsymbol {\varPhi }^{(1)}$) is presented in figure 22 for representative planes, and the paired mode ($\boldsymbol {\varPhi }^{(2)}$) is not shown for brevity. The largest magnitudes of $\boldsymbol {\varPhi }^{(1)}$ occur in the aft portion of the LSB for $z/c\leq 2.15$, where shear layer vortices are observed in figure 17. A typical periodic spatial pattern of positive and negative wall-normal velocity seen in the aft portion of the field of view in figure 22(a–c) confirms the association between the dominant mode pair and the shear layer vortices within this spanwise region. The results show that the spanwise mode component is notably lower than the streamwise and wall-normal components at the onset of shear layer roll-up, supporting the earlier observation of nominally two-dimensional initial shear layer roll-up. However, an increase in the spanwise component of $\boldsymbol {\varPhi }^{(1)}$ can be seen at the same $x/c$ locations for $z/c=2.00$ and $2.15$, which is attributed to mixing between the fluid with greater negative spanwise velocity below the shear layer and the fluid with lesser negative spanwise velocity above the shear layer (figures 18b and 18c) induced by vortex shedding in the intermediate region. Although the r.m.s. spanwise velocity fluctuations associated with the breakdown of shear layer vortices are greater in the inboard region than in the intermediate region (figure 14c), coherent spanwise velocity fluctuations are not associated with the shear layer vortices in the inboard region because the difference in spanwise velocity between the fluid above and below the shear layer is much smaller (figure 18a). Mode $1$ at $z/c=2.25$ does not display a spatially periodic pattern associated with vortex formation, and modes $1$ and $2$ are no longer paired at this location, consistent with the absence of shear layer vortices in the tip region. Here, mode $1$ is mostly associated with the $u$-component fluctuations ascribed to shear layer flapping, which involves low frequency oscillations of the separated shear layer (Lengani et al. Reference Lengani, Simoni, Ubaldi, Zunino and Bertini2017b).

Figure 22.

First spatial POD modes from side-view PIV. Masked flow area within $0.21< x/c<0.30$ pertains to region of relatively high random errors near the wall.

The first four POD modes of $(u',v')$ from the top-view measurements form two mode pairs that contain substantially higher modal energy content than higher modes (figure 20b). The chordwise component of the first ($\boldsymbol {\varPhi }^{(1)}$) and third ($\boldsymbol {\varPhi }^{(3)}$) spatial modes are presented in figure 23, with the paired modes $2$ and $3$ not shown for brevity. The results indicate that the first mode pair is associated with the shear layer vortices, with the characteristic streamwise wavelength matching that seen in the side-view mode projections of figure 22. In the inboard region, the spatial mode shapes of mode $1$ reflect strong two-dimensionality of the roll-up vortices. This is followed by progressive spanwise deformations for $Z/c>1.7$, with the shapes of the initial spanwise deformations following that of the mean reattachment line (dashed line). The contours of mode $3$ indicate that the second mode pair is associated with velocity fluctuations at twice the fundamental wavelength, i.e. at the subharmonic of the shedding frequency. The relative magnitude of the modal energy increases notably for $Z/c>2.00$, suggesting that the increased growth of the subharmonic mode is a result of wingtip effects. The amplification of disturbances at the subharmonic of the fundamental shear layer frequency has been associated with vortex merging in experiments on LSBs on two-dimensional airfoils (Yarusevych et al. Reference Yarusevych, Sullivan and Kawall2009; Kurelek, Yarusevych & Kotsonis Reference Kurelek, Yarusevych and Kotsonis2019). The merging of vortices in LSBs is often a localised phenomenon along the span, and has been speculated to be linked to cycle-to-cycle variations in the vortex strength or spacing in nominally two-dimensional LSBs (Kurelek et al. Reference Kurelek, Yarusevych and Kotsonis2019). This is consistent with the increased vortex deformations seen near the wingtip, which can promote localised vortex merging.

Figure 23.

The $u$ component of first and third top-view spatial POD modes of $(u',w')$. Dashed line: reattachment line from side-view stereo-PIV volumetric reconstruction.

