2.1. Experimental set-up

The experiments are conducted at the low-speed recirculation water channel in Beijing University of Aeronautics and Astronautics, China. The test section is $600\ {\rm mm}\times 600 \ {\rm mm}\times 3000\ {\rm mm}$ (height $\times$ width $\times$ length). The separated flow over a low-$A\!R$ trapezoidal plate is investigated, as shown in figure 1. The rigid flat plate is constructed from a $1$ mm thick aluminium sheet; all edges are left square. The planform shape is depicted in figure 1(a). In the natural world, similar trapezoidal planform shapes are utilized by numerous animals when gliding, including the flying squirrel and the flying lemur bat. The study of the trapezoidal plate has value in advancing our understanding of the remarkable gliding capacities of these animals. Furthermore, the vortex dynamics for the trapezoidal plate are influenced significantly by the tip effects, as demonstrated via flow visualization in Appendix A. Recalling the motivation of this study, which is to explore in depth the tip effects for low-$A\!R$ plates, the flow over the trapezoidal plate is a valuable subject to investigate. The chord length of the plate is $c=58$ mm, giving thickness-to-chord ratio $1.7\,\%$. The leading- and trailing-edge lengths are $L_l=80$ mm and $L_t=58$ mm (equal to the chord length), respectively. The taper angle of the side edges is $\beta =12^\circ$. In this paper, $A\!R$ is defined as the ratio of maximum span to chord, $A\!R\equiv L_{max}/c = L_l/c=1.38$. The freestream velocity is set at $U_\infty =96$ mm s$^{-1}$, and the chord-based Reynolds number is approximately $Re_c=5800$. This value falls within the operating range for flying insects ($Re_c=10^3\unicode{x2013}10^4$).

Figure 1. (a) Planform shape and dimensions of the plate. The dashed circle depicts the position where the plate is glued on the round rod. (b) Experimental set-up for the 3-C PIV configuration. (c) Plate support system.

As shown in figure 1(b), the plate is hung upside down into the water channel test section with a square strut connected to a translator. The orientation of the translator can be modified to shift the plate in either the freestream or the spanwise direction with resolution 0.1 mm. The plate is covered with matt black paint to attenuate the undesired light reflections. The detailed plate support system is sketched in figure 1(c). The plate is glued on a round rod (4 mm diameter, 100 mm length) with acrylate adhesive. The round rod is connected to the square strut with a square rod. Meanwhile, a mechanical inclinometer is designed to manually adjust the angle between the strut and the rods with resolution $0.5^\circ$. The support system is firmly attached with two bolts B1. By adjusting the orientation of the rods, four angles of attack are tested in this study, namely $\alpha =4^\circ$, $6^\circ$, $8^\circ$ and 10$^\circ$. After each angle adjustment, a bolt B2 is used to lock the rods.

Two coordinate systems are used for data presentation, as shown in figure 2. One is the natural coordinate system $O$-$xyz$, where the origin $O$ is defined so that the point at the trailing edge at the midspan is $(c,0,0)$ (as illustrated with the red point in figure 2b), and the three axes $(x,y,z)$ are along the streamwise, vertical and spanwise directions, respectively. The other is the plate-based coordinate system $\hat {O}$-$\hat {x}\hat {y}\hat {z}$, where the origin $\hat {O}$ is set at the leading edge at the midspan, and the three axes $(\hat {x},\hat {y},\hat {z})$ are along the chordwise, wall-normal and spanwise directions, respectively. Since the $z$-axis and the $\hat {z}$-axis have the same direction and scale, only the $z$-axis is retained to describe the spanwise position, for simplicity.

Figure 2. Sketch of the coordinate systems and the PIV measurement planes: (a) plan view; (b) side view (midspan plane).

2.1.1. Particle image velocimetry

Two PIV configurations are set up to conduct the quantitative flow-field measurements, i.e. the three-component (3-C) and two-component (2-C) PIV configurations. The detailed parameters for both PIV configurations are listed in table 1.

Table 1. PIV measurement parameters.

