2.1 The 30P30N multi-element airfoil

The 30P30N multi-element airfoil is a classical airfoil in the family of high-lift configuration. Extensive research has been conducted with this typical configuration (Paschal, Jenkins & Yao Reference Paschal, Jenkins and Yao2000; Jenkins et al. Reference Jenkins, Khorrami and Choudhari2004; Choudhari & Khorrami Reference Choudhari and Khorrami2007; Choudhari & Lockard Reference Choudhari and Lockard2015; Lockard et al. Reference Lockard, Choudhari and Buning2015; Souza et al. Reference Souza, Rodríguez, Simões and Medeiros2015). The 30P30N airfoil utilized in the current experiment is a scaled model from the AIAA BANC-II Workshop (Choudhari & Yamamoto Reference Choudhari and Yamamoto2012). As shown in figure 1(a), the stowed chord length $c$ is 150 mm. The aspect ratio is 3.33. According to Jenkins et al. (Reference Jenkins, Khorrami and Choudhari2004), this aspect ratio could ensure the quasi-two-dimensionality of the flow in the middle region of the airfoil. The angle of attack of the 30P30N airfoil ( $\unicode[STIX]{x1D6FC}$ ) is defined as the angle between the chord line of the main element and the horizontal line. Following this definition, the angle of attack in figure 1(a) is $\unicode[STIX]{x1D6FC}=0^{\circ }$ . Except for a few special definitions for local coordinate systems, the origin of the coordinate system ( $O$ in figure 1 a) is located at the leading edge of the slat with $\unicode[STIX]{x1D6FC}=0^{\circ }$ . $X$ , $Y$ and $Z$ denote the streamwise, vertical and spanwise directions, respectively. As shown in figure 1(b), two circular acrylic endplates (radii equal to 250mm) with cut corners are applied at two sides of the airfoil. These two endplates could improve the mean spanwise uniformity of the airfoil flow (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012a ).  $\unicode[STIX]{x1D6FC}$ is adjusted by rotating the airfoil around the pivot.

Figure 1.

(a) The 30P30N multi-element airfoil; (b) three-dimensional sketch for the installation of 30P30N airfoil.

Figure 2.

Sketch of hydrogen bubble flow visualization at $\unicode[STIX]{x1D6FC}=4^{\circ }$ : (a) plane view with volume illumination $(\text{FV}_{S})$ ; (b) side view with a vertical laser sheet; (c) oblique view with volume illumination; (d) end view with a transverse laser sheet $(\text{FV}_{1})$ ; (e) bottom view with two parallel horizontal laser sheets ( $\text{FV}_{2}$ and $\text{FV}_{3}$ ); (f) plane view with volume illumination $(\text{FV}_{M})$ .

2.2 Experimental set-up

The experiments are conducted in a water tunnel of the Institute of Fluid Mechanics at the Beijing University of Aeronautics and Astronautics. The test section is $600\times 600\times 3000~\text{mm}$ (height  $\times$  width  $\times$  length). The free-stream velocity is $U_{\infty }=60~\text{mm}~\text{s}^{-1}$ , with turbulence intensity less than 1 %. The Reynolds number based on the stowed chord length is $Re_{c}=8320$ . The flows over the multi-element airfoil with $\unicode[STIX]{x1D6FC}=0{-}16^{\circ }$ (the angle interval is $2^{\circ }$ ) are investigated. To keep the description briefly, only the experimental set-up at $\unicode[STIX]{x1D6FC}=4^{\circ }$ is introduced in this section.

2.3 Post-processing tools

In order to facilitate a deep comprehension of the flow around the multi-element airfoil, a few post-processing tools are applied to the velocity fields. This section provides a brief introduction to these tools.

3.1 Dominant flow structures

According to the evolution of dominant flow structures around the main element with angles of attack ( $\unicode[STIX]{x1D6FC}=0^{\circ }{-}16^{\circ }$ with an interval $2^{\circ }$ ), three regimes can be classified in the current work, as shown in figure 4: (i) irregular streamwise vortex at $\unicode[STIX]{x1D6FC}=0^{\circ }$ (figure 4 a); (ii) regular streamwise vortex, named the Görtler vortices, at $\unicode[STIX]{x1D6FC}=2{-}12^{\circ }$ (figure 4 b–e); (iii) two-dimensional spanwise shed vortex at $\unicode[STIX]{x1D6FC}=14^{\circ }{-}16^{\circ }$ (figure 4 f). It should be emphasized that the Görtler vortices are first observed in the flow around multi-element airfoil and are the focus of the current work. In this section, $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D6FC}=4^{\circ }$ and $\unicode[STIX]{x1D6FC}=16^{\circ }$ are respectively taken as the representative cases of the three regimes to be further explored. The quantities associated with length are normalized by the stowed chord length $c$ , and those associated with velocity are normalized by the free-stream velocity $U_{\infty }$ .

