2.1. Experimental set-up

Figure 1.

Plate geometry and coordinate system definition.

The experimental apparatus used in this study is identical to that described by Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023). As shown in figure 1, the plate model is machined from a 1 mm thick aluminium sheet with all edges maintaining a square profile. It has a chord length of $c = {58}$ mm, a leading-edge length of 80 mm and a trailing-edge length of 58 mm. The aspect ratio is defined as the ratio of maximum span to chord, yielding . The free stream velocity $U_\infty$ is set to ${96}\,\mathrm{mm \,s^{-1}}$ , corresponding to a chord-based Reynolds number $ \textit{Re}_c$ of $5800$ . This value falls within the typical operating regime of flying insects, whose exceptional flight performance is worthy of further exploration. The free stream turbulence intensity remains below $0.8\,\%$ . To facilitate data presentation, a Cartesian coordinate system $O$ - $\textit{xyz}$ is adopted. The origin $O$ is defined such that the coordinates of the trailing-edge point in the midspan plane (magenta point in figure 1) are $(c, 0, 0)$ , with the $x$ , $y$ and $z$ axes aligned in the streamwise, vertical and spanwise directions, respectively.

Figure 2.

Overview of the stereo-PIV set-up: (i) plate model, shown in an enlarged view for clarity; (ii) plate support system; (iii) high-resolution translator (iv) CCD cameras; (v) water prisms; (vi) laser sheet.

Stereoscopic particle image velocimetry (stereo-PIV) is employed to measure the flow fields over the plate and in its wakes, with the configuration depicted in figure 2. Experiments are conducted in the low-speed recirculation water channel at Beihang University. The water channel is seeded with hollow glass beads of $5$ – ${20}\,{\unicode{x03BC}}\mathrm{m}$ diameter and illuminated by a dual-head Nd:YAG laser (Beamtech Vlite-500). The laser light sheet remains stationary during experiments and has a thickness of 2 mm. Two charged-coupled device (CCD) cameras (IMPERX ICL-B2520M, $2456\times 2058$ pixels) are positioned on opposite sides of the water channel. Equipped with Nikkor 85 mm tilt-shift lenses, these cameras satisfy the Scheimpflug condition. Synchronisation between the cameras and the laser is achieved using a MicroVec Micropulse-725 synchroniser. Particle images are acquired in double-frame mode at a sampling frequency of 3 Hz. Furthermore, the plate model is hung upside down and shifted streamwise through the stationary light sheet via a high-resolution translator. In this manner, a total of $31$ closely spaced $y$ – $z$ measurement planes are obtained, spanning from $x/c = -0.03$ to $x/c = 1.52$ with an increment of $\Delta x/c \approx 0.052$ (3 mm). For each plane, $2000$ image pairs are recorded.

All particle images are pre-processed using a background removal method based on proper orthogonal decomposition (POD) (Sirovich Reference Sirovich1987; Mendez et al. Reference Mendez, Raiola, Masullo, Discetti, Ianiro, Theunissen and Buchlin2017). A self-calibration method (Wieneke Reference Wieneke2005) is applied to correct misalignment between the calibration target and the light sheet. Velocity fields are computed with an in-house stereo-PIV code based on Soloff reconstruction (Soloff, Adrian & Liu Reference Soloff, Adrian and Liu1997) and the optical flow method (Pan et al. Reference Pan, Xue, Xu, Wang and Wei2015). The interrogation window size is set to $48 \times 48$ pixels with $75\,\%$ overlap, yielding a vector spacing of approximately 0.3 mm. Raw velocity data are further validated using robust principal component analysis method (Scherl et al. Reference Scherl, Strom, Shang, Williams, Polagye and Brunton2020). The uncertainty relative to the free stream velocity is approximately $1.5\,\%$ for the velocity fields (Van Doorne & Westerweel Reference Van Doorne and Westerweel2007). Further experimental details are available from Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023). Distinct from the previous study, the current work focuses on wake characteristics and their correlations with upstream flow structures and aerodynamic forces.

2.2. Estimating forces based on wake integral

Given that aerodynamic forces under low Reynolds numbers and other operating conditions are often too small to be measured accurately, several alternative control volume methods have been developed. Among these, the wake-integral approach is widely adopted because it requires data only from a wake plane and provides acceptable accuracy (Roeland & Robert Reference Roeland and Robert2016; Wang, He & Liu Reference Wang, He and Liu2019; Baker et al. Reference Baker, Diaz, Bailly, David and Monnier2019; Tong, Wang & Yang Reference Tong, Wang and Yang2022).

Figure 3.

Computational domain $V_{\!f}$ for flow past the trapezoidal plate. The wake surface, denoted as $\varSigma _{w\textit{ake}}$ , is positioned at $x/c = x^\ast$ . The dashed rectangular region on $\varSigma _{w\textit{ake}}$ is the effective area used for force estimation.