For the 3-C PIV configuration, the stereoscopic PIV technique is adopted to obtain the statistical characteristics of the flow fields. As sketched in figure 1(b), the water channel is seeded with hollow glass beads of 5–20 $\mathrm {\mu }$m diameter and illuminated with a dual-head Nd:YAG laser (Beamtech Vlite-500). The thickness of the light sheet is 2 mm. Two charged-coupled device (CCD) cameras (IMPERX ICL-B2520M, $2456\times 2058$ pixels) are arranged on each side of the water channel. These cameras are equipped with Nikkor 85 mm tilt-shift lenses and satisfy the Scheimpflug condition. The viewing angles of these cameras are approximately ${\pm }65^\circ$ (Nobes, Wieneke & Tatam Reference Nobes, Wieneke and Tatam2004). The cameras and the laser are synchronized using a MicroVec Micropulse-725 synchronizer. Particle images are acquired in a double-frame mode at frequency 3 Hz. In this configuration, 3-C velocity fields in $21$ closely spaced $y$–$z$ planes are measured by shifting the plate in the streamwise direction through the stationary light sheet. The measurement planes range from $x/c=-0.03$ to $x/c=1.00$ with increment $\Delta x/c\approx 0.052\ (3\ {\rm mm})$, as illustrated with the red lines in figure 2(a). For each plane, a total of $2000$ image pairs are obtained.

For the 2-C PIV configuration, the time-resolved PIV measurements are performed with a focus on the dynamics of the LEVs. The flow is illuminated by the $1$ mm thick light sheet, which is generated by a semiconductor continuous laser (8 W, 532 nm). Particle images are recorded using a high-speed CMOS camera (Photron Fastcam SA2 86K-M3) at sampling rate 288 Hz. The images are cropped to $832 \times 2048$ pixels, and the field of view is approximately $0.52c\times 1.31c$. Considering the symmetry of the flow fields, 2-C velocity fields in four $x$–$y$ planes, $z/c=0,0.05,0.12,0.24$ (blue lines in figure 2a), are measured by shifting the plate along the $z$-axis. For each plane, a repetition of $13\,432$ images is obtained, continuously spanning approximately $240$ LEV shedding cycles.

All particle images are pre-processed using the POD-based background removal method (Mendez et al. Reference Mendez, Raiola, Masullo, Discetti, Ianiro, Theunissen and Buchlin2017), and the 2-D2-C velocity vectors are calculated using the multi-pass iterative Lucas–Kanade (MILK) algorithm (Champagnat et al. Reference Champagnat, Plyer, Le Besnerais, Leclaire, Davoust and Le Sant2011; Pan et al. Reference Pan, Xue, Xu, Wang and Wei2015) with four pyramid levels and three Gauss–Newton iterations per level. The interrogation window sizes are set to $48 \times 48$ and $32 \times 32$ pixels for the 3-C and 2-C PIV configurations, respectively. Regarding the stereoscopic PIV scenario, the stereoscopic reconstruction is then performed to obtain the 2-D3-C velocity vectors using the Soloff method (Soloff, Adrian & Liu Reference Soloff, Adrian and Liu1997). Additionally, the misalignment of the calibration target with the light sheet is corrected using the self-calibration method (Wieneke Reference Wieneke2005). For both configurations, the velocity data have a similar vector pitch of approximately 0.3 mm. The raw velocity data are validated using the robust principal component analysis method (Scherl et al. Reference Scherl, Strom, Shang, Williams, Polagye and Brunton2020). The uncertainty relative to the freestream velocity is approximately $1.5\,\%$ and $1.0\,\%$ for the 3-C and 2-C velocity fields, respectively (van Doorne & Westerweel Reference van Doorne and Westerweel2007). The spanwise vorticity $\omega _z$ is computed from the 2-C PIV data using the central-difference scheme. Therefore, the vorticity calculation is not affected by the coarse streamwise resolution for the 3-C PIV configuration. Following the strategies of Qu et al. (Reference Qu, Wang, Feng and He2019), the normalized uncertainty is estimated to be $\varepsilon (\omega _z)c/U_\infty =1.35$, which is acceptable to resolve the focused flow structures.