3.2 Travelling motion of the Görtler vortices

In this section, the characteristics of the Görtler vortices at $\unicode[STIX]{x1D6FC}=4^{\circ }$ are discussed in detail. Travelling motion of the Görtler vortices in the spanwise direction and secondary vortices underneath the Görtler vortices are presented. The spanwise wavelength of the Görtler vortices is also provided in this part.

In figure 11, the classical mushroom-shaped patterns are captured, which are typical topological structures of Görtler vortices (Floryan Reference Floryan1991) or Dean vortices (Ligrani & Niver Reference Ligrani and Niver1988). These patterns are also indicators of the nonlinear evolution of Görtler vortices (Winoto & Shah Reference Winoto and Shah2008). The upwash and downwash motions induced by the counter-rotating vortices make the horizontal hydrogen bubble curtain warp (MT and DB in figure 11). Four time-evolved patterns of Görtler vortices are marked in figure 11. Pattern 1 and pattern 2 show that the Görtler vortices travel right and left along the spanwise direction respectively. It is revealed in pattern 3 that two pairs of counter-rotating vortices could be replaced by one pair of counter-rotating vortices. Meanwhile, one pair of counter-rotating vortices could be replaced by two pairs of counter-rotating vortices, as shown in pattern 4. Pattern 4 has been reported on the Dean vortices over a concave wall (Ligrani & Niver Reference Ligrani and Niver1988).

Figure 11.

End view FV results ( $\text{FV}_{1}$ , $X/c=0.247$ ). The trace lines display different flow patterns. The bottom of each image is located on the surface of the main element. The top of the mushroom-shaped pattern is marked MT. The streaky line between two mushroom-shaped patterns is marked DB. The vortex tube of the counter-rotating vortices is marked VT. The main flow direction points outside of the measurement plane.

Figure 12.

(a) Oseen vortex fitting of an instantaneous end view PIV snapshot at $X/c=0.313\,(\text{PIV}_{4})$ ; (b) space–time diagram of $\unicode[STIX]{x1D706}_{ci}$ built from the sample position at (a); (c) space–time diagram of $\unicode[STIX]{x1D706}_{ci}$ built from the sample position at (a). The $\unicode[STIX]{x1D706}_{ci}$ contour with the sign of the vorticity is shown as the background in (a). The fitting vortex cores are marked by $\times$ in (a). The fitting vortices are denoted by black circles in (a). The origin of the local coordinate system is located at the bottom left corner of the field of view. The local vertical direction and the local spanwise direction are denoted $Y_{L2}$ and $Z_{L2}$ respectively. The line $Y_{L2}/c=0$ denotes the surface of the main element. Similarly, subscript $L1$ is utilized for the local coordinate system of the end view PIV measurement at $X/c=0.213$ ( $\text{PIV}_{3}$ ). The main flow direction points outside of the measurement plane.

In figure 12(a), the instantaneous $\unicode[STIX]{x1D706}_{ci}$ contour is shown with the results of Oseen vortex fitting. The counter-rotating vortices are clearly captured by this fitting method. The spanwise wavelength of the Görtler vortices is not identical. This non-uniform phenomenon is quite normal for naturally developed (no pre-set disturbance) Görtler vortices (Swearingen & Blackwelder Reference Swearingen and Blackwelder1987; Finnis & Brown Reference Finnis and Brown1997) and could be attributed to the non-uniform initial disturbances (Winoto, Mitsudharmadi & Shah Reference Winoto, Mitsudharmadi and Shah2005). Pattern 2 and pattern 1 can also be captured in figure 12(b,c). The unsteady travelling motion of the Görtler vortices is quite different from the classical stationary Görtler vortices (Floryan Reference Floryan1991; Saric Reference Saric1994). The interpretation of this unsteady phenomenon will be provided in § 3.5. It can be seen that the secondary counter-rotating vortices (N and P in figure 12 a) emerge beneath the main Görtler vortices. These secondary vortices have opposite rotation compared with the main Görtler vortices. In order to verify the relationship between the main Görtler vortices and the secondary vortices, the conditional average results are shown in figure 13. According to the position of the left vortex core (with positive vorticity), a resample region associated with one pair of counter-rotating vortices is defined in figure 12(a). The velocity data within all the resample regions are extracted to conduct the conditional average by aligning the positions of the vortex cores (point 2 in figure 13 b). Accordingly, the conditional average of the end view PIV result at $X/c=0.213$ is conducted in the similar way (figure 13 a). Ultimately, the secondary vortices with opposite rotation direction to the main Görtler vortices are clearly captured in the vorticity contours of the conditional average results (N and P in figure 13). The influences of these secondary vortices on $\text{SL}_{3}$ will be discussed in § 3.4. It should be emphasized that the term Görtler vortices is utilized to refer to the primary vortex.

Figure 13.

Vorticity contours of the conditional average of the end view PIV results: (a) plane at $X/c=0.213$ ; (b) plane at $X/c=0.313$ . In (b), the origin of the local coordinate system is located at the bottom left corner of the resampling region in figure 12(a). The local vertical direction and the local spanwise direction are denoted as $Y_{L4}$ and $Z_{L4}$ respectively. Similarly, subscript $L3$ is utilized for the local coordinate system of the plane at $X/c=0.213$ .