The foundation of the wake-integral approach lies in a global momentum balance over the control volume $V_{\!f}$ , which fully encloses the plate model (see figure 3). The volume $V_{\!f}$ is bounded by six surfaces $\varSigma$ , including the inlet surface $\varSigma _\infty$ , the exit wake surface $\varSigma _{w\textit{ake}}$ and the four peripheral surfaces. The outward-pointing normal vectors of these surfaces are denoted as $\boldsymbol{n}$ . The aerodynamic force exerted on the plate can be expressed as

(2.1) \begin{equation} \boldsymbol{F}=-\int _{V_{\!f}}{\frac {\partial \left ( \rho \boldsymbol{u} \right )}{\partial t}\,\mathrm{d}V}+\int _{\varSigma }{\left ( -\rho \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{uu} \right )\,\mathrm{d}S}+\int _{\varSigma }{\left ( -p\boldsymbol{n}+\boldsymbol{\tau }:\boldsymbol{n} \right )\,\mathrm{d}S}, \end{equation}

where $\rho$ denotes fluid density, $p$ denotes fluid pressure and $\boldsymbol{\tau }$ denotes viscous stress tensor. The first term on the right-hand side represents unsteady effects. Given that the flow investigated in this study is statistically stationary, this term becomes zero in the evaluation of time-averaged forces and is therefore omitted. Assuming that the four peripheral surfaces are sufficiently far from the plate, the velocity on these surfaces approaches that of the uniform free stream $U_\infty$ , allowing the global momentum balance to be simplified to a comparison between $\varSigma _\infty$ and $\varSigma _{w\textit{ake}}$ . Consequently, the aerodynamic force acting on the flat plate can be determined solely from the integral over the wake surface $\varSigma _{w\textit{ake}}$ . The time-averaged lift and drag coefficients are then expressed as

(2.2) \begin{equation} \overline {L}=\int _{\varSigma _{w\textit{ake}}}{-\rho \overline {uv}\,\mathrm{d}S}, \end{equation}

(2.3) \begin{equation} \overline {D}=\int _{\varSigma _{w\textit{ake}}}{\rho \big ( U_\infty ^2-{\overline {u}^2} \big)\mathrm{d}S} +\int _{\varSigma _{w\textit{ake}}}{-\rho \overline {u^\prime u^\prime }\,\mathrm{d}S}+\int _{\varSigma _{w\textit{ake}}}{( p_\infty -\overline {p})\mathrm{d}S} +\int _{\varSigma _{w\textit{ake}}}{\overline {\tau }_{xx}\,\mathrm{d}S}. \end{equation}

According to Stokes’ theorem, (2.2) can be transformed into

(2.4) \begin{equation} \overline {L}=\int _{\varSigma _{w\textit{ake}}}{\overline {l}\,\mathrm{d}S}=\int _{\varSigma _{w\textit{ake}}}{-\rho z\overline {u}{\overline {\omega _x}}-\rho z\left ( \overline {w}\frac {\partial \overline {u}}{\partial y}-\overline {v}\frac {\partial \overline {u}}{\partial z} \right )-\rho z\left ( \frac {\partial \overline {{u}^\prime {w}^\prime }}{\partial y}-\frac {\partial \overline {{u}^\prime {v}^\prime }}{\partial z} \right )\mathrm{d}S}, \end{equation}

where $\overline {l}$ denote lift element.

As for drag, given that $\overline {\dot {m}}=\int _{\varSigma _{w\textit{ake}}}{\rho ( \overline {u}-U_\infty )\,\mathrm{d}S}$ represents the net mass of fluid generated within the control volume, this term can be reasonably set to zero for incompressible flow. Accordingly, the first term on the right-hand side of (2.3) is transformed to $\int _{\varSigma _{w\textit{ake}}}{\rho \overline {u} ( \overline {u}-U_\infty )\,\mathrm{d}S}$ . Moreover, as established in the literature (van Dam Reference van Dam1999), the viscous stress term can be written as

(2.5) \begin{equation} \overline {\tau }_{xx} = 2 \mu \frac {\partial \overline {u}}{\partial x}-\rho \overline {u^\prime u^\prime }, \end{equation}

where $\mu$ denotes dynamic viscosity. The viscous component $2 \mu {\partial \overline {u}}/{\partial x}$ is calculated using the in-plane velocity gradients, given that the flow divergence is zero. The second term on the right-hand side of (2.3) is obtained via the pressure reconstruction based on the Poisson equation,