2.2. Post-processing tools

2.2.1. Proper orthogonal decomposition

Proper orthogonal decomposition (POD) is a powerful tool for detecting the large-scale coherent structures embedded in the flow. Using POD, the fluctuating part of given velocity fields can be projected on a set of orthogonal bases $\{\varPhi _k\}$, which is optimal in an energetic sense (van Oudheusden et al. Reference van Oudheusden, Scarano, van Hinsberg and Watt2005):

(2.1)\begin{equation} \boldsymbol{u}( \boldsymbol{x},t )=\bar{\boldsymbol{u}}( \boldsymbol{x} )+\boldsymbol{u}^{\prime}( \boldsymbol{x},t ) =\bar{\boldsymbol{u}}( \boldsymbol{x} )+\sum_{k}{{{a}_{k}}(t)\,{{\varPhi}_{k}}( \boldsymbol{x} )}, \end{equation}

where $\bar {\boldsymbol {u}}( \boldsymbol {x} )$ and $\boldsymbol {u}^{\prime }( \boldsymbol {x},t )$ represent the mean and the fluctuating components of the velocity fields, respectively. Also, ${{a}_{k}}(t)$ denotes the mode coefficient at a snapshot instant $t$ corresponding to the $k$th POD mode $\varPhi _k$. The POD modes are obtained by solving the eigenvalue problem of the time-averaged two-point spatial correlation matrix $\boldsymbol {C}$:

(2.2)\begin{equation} \boldsymbol{C}{{\varPhi }_{k}}={{\lambda }_{k}}{{\varPhi }_{k}}, \quad \text{with}\ {{C}_{m,n}}=\overline{\boldsymbol{u}^{\prime}( {{x}_{m}},t )\,\boldsymbol{u}^{\prime}( {{x}_{n}},t )}. \end{equation}

The eigenvalue ${\lambda }_{k}$ measures the energy contained in the $k$th POD mode. When the POD modes are ranked in descending order of the eigenvalues, the first few orthogonal modes usually contain the most energy. Thus these low-rank POD modes provide an efficient way to build a reduced-order model (ROM) for the flow fields.

Another great virtue of POD is its capability to identify the phase information for quasi-periodic flows. When flows are dominated by periodical travelling coherent structures, the corresponding phase angle $\theta ( t )$ can be obtained with the aid of the first mode pair (Legrand, Nogueira & Lecuona Reference Legrand, Nogueira and Lecuona2011a; Toppings & Yarusevych Reference Toppings and Yarusevych2021):

(2.3)\begin{equation} \theta ( t )=\tan^{{-}1}\left( \frac{{{a}_{1}}( t )}{{{a}_{2}}( t )}\, \frac{\sqrt{2{{\lambda }_{2}}}}{\sqrt{2{{\lambda }_{1}}}} \right). \end{equation}

According to Lengani et al. (Reference Lengani, Simoni, Ubaldi and Zunino2014), the wall-normal velocity component $\hat {v}$ is closely related to the vortex shedding phenomenon. To identify the phase information of the shed LEVs, POD is performed on this velocity component for the 2-C PIV data.

Once the phase information is determined, the instantaneous velocity vectors $\boldsymbol {u}=(u,v)$ can be sorted along the phase angle $\theta$ and averaged using the curve-fitting method (Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014; Toppings & Yarusevych Reference Toppings and Yarusevych2021). Specifically, this involves applying an $8$-term Fourier model fit for the phase-sorted velocity vector at every spatial point, as illustrated in figure 3.

Figure 3. Phase averaging using Fourier series model fitting. The velocity vectors $\boldsymbol {u}=(u,v)$ at the spatial point $(x/c,y/c,z/c)=(0.36,0.15,0)$ are extracted from the 2-C PIV data at $\alpha =6^\circ$. Contours show the distributions of the velocity components.

2.2.2. 3-D3-C flow-field reconstruction and linear stochastic estimation

To characterize the dynamical evolution of the flow structures, a 3-D3-C flow-field reconstruction method is proposed. This method links the velocity data from the 3-C and 2-C PIV measurements to restore the phase-averaged 3-D coherent structures.

The spatial consistency between the PIV measurement planes is the key to the proposed reconstruction method. The spatial distribution of the planes is sketched in figure 4(a). The reconstruction domain is indicated by the dashed box. Due to the light sheet reflections, flow fields under height $\hat {y}/c=0.016\ (0.9\ {\rm mm})$ are excluded from the reconstruction. The 2-C velocity fields in the plane $z/c=0$ (blue) are denoted as ${\boldsymbol {u}_{2\text {-}C}}( \boldsymbol {x}_{( 0 )},t )$; the 3-C velocity fields in the given plane $x/c=x^\ast$ (one of the red planes) are denoted as ${{\boldsymbol {u}}_{3\text {-}C}}( {{\boldsymbol {x}}_{( \ast )}},\tau )$. The planes $\boldsymbol {x}_{( 0 )}$ and $\boldsymbol {x}_{( \ast )}$ intersect in the line (the matched one of the yellow lines) denoted as $\boldsymbol {x}_{line}$. Notably, these intersection lines provide an opportunity to restore the global flow information from the asynchronously measured local velocity fields. Specifically, the linear stochastic estimation (LSE) method (Adrian Reference Adrian1994; Podvin et al. Reference Podvin, Nguimatsia, Foucaut, Cuvier and Fraigneau2018) is adopted to link the asynchronous data in every two orthogonal planes, inspired by the work of Dellacasagrande et al. (Reference Dellacasagrande, Verdoya, Barsi, Lengani and Simoni2021).