Figure 14.

(a) Contour of two-point correlation $R$ at $X/c=0.313$ ; (b) profile of the correlation coefficient at the sample position in (a). The local peaks in (a) and (b) are marked as black $\times$ .

Two-point correlation is utilized to provide the spanwise wavelength information of the Görtler vortices at $X/c=0.313$ . This method has been utilized to conduct the correlation between velocity and $\unicode[STIX]{x1D706}_{ci}$ to capture the inclined hairpin packets in turbulent flow (Christensen & Adrian Reference Christensen and Adrian2001). However, this method is used to conduct correlation between $\unicode[STIX]{x1D706}_{ci}$ with positive sign ( $\unicode[STIX]{x1D6EC}_{ci}$ ) in the current work. The details are as follows:

where $Z_{L2}$ and $Y_{L2}$ follow the definition in figure 12, $y_{ref}$ denotes the reference position of the correlation ( $Y_{L2}/c=0.0728$ ) and $\langle \bullet \rangle$ denotes the inner product. Finally, the two-point correlation coefficient $R$ is shown in figure 14. To display the results clearly, only a portion of $R$ is provided in figure 14(a) with a zoomed-in view. The spanwise periodic property of the Görtler vortices is captured by these correlation coefficients distinctly. Since the spanwise wavelength of the Görtler vortices is non-uniform, the wavelength extracted from the distance between the local peaks of the correlation coefficients ( $\unicode[STIX]{x1D706}_{G}$ in figure 14) could represent the most likely dominant wavelength. The local boundary layer displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ at $X/c=0.313$ is applied to normalize the wavelength (Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Schlatter et al. Reference Schlatter, Brandt, De Lange and Henningson2008):

where $\unicode[STIX]{x1D6FF}$ denotes the position of the separatrix between the slat wake and the separation bubble above the main element, $U_{\unicode[STIX]{x1D6FF}}$ represents the streamwise velocity at $\unicode[STIX]{x1D6FF}$ . Ultimately, the non-dimensional wavelength of the Görtler vortices at $\unicode[STIX]{x1D6FC}=4^{\circ }$ is approximately 1.72. This value is smaller than the streak spacing in the transitional boundary layer (with adverse streamwise pressure gradient) induced by free-stream turbulence (FST), which is approximately located at the range of $6{-}8\unicode[STIX]{x1D6FF}^{\ast }$ (Brinkerhoff & Yaras Reference Brinkerhoff and Yaras2015; Lengani et al. Reference Lengani, Simoni, Ubaldi, Zunino and Bertini2017). This difference in wavelength indicates that the generation mechanism of counter-rotating vortices in the current work is different from that of streaky structures in the FST-induced transitional boundary layer. The variation of the absolute value of the dominant wavelength is insensitive to the variation of angle of attack in the current measurement. The root-mean-square of the non-uniform wavelength is provided by Oseen vortex fitting. The distance between two left vortex cores (with positive vorticity) is defined as the wavelength between two pairs of Görtler vortices. The non-dimensional dominant wavelength calculated from Oseen vortex fitting is also around 1.72, which is nearly the same with the result of two-point correlation. The non-dimensional root-mean-square value of this dominant wavelength is approximately 0.31, which is 18 % of the dominant wavelength.

3.3 Footprints of the travelling Görtler vortices

In this section, the footprints of the travelling Görtler vortices are determined in the side view PIV measurements at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . These footprints reveal that the Görtler vortices reside on the separated shear layer originating from the leading-edge separation of the main element ( $\text{SL}_{3}$ in figure 7). The secondary counter-rotating vortices are found within the separated shear layer. Besides, the different statistical properties of several parameters between $\unicode[STIX]{x1D6FC}=4^{\circ }$ and $\unicode[STIX]{x1D6FC}=16^{\circ }$ , resulting from whether Görtler vortices exist in the flow or not, are also addressed in this part.

In figure 15, the probability density functions (PDF) of the vertical positions of the Görtler vortices cores are shown at two streamwise locations ( $X/c=0.213$ and $X/c=0.313$ ). The PDF calculated from the experimental data can be fitted well with a Gaussian function, except for the deviation at $Y_{L1}/c=0.04{-}0.043$ in figure 15(a). The peaks of Gaussian fittings are chosen as the dominant vertical positions at these two streamwise locations: $Y_{L1}/c=0.0503$ at $X/c=0.213$ and $Y_{L2}/c=0.0748$ at $X/c=0.313$ .

Figure 15.

Probability density functions of the vertical positions of Görtler vortices cores: (a) $X/c=0.213$ ; (b) $X/c=0.313$ . The original PDF is denoted by the black line with dark spots. The Gaussian fitting of the original PDF is denoted by the red line. The pink function provides the parameters of the Gaussian fitting.