For a statistical comparison of the wavelengths of shear layer disturbances in the LSB, the spatial power spectral density of the wall-normal component of velocity fluctuations ($\mathcal {F}_{v'v'}$) was calculated for each instantaneous velocity field from the side-view configuration. The data were sampled along $y=\delta ^*_x$, over the streamwise extent of the field of view. The resulting spectra of streamwise wavelength ($\lambda$) were then averaged over all instantaneous velocity fields at each measurement plane, and the results are presented in figure 24. In figure 24(a), all spectra have been normalised by total energy and stepped by an order of magnitude to facilitate the identification of the dominant spectral peaks, while the comparison of the spectral energy content is facilitated by figure 24(b). In the inboard and intermediate regions $1.25\leq z/c\leq 2.25$, the results reveal clear spectral peaks within $0.037\leq \lambda /c \leq 0.043$, which are consistent with the spacing between vortices seen in figures 17 and 18. The associated spectral energy contours in figure 24(b) show maximum energy content reached inside the inboard and intermediate regions, while a notable decrease in the peak energy is seen at the boundaries of the intermediate region. The results also show a relative increase in the spectral energy of the subharmonic mode for $2.05\leq z/c \leq 2.25$, agreeing with the increase in the magnitude of the top-view subharmonic POD spatial mode (figure 23) at the outboard end of the intermediate region. However, the energy content associated with these fluctuations is relatively low, as can be seen in figure 24(b), and thus the overall impact on the LSB topology is not expected to be significant. As discussed earlier, the delayed and eventually suppressed separation and the associated reduction in the growth rates of disturbances delay transition near the wingtip, producing insignificant spectral content in the tip region in figure 24.

Figure 24.

Streamwise wavelength spectra from side-view stereo-PIV at $y=\delta ^*_x$. Dashed line: $\lambda /c=0.04$; dotted line: $\lambda /c=0.08$. Each spectrum in (a) is shifted by an order of magnitude at consecutive $z/c$ locations for clarity. (a) Normalised power spectral density and (b) power spectral density.

The spanwise variation of the fundamental wavelength ($\lambda _0$) from the spectral analysis is presented in figure 25(a), along with the mean streamwise velocity at the mean location of maximum shape factor ($x=x_{H_{max}}$, $y=\delta ^*_x$) which approximates the average convective velocity of shear layer vortices (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012b) at a given $z$ plane. The estimates are then used to obtain the shedding Strouhal number ($St_0=f_0c/U_\infty$, where $f_0\approx \bar {u}/\lambda _0$) which is presented in figure 25(b). The results confirm that the dominant streamwise wavelength of the vortices does not change appreciably across the span of the wing, despite notable changes to the mean bubble topology and vortex dynamics in the proximity of the wingtip. The associated shedding frequency shows minor variations across the span driven largely by the variations in the convective velocity estimates. The shedding frequency estimates agree to within the experimental uncertainty with the direct estimates from time-resolved measurements obtained by Toppings et al. (Reference Toppings, Kurelek and Yarusevych2021).

Figure 25.

(a) Convection velocity at $x=x_{H_{max}}$, $y=\delta ^*_x$ and fundamental wavelength. (b) Estimated fundamental vortex shedding frequency.

It is important to note that, based on measurements of two-dimensional LSBs on the same airfoil profile, a change in the effective angle of attack from approximately $5^\circ$ at $z/c=1.25$ to $3^\circ$ at $z/c=2.15$ would result in a decrease in $St_0$ of approximately $20\,\%$ in a two-dimensional LSB (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012a). In contrast, the variation of $St_0$ seen in figure 25 is non-monotonic and is confined primarily to within approximately $10\,\%$ of the inboard value largely due to experimental uncertainty. Thus, such spanwise uniformity in the mean LSB structure and transition process are surprising given the substantial change in the mean streamwise adverse pressure gradient along the span (figure 7b). Because the characteristics of the mean LSB and the dynamics of the transition process are interdependent, the relative spanwise uniformity of both the mean LSB topology and the transition statistics can be interpreted as an indication of the spanwise homogeneity of the transition process. Thus, it can be conjectured that shear layer transition within an LSB formed on a finite wing locks onto a common unstable mode across the span, with relatively small variations in the vortex dynamics compared with those expected due to the change in the effective angle of attack across the span.