Figure 4. (a) Schematic of the PIV measurement planes used for reconstruction. (b) Flowchart of the reconstruction method.

The proposed reconstruction method contains three main steps, as illustrated in figure 4(b). The first step is to establish a ROM for the 3-C velocity fields ${{\boldsymbol {u}}_{3\text {-}C}}( {{\boldsymbol {x}}_{( \ast )}},\tau )$ using POD:

(2.4)\begin{equation} {{\boldsymbol{u}}_{3\text{-}C}}( {{\boldsymbol{x}}_{({\ast} )}},\tau ) ={{\bar{\boldsymbol{u}}}_{{3\text{-}C}({\ast} )}}+{\boldsymbol{u}^{\prime}_{3\text{-}C}}( {{\boldsymbol{x}}_{({\ast} )}},\tau ) ={{\bar{\boldsymbol{u}}}_{{3\text{-}C}({\ast} )}}+\sum_{k=1}^{R}{{{a}_{k}}( \tau )\,{{\varPhi }_{k}}}. \end{equation}

Here, the truncation rank (TR) $R$ is determined using the cumulative energy criteria with cutoff threshold $90\,\%$.

Next, the LSE method is utilized to conveniently calibrate the linear mapping factor $\boldsymbol {M}_k$ between the $k$th POD mode coefficients ${a}_{k}( \tau )$ and the fluctuating velocity fields ${\boldsymbol {u}^{\prime }_{2\text {-}C}}( \boldsymbol {x}_{line},\tau )$:

(2.5)\begin{equation} \overline{{\boldsymbol{u}^{\prime}_{2\text{-}C}}( m,\tau )\,{\boldsymbol{u}^{\prime}_{2\text{-}C}}( n,\tau )}\,{{M}_{km}} =\overline{{{a}_{k}}( \tau )\,{\boldsymbol{u}^{\prime}_{2\text{-}C}}( n,\tau )}, \quad \text{with}\ m,n\in {{\boldsymbol{x}}_{{line}}}. \end{equation}

Here, the ${\boldsymbol {u}^{\prime }_{2\text {-}C}}( \boldsymbol {x}_{line},\tau )$ are composed of the $(u,v)$ components and collected from the 3-C velocity fields ${\boldsymbol {u}_{3\text {-}C}}( \boldsymbol {x}_{(\ast )},\tau )$. Once the factor $\boldsymbol {M}_k$ is calibrated, the pseudo-mode coefficient at a given reference instant $t_{ref}$ can be estimated as

(2.6)\begin{align} {{a}_{k,est.}}( {{t}_{{ref}}} ) =\text{linear estimate of }\mathbb{E}\{ {{a}_{k}}( {{t}_{{ref}}} )\mid {\boldsymbol{u}^{\prime}_{2\text{-}C}}( {{\boldsymbol{x}}_{{line}}},{{t}_{{ref}}} )\} ={{\boldsymbol{M}}_{k}}\,{\boldsymbol{u}^{\prime}_{2\text{-}C}}( {{\boldsymbol{x}}_{{line}}},{{t}_{{ref}}} ), \end{align}

where $\mathbb {E}\{ {\cdot } \}$ is the expectation operator, and ${\boldsymbol {u}^{\prime }_{2\text {-}C}}( {{\boldsymbol {x}}_{{line}}},{{t}_{{ref}}} )$ are the reference signals for estimation. Intuitively, the reference signal can be extracted from the original 2-C velocity fields ${\boldsymbol {u}_{2\text {-}C}}( \boldsymbol {x}_{( 0 )},t )$.