Figure 16.

(a) Mean velocity profiles at $X/c=0.213$ ; (b) mean velocity profiles at $X/c=0.313$ ; (c) root-mean-square of velocity fluctuations at $X/c=0.213$ ; (d) root-mean-square of velocity fluctuations at $X/c=0.313$ . The black and red lines represent the profiles associated with streamwise and vertical velocities, respectively. The vertical positions of Görtler vortices cores are marked as blue pentagrams.

Figure 17.

(a) Root-mean-square contour of the vertical fluctuation velocity from the PIV results focused on the gap flow; (b) root-mean-square contour of the vertical fluctuation velocity from the PIV results focused on the flow around the main element; (c) time sequences of velocity at different virtual probe positions in (a). The streamlines of the mean flow are superimposed in (a) and (b). All the virtual probes in (a) are located at the local peaks of $v_{rms}/U_{\infty }$ . The fluctuation velocity within the black box in ( $a$ ) is extracted for the POD in figure 20. Only the vertical components of the velocity are shown at P2, P3 and P4 in (c). Both the vertical and the streamwise components of the velocity are shown at P1 in (c). The instants I1, I2, I3 and I4 are chosen for the FTLE calculation in figure 19.

Figure 18.

Time-shift correlation coefficients between different velocity sequences. Only the correlation coefficients with $-6<\unicode[STIX]{x0394}t<6$ are provided in this figure to ensure clarity. Both the time shifts and the correlation coefficients related to the maxima are denoted by arrows.

Figure 19.

FTLE fields of backward integral: (a) I1 in figure 17(c); (b) I2 in figure 17(c); (c) I3 in figure 17(c); (d) I4 in figure 17(c).

The mean velocity profiles and the root-mean-square of the velocity fluctuations extracted from the side view PIV results at $X/c=0.213$ and $X/c=0.313$ are displayed in figure 16. Mean streamwise velocity profiles ( $U/U_{\infty }$ ) in figure 16(a,b) demonstrate the classical velocity profiles of a separation bubble (Lang et al. Reference Lang, Rist and Wagner2004; Simoni et al. Reference Simoni, Ubaldi, Zunino, Lengani and Bertini2012; Boutilier & Yarusevych Reference Boutilier and Yarusevych2012b ). The dominant vortex core positions calculated from the Gaussian fittings are shown in figure 16(c,d) as blue pentagrams. These two vortex core positions are located at the first peaks of the two root-mean-square profiles of vertical fluctuation velocity ( $v_{rms}/U_{\infty }$ ), respectively. The coincidence between the first peak of $v_{rms}/U_{\infty }$ and the vortex core position indicates that the $v_{rms}/U_{\infty }$ band initiated from the cove in figure 17(a) is the footprint of the travelling Görtler vortices. This footprint can also be captured in figure 17(b). According to the local peaks of $v_{rms}/U_{\infty }$ , four virtual probes are set in the footprint (P1, P2, P3, P4 in figure 17 a) to extract the time sequences of velocity in figure 17(c). The high $v_{rms}/U_{\infty }$ at these virtual probe positions can be attributed to these undulatory velocity sequences. A time-shift correlation is utilized to these velocity sequences:

where $m_{a}$ and $m_{b}$ represent the velocity sequences at two different positions with the same velocity component or at the same position with different velocity components, $\unicode[STIX]{x1D70E}_{m_{a}}$ and $\unicode[STIX]{x1D70E}_{m_{b}}$ represent the standard deviation of $m_{a}$ and $m_{b}$ , $\overline{(\bullet )}$ denotes the average over time and $\unicode[STIX]{x0394}t$ denotes the time shift. The maxima of correlation coefficients ( $R_{T}$ ) and the time shifts are shown in figure 18. The high values of these maxima indicate that these undulatory velocity signals are highly related to the travelling motion of the Görtler vortices. The time shifts reveal that the travelling Görtler vortices pass through the measurement plane with the vortex tube slightly inclined to the plane. To further explore the physical significance of these undulatory sequences, the FTLE fields with backward integral at four characteristic instants (I1–I4 in figure 17 c) are provided in figure 19. At the trough region (I1 and I3 in figure 17 c), the ridge originating from the slat cusp converges to the ridge originating from the leading edge of the main element (figure 19 a,c). The ridge originating from the leading edge of the main element could be treated as the Lagrangian coherent structure (LCS) associated with $\text{SL}_{3}$ . However, at the crest region (I2 and I4 in figure 17 c), the ridge originating from the slat cusp will diverge from the ridge originating from the leading edge of the main element (figure 19 b,d). A few ridges (denoted braid structures in figure 19 b,d) emerge above the ridge originating from the leading edge of the main element. The discrepancies between figures 19(a,c) and 19(b,d) can be attributed to the travelling motion of Görtler vortices. Fixing the travelling Görtler vortices as the new reference frame (NRF), the laser sheet travels in the new reference frame. When the laser sheet cuts the DB region (shown in figure 11), the downwash motion around DB extrudes the ridge originating from the slat cusp to converge to the ridge originating from the leading edge of the main element. When the laser sheet cuts the MT region (shown in figure 11), the upwash motion between the counter-rotating vortices attracts the ridge originating from the slat cusp to diverge from the ridge originating from the leading edge of the main element. The braid structures in figure 19(b,d) are deemed to be the footprint of the upwash motion between the counter-rotating vortices. After integrating the results of figures 16–19, it could be clearly confirmed that the Görtler vortices really reside above $\text{SL}_{3}$ . The upwash and downwash motions associated with the travelling Görtler vortices are responsible for the undulatory velocity signals in figure 17(c). This undulatory property results in the existence of the $v_{rms}/U_{\infty }$ band initiated from the cove in figure 17(a).