(2.6) \begin{equation} {\nabla} ^2\overline {p}=-\boldsymbol{\nabla }\boldsymbol{\cdot }\left \{ \rho \left ( \left ( \overline {\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{\nabla }\right ) \overline {\boldsymbol{u}}+\boldsymbol{\nabla }\overline {\boldsymbol{u}^\prime \boldsymbol{u}^\prime } \right ) \right \}\!. \end{equation}

Specifically, velocity field data from three adjacent planes are employed to reconstruct the time-averaged pressure distribution using the spectral decomposition-based fast pressure integration algorithm proposed by Wang et al. (Reference Wang, Gao, Wei, Li and Wang2017). The divergence of the volumetric velocity fields is corrected using the divergence-free smoothing method developed by Wang et al. (Reference Wang, Gao, Wang, Wei, Li and Wang2016). It should be noted that the drag evaluation depends on the pressure difference between the exit wake surface $\varSigma _{w\textit{ake}}$ and the inlet surface $\varSigma _\infty$ . Therefore, the absence of static pressure measurements does not affect the results. The final expression for drag estimation is given by

(2.7) \begin{equation} \overline {D}=\int _{\varSigma _{w\textit{ake}}}{\overline {d}\,\mathrm{d}S}=\int _{\varSigma _{w\textit{ake}}}{\rho \overline {u}( U_\infty -\overline {u} )\,\mathrm{d}S} + \int _{\varSigma _{w\textit{ake}}}{( p_\infty -\overline {p})\,\mathrm{d}S}+ \int _{\varSigma _{w\textit{ake}}}2{\left ( \mu \frac {\partial \overline {u}}{\partial x}-\rho \overline {u^\prime u^\prime } \right )\mathrm{d}S}, \end{equation}

where $\overline {d}$ denote drag element. The corresponding force coefficients are computed by

(2.8) \begin{equation} \left \{ \begin{matrix} \overline {C}_L & \hspace {-0.75em} = &\hspace {-0.75em} \overline {L}/\left ( 0.5\rho U_\infty ^2 S \right )\!,\quad \overline {C}_D & \hspace {-0.75em} = &\hspace {-0.75em} \overline {D}/\left ( 0.5\rho U_\infty ^2 S \right )\!; \\[7pt] \overline {C}_l & \hspace {-0.75em} = &\hspace {-0.75em} \overline {l}/\left ( 0.5\rho U_\infty ^2 S \right )\!,\quad \overline {C}_d & \hspace {-0.75em} = &\hspace {-0.75em} \overline {d}/\left ( 0.5\rho U_\infty ^2 S \right )\!. \\ \end{matrix} \right . \end{equation}

Here, $S$ denotes wing area.

In practice, since the force elements are significant only within a localised subregion of the wake surface, the integration can be restricted to this effective area, as illustrated by the dashed rectangular region in figure 3. It is noted that the flow fields beneath this rectangular region are influenced by the plate support system. Nevertheless, these flow fields are safely excluded from the force estimation, as their contribution to the overall forces is negligible and does not compromise the validity of this study. For ease of description, the streamwise location of the wake surface is denoted as $x^\ast$ .

3.1. Separation bubble and its associated wake topology

Figure 4.

Overview of the flow structures at $\alpha = = 4^\circ$ – $10^\circ$ . Separation bubbles are represented by grey iso-surfaces of $\overline {u}/U_\infty = 0$ , while colour contours depict the distribution of time-averaged streamwise vorticity $\overline {\omega }_x$ . For enhanced visualisation, contours are blanked over the plate surface upstream of $x/c = 0.74$ .

Figure 4 provides an overview of the dominant flow structures. The first structure discussed herein is the separation bubble, represented by the grey iso-surface of $\overline {u}/U_\infty =0$ . As previously discussed by Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023), the separation bubble emerges at $\alpha = 6^\circ$ , featuring a distinct concavity near the midspan. This particular bubble structure is designated as the swallow-tailed structure. With the increase of $\alpha$ , the concavity gradually diminishes and the separation bubble transforms into a conventional single-tailed structure at $\alpha = 10^\circ$ . Notably, the swallow-tailed bubble structure is also observed for rectangular plates with ranging from $1.00$ to $1.50$ at $\alpha = 6^\circ$ (Zhu, Wang & Liu Reference Zhu, Wang and Liu2024). The reader is referred to Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023, Reference Zhu, Wang and Liu2024) for the relevant evolution of LEVs over the swallow-tailed separation bubble.