Finally, to take advantage of the linearity of the mapping matrix $\boldsymbol {M}_k$, the pseudo-mode coefficients are estimated using the phase-averaged velocity fields ${\boldsymbol {u}_{2\text {-}C}}( \boldsymbol {x}_{( 0 )},\theta )$ rather than the original ones ${\boldsymbol {u}_{2\text {-}C}}( \boldsymbol {x}_{( 0 )},t )$. This modification filters out the secondary cycle-to-cycle variations to reveal the primary evolution of LEVs, as demonstrated in § 5.1. Note that the pseudo-mode coefficients for different planes $\boldsymbol {x}_{( \ast )}$ are synchronized with the reference phase angle $\theta$. Thus the phased-averaging 3-D3-C flow fields are reconstructed as follows:

(2.7)\begin{equation} {\boldsymbol{u}}_{{est.}}( {{\boldsymbol{x}}_{({\ast} )}},\theta ) ={{\bar{\boldsymbol{u}}}_{{3\text{-}C}({\ast} )}}+\sum_{k=1}^{R}{{{\boldsymbol{M}}_{k({\ast} )}}\,{\boldsymbol{u}^{\prime}_{2\text{-}C}}( {{\boldsymbol{x}}_{{line}}},\theta )\,{{\varPhi }_{k}}} \quad \text{for}\ 0.17 \leq {{x}^{{\ast}}} \leq 0.95. \end{equation}

5.1. Primary evolution path of leading-edge vortices

To track quantitatively the evolution of the LEVs, the positions of the vortex heads are determined first. In practice, the central positions $(x_c,y_c)$ of the LEV heads are extracted from the phase-averaged 2-C velocity fields at $z/c=0$, ${\boldsymbol {u}_{2\text {-}C}}( \boldsymbol {x}_{( 0 )},\theta )$, which are exactly the reference signals for reconstruction. Specifically, the central positions are calculated by $x_c={\iint _{C}{x{\omega _z}\,{\rm d}s}}/{\iint _{C}{c{{\omega }_{z}}\,{\rm d}s}}$, $y_c={\iint _{C}{y{{\omega }_{z}}\,{\rm d}s}}/{\iint _{C}{c{{\omega }_{z}}\,{\rm d}s}}$, where $C$ is the integral region. Here, the integral region is defined as where $Q_{2\text {-}C}\,c^2/U_\infty ^2\geq 30$ (with $Q_{2\text {-}C}$ computed using the in-plane velocity components). The spanwise vorticity field at $\theta =0^\circ$ is shown, for example, in figure 9. The LEV heads and their central positions are illustrated with the yellow lines and crosses. The benefit of this vorticity integral method is that the LEVs can be located despite the vortex transformations along the streamwise direction.

Figure 9. Spanwise vorticity field in the plane $z/c=0$ at $\theta =0^\circ$. The LEV heads are presented by the contour lines of $Q_{2\text {-}C}\,c^2/U_\infty ^2=30$ in yellow, with the central positions marked by the yellow crosses.

Figure 10 illustrates the LEV evolution as a function of the extended phase angle $\theta _c$ (or the streamwise position of the LEV head $x_c$). For $\theta _c\leq 240^\circ$ (or $x_c\leq 0.4$), the spanwise extent of the spanwise vortex increases with $\theta _c$. Meanwhile, the newly formed lateral arms are located further downstream relative to the vortex head. As a result, the C-shape vortex forms. For $240^\circ <\theta _c\leq 720^\circ$, the LEV head convects downstream faster than its arms, i.e. the C-shape vortex transforms into the M-shape vortex. With a further increase in $\theta _c$, the M-shape vortex quickly breaks down into small substructures. It is noteworthy that the M-shape vortex and the corresponding LEV evolution path (i.e. spanwise vortex $\rightarrow$ C-shape vortex $\rightarrow$ M-shape vortex) are detected for the first time.

Figure 10. LEV evolution path. The initial spanwise vortex is extracted at $\theta =0^\circ$. The red iso-surfaces correspond to $Qc^2/U_\infty ^2=30$. The grey mask region highlights the streamwise range $x/c\in [0.38,0.59]$, which is the same as in figure 7(a).