POD is applied to the fluctuation velocity field (both streamwise and vertical components) within the black box in figure 17(a). The vertical components of the first two POD modes are shown in figures 20(a) and 20(b), respectively. The velocity signals reconstructed from these two modes coincide with the original velocity signals well in figure 20(c). This accordance means that these two modes (capturing approximately 73 % of the total energy) are the most dominant modes of the flow within this region. The vertical fluctuations within the footprint of the Görtler vortices (red contour in figure 20 a,b) could represent both the upwash and the downwash motions induced by the Görtler vortices, which depend on the sign of the time coefficients of these two POD modes (Mandal & Dey Reference Mandal and Dey2011). The relative position between the Görtler vortices and the secondary counter-rotating vortices is calculated from figure 13. It is found that the second peaks of $v_{rms}/U_{\infty }$ in figure 16(c,d) (marked by the black dashed lines) are the footprints of the travelling secondary counter-rotating vortices. These secondary counter-rotating vortices are located within $\text{SL}_{3}$ . Due to the existence of the secondary counter-rotating vortices, both downwash and upwash motions exist within $\text{SL}_{3}$ (blue contour in figure 20 a,b). As indicated by Hain et al. (Reference Hain, Kähler and Radespiel2009), the high velocity gradient within $\text{SL}_{3}$ could transform the vertical fluctuations induced by the secondary counter-rotating vortices into large streamwise fluctuations. These large streamwise fluctuations could be shown as the streaky structures in figure 21. Although the velocity gradient within the slat wake could also visualize Görtler vortices as streaky structures, the streaky structures in figure 21 cannot be the representation of Görtler vortices according to the difference of the vertical positions between the Görtler vortices and the secondary counter-rotating vortices. Therefore, the travelling secondary vortices are located within $\text{SL}_{3}$ and generate streaky structures within $\text{SL}_{3}$ . This scenario is sketched in figure 22.

Both $v_{rms}/U_{\infty }$ and $u_{rms}/U_{\infty }$ contours at $\unicode[STIX]{x1D6FC}=16^{\circ }$ are displayed in figure 23 to show the statistical property under the absence of Görtler vortices. The $v_{rms}/U_{\infty }$ band initiated from the cove in figure 17(a), resulting from the travelling motion of Görtler vortices, vanishes in figure 23(a,b). The only high $v_{rms}/U_{\infty }$ concentrated region results from the transition of $\text{SL}_{3}$ . The $u_{rms}/U_{\infty }$ contours also demonstrate the classical transition scenario of a separated shear layer (Marxen & Henningson Reference Marxen and Henningson2011; Simoni et al. Reference Simoni, Ubaldi, Zunino, Lengani and Bertini2012).

Figure 20.

(a) Contour of the vertical components of the first POD mode; (b) contour of the vertical components of the second POD mode; (c) time sequences of velocity at virtual probes P3 and P4 in figure 17(a). Both the streamwise and the vertical components are shown in (a) and (b) as velocity vectors. Every fourth vector is shown for clarity. Both the original vertical velocity (red line for P3 and blue line for P4) and the vertical velocity reconstructed from the first two POD modes (black line for P3 and pink line for P4) are provided in (c).

Figure 21.

Plane view flow visualization with volume illumination ( $\text{FV}_{M}$ ). This result captures the flow structures within $\text{SL}_{3}$ . The high-concentration regions of hydrogen bubbles (blue colour in the image) represent the low-speed streaky structures. The VS of figure 7 is marked in this figure.

Figure 22.

Sketch of the relative positions of the Görtler vortices, the secondary counter-rotating vortex and the low-speed streak.

Figure 23.

Root-mean-square contours of the fluctuation velocity from the PIV results at $\unicode[STIX]{x1D6FC}=16^{\circ }$ : (a) $v_{rms}/U_{\infty }$ from the PIV results focused on the gap flow; (b) $v_{rms}/U_{\infty }$ from the PIV results focused on the flow around the main element; (c) $u_{rms}/U_{\infty }$ from the PIV results focused on the gap flow; (d) $u_{rms}/U_{\infty }$ from the PIV results focused on the flow around the main element.