Interestingly, the evolution of a C-shape vortex into an M-shape vortex is different from the results of Burgmann et al. (Reference Burgmann, Dannemann and Schröder2008), where a C-shape vortex transforms into a screwdriver vortex pair. This difference is believed to originate from the occurrence of the near-wall spanwise flow for the trapezoidal plate. The streamwise range under the strong spanwise flow influence is highlighted by the grey mask region in figure 10. It can be seen that the downstream evolution of the C-shape vortex is affected by the spanwise flow. According to the strength of the encountered spanwise flow, two processes are worth noting during the LEV evolution: (i) the LEV formation process ($\theta _c\leq 240^\circ$, $x_c\leq 0.4$), where the spanwise vortex forms and develops into the C-shape vortex; (ii) the LEV transformation process, where the C-shape vortex transforms into the M-shape vortex under the influence of the near-wall spanwise flow.

Before further exploration of the LEV evolution, the cycle-to-cycle variations are characterized. The cyclic variations are evaluated via the probability distribution of the central positions of the LEV heads, as shown in figure 11. The central positions are extracted from the instantaneous flow fields (2-C PIV data) at $z/c=0$, and the joint probability density function (PDF) is retrieved by applying the kernel density estimation method (Sheather Reference Sheather2004). Subsequently, the conditional PDF is extracted at every $x/c$ based on the joint PDF and presented in figure 11(a). The yellow solid line represents the phase-averaged trajectory of the LEV heads. To further quantify the cyclic variations, the standard deviation $\sigma$ is calculated from the conditional PDF at every $x/c$, as illustrated in figure 11(b). The cyclic variations become increasingly noticeable downstream of $x/c\approx 0.4$. This is manifested by an increased probability of the vortex centre deviating from the yellow line (figure 11a) and a rise in $\sigma$ (figure 11b). Nonetheless, even in the region where the cyclic variations are quite pronounced (at approximately $x/c=0.55$), most LEV heads remain distributed around the yellow line, as illustrated in figure 11(a). This indicates that the phase-averaged evolution of LEVs represents the primary evolution path, and the obtained $\sigma$ level is acceptable.

Figure 11. (a) Conditional probability density function (PDF) of the LEV heads for all given $x/c$. The LEV heads are obtained from the instantaneous flow fields at $z/c=0$. The yellow solid line represents the phase-averaged trajectory of the LEV heads. (b) Standard deviation $\sigma$ as a function of $x/c$.

5.2. Vortex formation process and Kelvin–Helmholtz instability

As is known, the formation of LEVs originates from the roll-up of shear layer. The discrepancies in the roll-up process across the span are explored in figure 12. This figure presents both the root-mean-square (r.m.s.) of the wall-normal velocity and the corresponding normalized power spectra in four $x$–$y$ planes, which are obtained via 2-C PIV measurements. In the $\hat {v}_{rms}/U_\infty$ contours, the displacement thickness $\hat {\delta }_1$ is presented by the red solid lines and calculated from the wall-normal integration of the spanwise vorticity $\omega _z$ (Spalart & Strelets Reference Spalart and Strelets2000; Marxen et al. Reference Marxen, Lang, Rist, Levin and Henningson2009). One distinct feature of the $\hat {v}_{rms}/U_\infty$ distributions is the emergence of a single peak corresponding to the formation and subsequent vortex shedding. This phenomenon is consistent with the previous studies on different geometries, such as 2-D flat plates under adverse pressure gradients (Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014), aerofoils (Burgmann et al. Reference Burgmann, Dannemann and Schröder2008), and finite-$A\!R$ wings (Toppings & Yarusevych Reference Toppings and Yarusevych2021). Furthermore, the onset for discernible disturbance amplification is defined as the location where the $\hat {v}_{rms}/U_\infty$ amplitude reaches a threshold $0.025$. It is denoted as $\hat {x}_{amp}$ and shown by the grey vertical lines in figure 12. One can observe that $\hat {x}_{amp}$ shifts downstream with increasing $z$, suggesting a lower rate of disturbance amplification.

Figure 12. Root-mean-square (r.m.s.) of the wall-normal velocity (top) and the normalized power spectra (bottom) at $z/c=0,0.05,0.12,0.24$. The red lines represent the displacement thickness $\hat {\delta }_1$. The grey vertical lines indicate the onset location $\hat {x}_{amp}$ for discernible disturbance amplification. The dark grey areas ($\hat {x}/c\leq 0.12$) are masked due to the relatively high measurement errors near the leading edge.

Spectral analysis is performed to evaluate the spanwise variation of the shear layer roll-up in more detail. The temporal signal of the wall-normal velocity is extracted at $y/c=\hat {\delta }_1$ for each chordwise position. The power spectral density (PSD) is estimated via the Welch method (Welch Reference Welch1967) and normalized with its maximum value in each $x$–$y$ plane. Here, the Strouhal number is defined as $St\equiv f c \sin \alpha /U_\infty$, and the frequency resolution is $\Delta St = 0.004\ (0.07\ {\rm Hz})$. A salient feature of the PSD distribution in each plane is that two distinct spectral peaks appear at approximately $\hat {x}_{amp}$, i.e. $St_0=0.294\ (5.146\ {\rm Hz})$ and $St_1=0.245\ (4.288\ {\rm Hz})$. As $\hat {x}$ increases, the relative spectral energy contained in the $St_0$ component decreases for $z/c=0,0.05$, but the energy corresponding to $St_1$ increases from a low initial level for $z/c=0.12,0.24$. As a result, $St_1$ is the dominant frequency for $z/c=0,0.05$, whereas $St_0$ becomes dominant for $z/c=0.12,0.24$.

To further reveal the underlying mechanism behind the spectral characteristics, the peak frequencies are scaled as $\omega ^\ast ={\rm \pi} \hat {\delta }_\omega \,St \csc \alpha /\hat {u}_e$, where $\hat {u}_e$ represents the velocity at the top of the separated shear layer, and $\hat {\delta }_\omega$ represents the vorticity thickness. This scaling method is proposed by Monkewitz & Huerre (Reference Monkewitz and Huerre1982) and used widely to verify the occurrence of Kelvin–Helmholtz (KH) instability (Simoni et al. Reference Simoni, Ubaldi, Zunino and Bertini2012; Michelis, Yarusevych & Kotsonis Reference Michelis, Yarusevych and Kotsonis2018). The expected value for KH instability in the free shear layer is approximately $\omega ^\ast =0.21$. Actually, the peak frequency $St_0$ can be scaled within a range $\omega ^\ast _0=0.2\unicode{x2013}0.22$ for $0.17 \leq \hat {x}/c \leq 0.22$ in all $x$–$y$ planes, reflecting its association with the KH instability. Also, the peak frequency $St_1$ is supposed to be associated with the nonlinear development of absolute instability. This is because the maximum reverse-flow velocity exceeds $15\,\%$ of $U_\infty$ in the planes $z/c=0,0.05$, as shown in figure 7(a) (Alam & Sandham Reference Alam and Sandham2000).

As is evident from the previous discussion, the shear layer roll-up appears to be unaffected by the near-wall spanwise flow and largely driven by the KH instability. For this instability (Rist & Maucher Reference Rist and Maucher2002), the reductions of the maximum reverse-flow intensity and the distance from the separated shear layer to the plate surface result in a lower rate of disturbance amplification. Thus the vortex formation process delays in the spanwise direction away from the midspan. In other words, the spanwise vortex forms first near the midspan and then develops into the C-shape vortex.

5.3. Vortex transformation process and near-wall spanwise flow

To investigate the influence of the near-wall spanwise flow, its interactions with the C-shape vortices are explored.

The spanwise interval of the near-wall spanwise flow region, $d_w$, as a function of streamwise position $x$ and phase angle $\theta$, is plotted in figure 13. The spanwise interval $d_w$ appears to be independent of $\theta$ for $x/c \leq 0.38$, whereas it oscillates with $\theta$ in the downstream. Considering the mass transport mechanism mentioned in § 4, a reduction in $d_w$ implies a stronger influence of the near-wall spanwise flow. To characterize the evolution of the near-wall spanwise flow, the phase angle, where $d_w$ reaches its minimum value at a given streamwise position $x/c=x_w$, is extracted and denoted as $\theta _w(x_w) = \arg \min d_w (x_w,\theta )$. The resulting mappings $\theta _w$–$x_w$ are essentially the valley lines of the $d_w(x,\theta )$ distribution, as shown by the red solid lines in figure 13. Moreover, to elucidate the spatial relationship between the near-wall spanwise flow and the LEVs, the LEV evolution is characterized by their streamwise positions $x_c$. The mappings $\theta$–$x_c$ are presented by the yellow dash-dotted lines. Interestingly, the streamwise distance between the mappings $\theta _w$–$x_w$ and $\theta$–$x_c$ remains fairly constant at $\Delta x=0.06$ for $x/c\geq 0.38$ (or for $x/c\geq 0.43$ from the view of the LEVs). In other words, despite the variations in phase angle $\theta$, $d_w$ reaches its local minimum value at a constant distance upstream from the C-shape vortex heads. For instance, when $d_w$ reaches its minimum value at $x/c=0.43$ and $\theta =70^\circ$ (indicated by the red dot), the C-shape vortex head locates at $x_c=0.49$ (yellow dot). The corresponding sectional flow field is displayed in the upper plot of figure 13, which indicates that the position $x/c=0.43$ (red dashed line) is just upstream of the C-shape vortex head. Therefore, as LEVs evolve downstream, the flow on the windward side of the C-shape vortex heads is affected persistently by the near-wall spanwise flow.