3.4 Effect of Görtler vortices on the leading-edge separated shear layer

As indicated in § 3.3, the travelling Görtler vortices could induce the travelling secondary vortices within $\text{SL}_{3}$ at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . The vertical disturbances resulting from the upwash–downwash motions of these secondary vortices could generate large streamwise disturbances within $\text{SL}_{3}$ . These streamwise disturbances engender the scenario that the low-speed and high-speed streaks co-exist within $\text{SL}_{3}$ in the spanwise direction. This scenario is quite similar to the bypass transition of the separated shear layer under elevated free-stream turbulence (Mcauliffe & Yaras Reference Mcauliffe and Yaras2009). However, in the current investigation, the vortex shedding originating from $\text{SL}_{3}$ indicates that the modal growth associated with the K–H instability within $\text{SL}_{3}$ is not fully bypassed. Accordingly, the not-bypassed K–H instability is consistent with the transition of the separated shear layer under a steady three-dimensional disturbance (Lang et al. Reference Lang, Rist and Wagner2004). In order to determine the effects of Görtler vortices on the transition process of $\text{SL}_{3}$ , FMD is utilized to track the growth of the disturbances within different frequency bands in this section.

In figure 24(a), the $u_{rms}/U_{\infty }$ band with large amplitude is located within $\text{SL}_{3}$ . This is the classical scenario associated with the transition of the separated shear layer (Marxen & Henningson Reference Marxen and Henningson2011; Simoni et al. Reference Simoni, Ubaldi, Zunino, Lengani and Bertini2012). Here, POD is utilized to the fluctuation velocity field within the black box in figure 24(a). As indicated by Lengani et al. (Reference Lengani, Simoni, Ubaldi and Zunino2014), only the vertical components are contained in POD to extract the vortex shedding phenomenon, which could eliminate the contamination from the streamwise components. The first two POD modes clearly capture the vortex shedding associated with $\text{SL}_{3}$ (Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014).

In order to extract the frequency band associated with the vortex shedding at $\unicode[STIX]{x1D6FC}=4^{\circ }$ , all vertical velocity sequences in the black box (figure 24 a) are contained to conduct FMD (6171 velocity sequences). The global power spectral density (PSD) within this region is shown in figure 25(a). To reconstruct the original velocity fluctuations, no window function is used for the PSD. The frequency bands associated with the low frequency fluctuations ( $fc/U_{\infty }=0.0503{-}1.3068$ ) and the K–H instability ( $fc/U_{\infty }=2.7150{-}6.1325$ ) are marked as LF band and KH band in figure 25(a), respectively. As indicated by Simoni et al. (Reference Simoni, Ubaldi, Zunino, Lengani and Bertini2012), the fluctuation energy within different frequency bands can be extracted from the power spectral density. In the current work, these two frequency bands (LF and KH) are applied to the local power spectral densities ( $\text{PSD}_{local}$ ) to track the evolution of the fluctuation energy associated with different flow structures. The $\text{PSD}_{local}$ of the streamwise velocity is calculated at the local peak of $u_{rms}/U_{\infty }$ (black line with dots in figure 24 a). The definitions of the total fluctuation energy, the KH fluctuation energy and the LF fluctuation energy are as follows:

Finally, the spatial growth of the fluctuation energy associated with different frequency bands at $\unicode[STIX]{x1D6FC}=4^{\circ }$ is shown in figure 25(b). As with $\unicode[STIX]{x1D6FC}=4^{\circ }$ , the spatial growth of the fluctuation energy at $\unicode[STIX]{x1D6FC}=16^{\circ }$ is also displayed in figure 25(b). The total fluctuation energy at these two angles of attack increases along the streamwise direction. The total fluctuation energy at $\unicode[STIX]{x1D6FC}=16^{\circ }$ fits the exponential amplification better than that at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . The initial value of the total fluctuation energy at $\unicode[STIX]{x1D6FC}=4^{\circ }$ is one order of magnitude higher than that at $\unicode[STIX]{x1D6FC}=16^{\circ }$ . This deviation of the initial value between these two angles of attack is mainly attributed to the deviation of LF fluctuation energy. Due to the existence of Görtler vortices at $\unicode[STIX]{x1D6FC}=4^{\circ }$ , the travelling secondary vortices are generated within $\text{SL}_{3}$ . The travelling low-speed and high-speed streaks associated with the secondary vortices could increase the low frequency disturbances within $\text{SL}_{3}$ in the measurement plane. As indicated by Mcauliffe & Yaras (Reference Mcauliffe and Yaras2009), these streaks could represent the potential transient growth within $\text{SL}_{3}$ and bypass the modal growth associated with the K–H instability. However, the similarity of the spatial growth of the KH fluctuation energy between $\unicode[STIX]{x1D6FC}=4^{\circ }$ and $\unicode[STIX]{x1D6FC}=16^{\circ }$ reveals that the K–H instability is not strongly affected by the existence of the Görtler vortices in the current investigation. Therefore, it can be concluded that the separated shear layer can be receptive to the disturbances outside the separated shear layer (generated by the Görtler vortices) to generate streaky structures. The streaky structures cannot fully bypass the modal growth associated with the K–H instability. The potential transient growth associated with the streaky structures and the modal growth associated with the K–H instability may co-exist within $\text{SL}_{3}$ .