Figure 13. Distribution of the spanwise interval of the near-wall spanwise flow region, $d_w(x,\theta )$. The upper plot shows the corresponding spanwise vorticity field in the plane $z/c=0$ at $\theta =70^\circ$.

An oblique view of the whole flow field at $\theta =70^\circ$ is provided in figure 14, which presents the 3-D vortical structures, as well as the spanwise velocity fields in the planes $\hat {y}/c=0.026$ and $x/c=0.43$. As discussed previously, the near-wall spanwise flow region (indicated by the green solid arrows) gets its minimum spanwise interval at $x/c=0.43$. In the vicinity, an upwash flow (green dashed arrow) is possible to be strengthened by the transport of high-momentum fluid from the tips to the midspan. Considering that the strengthened upwash flow occurs on the windward side of the C-shape vortex heads, an increase in the strength of the vortex heads is expected.

Figure 14. Oblique view of the flow field at $\theta =70^\circ$. The vortical structures are visualized using the iso-surface of $Qc^2/U_\infty ^2=30$ in transparent grey.

To verify the presence of the interactions between the near-wall spanwise flow and the C-shape vortex heads, the strength of the LEV heads is then monitored and shown in figure 15. Here, the spanwise circulation $\varGamma _z =\iint _{C}{\omega _z\,{\rm d}s}$ is used as a surrogate for the LEV head strength, where the integral region $C$ follows the same definition as figure 9. The circulation magnitude $| \varGamma _z |$ of the vortex head located at $x_c$ is plotted by the black spots in figure 15. A smoothed spline fit is then applied for illustration purposes and shown by the yellow solid line. Remarkably, there are two distinct rapid growth regimes, i.e. regime I and regime II. The growth in regime I originates from the shear layer vorticity feeding. However, the growth rate decreases to a low level at approximately $x/c=0.35$, reflecting the weakening of the shear layer feeding. Downstream of $x/c=0.38$, a second rapid growth is visible in regime II, which is attributed to the influence of the near-wall spanwise flow.

Figure 15. Evolution of the spanwise circulation of the LEV heads and the minimum spanwise interval. The yellow line is the fitted smoothing spline for the black spots $(x_c,| \varGamma _z |)$.

To clarify the role of the near-wall spanwise flow in the second rapid growth of $| \varGamma _z |$, the evolution of the minimum spanwise interval $d_{w,min}(x_w) = d_w(x_w,\theta _w)$ is presented by the red dash-dotted line in figure 15. There is a steep drop of $d_{w,min}$ in the range $0.35< x/c<0.38$, which implies the stronger influence of the near-wall spanwise flow. Under this influence, the second rapid growth of $| \varGamma _z |$ is triggered. Beyond $x/c=0.38$, $d_{w,min}$ remains at a rather low level. In light of the $d_w$ oscillation with $\theta$ (figure 13), the interactions between the near-wall spanwise flow and the C-shape vortex heads take place. Presumably, these interactions affect the LEV heads located after $x/c=0.43$. Therefore, the $| \varGamma _z |$ growth in the region $x/c\in (0.43,0.51)$ (masked in green in figure 15) substantiates that the interactions of the flow structures strengthen the C-shape vortex heads as discussed previously in figure 14.

Furthermore, the strengthened vortex head can entrain more high-momentum fluid from the outer flow towards the plate surface, as moving downstream. This entrainment increases the momentum of the separated boundary layer near the midspan. Consequently, the downstream motion of the C-shape vortex head is accelerated, and the vortex core bends downstream near the midspan, i.e. a C-shape vortex transforms into an M-shape vortex.