POD modes of plane view PIV measurements provide additional information on the effect of Görtler vortices on the shed vortices. Both the streamwise and the spanwise components of the fluctuation velocity are contained to obtain the POD modes. The longitudinal structures, resulting from the existence of Görtler vortices, dominate the first eight POD modes. The streamwise components of the first two POD modes are provided in figure 26(a,b) to show these longitudinal structures. As indicated by Lengani & Simoni (Reference Lengani and Simoni2015), the coherent patterns in the spanwise direction (marked VS in figure 26 c,d) represent the footprints of shed vortices associated with the K–H instability. The spanwise periodic property of these coherent patterns indicates that the shed vortices are modified by the Görtler vortices to become undulatory in the spanwise direction.

Figure 24.

(a) Root-mean-square contour of the streamwise fluctuation velocity from the PIV results at $\unicode[STIX]{x1D6FC}=4^{\circ }$ ; (b) first POD mode of the vertical fluctuation velocity within the black box in (a); (c) second POD mode of the vertical fluctuation velocity within the black box in (a). The local peaks of $u_{rms}/U_{\infty }$ are provided in (a) as black line with dots.

Figure 25.

(a) Power spectrum density of the vertical velocity within the black box in figure 24(a); (b) spatial growth of the fluctuation energy associated with different frequency bands at $\unicode[STIX]{x1D6FC}=4^{\circ }$ and $\unicode[STIX]{x1D6FC}=16^{\circ }$ . The exponential fittings of KH fluctuation energy are marked as black dashed lines in (b).

Figure 26.

Streamwise components of the POD modes extracted from the plane view PIV result at $\unicode[STIX]{x1D6FC}=4^{\circ }$ : (a) first POD mode; (b) second POD mode; (c) ninth POD mode; (d) tenth POD mode. The footprints of shed vortices are marked VS in (c) and (d). The origin of the local coordinate system is located at the bottom left corner of the field of view. The local streamwise direction and the local spanwise direction are denoted $X_{L5}$ and $Z_{L5}$ , respectively.

3.5 Origin of the Görtler vortices

In this section, the Rayleigh discriminant combined with the FV results is used to explore the origin of the Görtler vortices at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . The position of the Görtler vortices, determined in § 3.3, also contributes to this exploration. A possible generation mechanism of Görtler vortices is proposed, which could provide evidence for the innovative generation mechanism of Inger (Reference Inger1987). Some further discussion is also given here to explain the travelling motion of Görtler vortices.

The Rayleigh discriminant applied in the current work is a modulated version (Beaudoin et al. Reference Beaudoin, Cadot, Aider and Wesfreid2004) according to the theoretical derivation (Sipp & Jacquin Reference Sipp and Jacquin2000). The formula is as follows:

where $U$ and $V$ represent the streamwise and the vertical components of the time-averaged velocity field, respectively. Both the spanwise vorticity $\unicode[STIX]{x1D714}_{Z}$ and the radius of curvature $K_{r}$ are calculated from the time-averaged velocity field. The formula is detailed in Beaudoin et al. (Reference Beaudoin, Cadot, Aider and Wesfreid2004). The flow in the region with positive Rayleigh discriminant is stable to the centrifugal force. Meanwhile, the region with negative Rayleigh discriminant is treated as the potential centrifugal instability region. When the stabilizing effect of viscosity cannot resist the inviscid centrifugal instability, the flow can ultimately undergo centrifugal instability.

Three potential centrifugal instability regions are labelled in figure 27(a) according to the sign of the Rayleigh discriminant ( $\text{R}_{1}$ , $\text{R}_{2}$ and $\text{R}_{3}$ in figure 27 a). For $\text{R}_{1}$ , the curved surface of the slat cove may lead to the occurrence of a centrifugal instability. However, if this is the case, the disturbance resulting from the centrifugal instability would modulate $\text{SL}_{2}$ into a wavy shape in the spanwise direction. This deduction conflicts with the results of bottom view flow visualization. The hydrogen bubble curtain keeps the two-dimensional shape well at $Y/c=0.071$ (figure 28 a). Therefore, the flow within this potential region is finally stable to the centrifugal force. The potential centrifugal instability region $\text{R}_{2}$ may result from the curved streamline around the front stagnation point of the main element. As indicated by Inger (Reference Inger1987) and Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009), the curved streamline above the separation point of the laminar separation bubble may play the same role as the curved wall to generate Görtler vortices. However, for the configuration with a concave wall followed with a convex wall, the convex wall could suppress the disturbance amplification resulting from the centrifugal instability around the concave wall (Benmalek & Saric Reference Benmalek and Saric1994). Therefore, with a tiny concave streamline followed with a consistent convex streamline, $\text{R}_{2}$ is still not deemed to be the origin region of the Görtler vortices at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . For $\text{R}_{3}$ , the potential centrifugal instability region is located around the upper right boundary of $\text{SB}_{S}$ . The wavy pattern of the hydrogen bubble curtain at $Y/c=0.126$ ( $\text{W}_{1}$ , $\text{W}_{2}$ and $\text{W}_{3}$ in figure 28 b) strongly indicates that $\text{R}_{3}$ is responsible for the generation of the centrifugal instability. Figure 19(b) is revisited in figure 27(b) superimposed by the instantaneous streamlines. Since the ridge within the FTLE field can be regarded as the virtual boundary (Shadden et al. Reference Shadden, Lekien and Marsden2005), the ridge enclosed $\text{SB}_{S}$ can be treated as a virtual curved boundary. This virtual curved boundary could be approximated by a special streamline, which splits the slat cove bubble and the gap flow. This special streamline based on the averaged velocity is marked in figure 27(a) as a black line. Around the slat cove, the free-stream flow will suffer from a virtual convex boundary followed with a virtual concave boundary. The centrifugal instability could be generated in the concave region $\text{R}_{3}$ . Since the centrifugal instability is generated from the virtual curved wall, the counter-rotating vortices are named ‘Görtler vortices’. It should be emphasized that the Rayleigh discriminant is based on the two-dimensional basic flow. Therefore, only the Rayleigh discriminant concentrated at the flow regions with small spanwise disturbances can make sense. Fortunately, this limitation does not affect the determination of the origin of Görtler vortices in the current investigation.

Figure 27.

(a) Contour of Rayleigh discriminant; (b) revisiting of figure 19(b). A special black streamline calculated from the averaged flow is superimposed on (a). The streamlines calculated from the instantaneous flow are superimposed on (b).

Figure 28.

Bottom view FV results: (a) $Y/c=0.071$ ; (b) $Y/c=0.126$ . The main flow direction points inside of the measurement plane.

Figure 29.

Side view FV results at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . The shadows of VT appear in (a) and (b) by image enhancement. The nomenclatures MT, VT and DB are same as those in figure 11. The nomenclatures $\text{SB}_{S}$ and RP are same as those in figure 7. The superposition among (a), (b) and (c) is shown in (d).

Side view flow visualization can provide additional evidence for the origin of Görtler vortices. Benefiting from the application of the horizontal hydrogen bubble curtain, the three-dimensional flow structures outside the laser sheet can still be illuminated by the scattered light of the hydrogen bubble inside the laser sheet. Following the definition of NRF, when the laser sheet cuts the MT or the DB region (shown in figure 11), a shadow of VT will be projected into the laser sheet plane (VT in figure 29 a,b). In figure 29(c), the laser sheet just passes the VT. The relative positions of MT, VT and DB in side view FV are clearly manifested in figure 29(d). These relative positions are consistent with the end view FV results. It deserves to be paid more attention to the fact that the vortex tube of the Görtler vortices emanates from the upper right boundary of $\text{SB}_{S}$ . This phenomenon is consistent with the result in figure 17(a) that the footprint of the travelling Görtler vortices initiates from the upper right boundary of $\text{SB}_{S}$ . In figure 30(a), the diverging averaged streaky line still initiates from $\text{R}_{3}$ (after arrow 1). The dissipation of the streaky line could represent the amplification of the disturbances associated with the centrifugal instability. In this view, the initial amplification of the disturbances emerges in $\text{R}_{3}$ at $\unicode[STIX]{x1D6FC}=4^{\circ }$ . After passing through the gap region, the disturbances are strongly amplified. On the contrary, for the case without Görtler vortices ( $\unicode[STIX]{x1D6FC}=16^{\circ }$ ), the diverging averaged streaky line emerges after arrow 2 in figure 30(b). This diverging pattern can be ascribed to the occurrence of RP.

Figure 30.

Averaged side view FV results: (a) $\unicode[STIX]{x1D6FC}=4^{\circ }$ ; (b) $\unicode[STIX]{x1D6FC}=16^{\circ }$ . A total of 4500 FV snapshots are utilized for the averaged process at these two angles of attack.

It should be emphasized that the wavy pattern in figure 28(b) also travels in the spanwise direction. The downstream travelling Görtler vortices stick to the travelling wavy pattern. Therefore, the travelling property of the Görtler vortices is just an inheritance of the travelling property of the wavy pattern in $\text{R}_{3}$ . As indicated in the open cavity flow (Faure et al. Reference Faure, Adrianos, Lusseyran and Pastur2007, Reference Faure, Pastur, Lusseyran, Fraigneau and Bisch2009; Faure Reference Faure2014; Douay et al. Reference Douay, Pastur and Lusseyran2016a ; Douay, Lusseyran & Pastur Reference Douay, Lusseyran and Pastur2016b ), the separation bubble within the slat cove could contain a spanwise travelling flow. This travelling flow could drive the wavy pattern to travel in the spanwise direction.