The distribution of time-averaged streamwise vorticity $\overline {\omega }_x$ in the wake exhibits a strong correlation with the separation bubble structure, as illustrated by the colour contours in figure 4. Near the midspan, $\overline {\omega }_x$ is relatively weak at $\alpha = 4^\circ$ (figure 4 a), whereas large magnitudes of $\overline {\omega }_x$ are observed downstream of the separation bubble for $\alpha = 6^\circ {-}10^\circ$ (figure 4 b–d). At these angles of attack, a pair of counter-rotating vortices appears on both lateral sides of the separation bubble, labelled as $A$ in figure 4(b–d). Moreover, an additional counter-rotating vortex pair forms when the separation bubble exhibits the swallow-tailed structure, denoted as $B$ in figures 4(b) and 4(c). This additional vortex pair persists downstream at $\alpha = 6^\circ$ , whereas it becomes indiscernible before reaching the trailing edge at $\alpha = 8^\circ$ . This behaviour is consistent with the progressive disappearance of the concavity as $\alpha$ increases. Accordingly, the formation of this additional vortex pair can be attributed to the sufficiently enhanced entrainment in the rear region of the separation bubble during its development into the swallow-tailed structure, as discussed by Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023, Reference Zhu, Wang and Liu2024). Overall, the streamwise vorticity concentrations (vortex pairs $A$ and $B$ ) form a quadrupole pattern at $\alpha = 6^\circ$ , originating from the swallow-tailed separation bubble. In contrast, the wake vorticity concentrations at $\alpha = 8^\circ$ and $10^\circ$ constitute a dipole configuration, as only vortex pair $A$ remains in the wake.

Figure 5.

Comparison of the stretching term $\overline {S}_{x}$ between $\alpha = 6^\circ$ and $10^\circ$ cases. The sectional shape of the separation bubble is presented using the black dash-dotted lines $\overline {u}/U_\infty = 0$ . The vortex pairs $A$ and $B$ are consistent with those in figure 4.

Figure 5. Long description

Panel A: A heat map compares stretching terms at an angle of 6 degrees. The x-axis represents the normalized spanwise coordinate z over c, and the y-axis represents the normalized vertical coordinate y over c. The color scale ranges from −8 to 8, indicating the normalized stretching term. Black dash-dotted lines outline the sectional shape of the separation bubble. Magenta and yellow dashed lines highlight specific vortex pairs. Panel B: A heat map compares stretching terms at an angle of 10 degrees. The x-axis represents the normalized spanwise coordinate z over c, and the y-axis represents the normalized vertical coordinate y over c. The color scale ranges from −8 to 8, indicating the normalized stretching term. Black dash-dotted lines outline the sectional shape of the separation bubble. Magenta and yellow dashed lines highlight specific vortex pairs.

To further shed light on the relationship between bubble structures and wake topology, vorticity transport analysis is conducted in the vicinity of the separation bubbles. Specifically, the contributions due to the stretching and tilting terms for the generation of $\overline {\omega }_x$ are evaluated,

(3.1) \begin{equation} \left \{ \begin{matrix} \overline {S}_x = & \overline {\omega }_x\displaystyle \frac {\partial \overline {u}}{\partial x}; \\[11pt] \overline {T}_x = & \overline {\omega }_y\displaystyle \frac {\partial \overline {u}}{\partial y}+\overline {\omega }_z \displaystyle \frac {\partial \overline {u}}{\partial z}. \\ \end{matrix} \right . \end{equation}

Here, $\overline {S}_{x}$ and $\overline {T}_{x}$ denote the time-averaged stretching and tilting terms, respectively. Near the separation bubble, the stretching term is found to be dominant and is further compared between the $\alpha = 6^\circ$ and $10^\circ$ cases, as shown in figure 5. At $\alpha = 6^\circ$ , $\overline {S}_{x}$ is evidently concentrated above the swallow-tailed separation bubble. Particularly, accompanied by the formation of the additional vortex pair $B$ , $\overline {S}_{x}$ contributes to an increase in local vorticity magnitude. In contrast, no obvious $\overline {S}_{x}$ distribution is captured near the midspan plane at $\alpha = 10^\circ$ . This evidence suggests that vortex stretching plays a critical role in the formation of the quadrupole wake topology. Combined with the evolution of LEVs (Zhu et al. Reference Zhu, Wang, Xu, Qu and Long2023), the emergence of $\overline {S}_{x}$ is presumably associated with the LEV transformation from a C-shape to an M-shape structure. However, the detailed physical mechanism still requires further investigation.

3.2. Characteristics of wingtip vortices

Another dominant flow structure for the low- plate is the wingtip vortex. As shown in figure 4, a pair of counter-rotating vortices trails from the two side edges of the plate, forming the wingtip vortices. Since the rollup of the wingtip vortices is incomplete in the near field, they exhibit the ‘arm-and-fist’ pattern, consistent with previous studies (Devenport et al. Reference Devenport, Rife, Liapis and Follin1996; Skinner, Green & Zare-Behtash Reference Skinner, Green and Zare-Behtash2020).

The characteristics of wingtip vortices are investigated to explore the influence of separation bubble development. The instantaneous locations of the vortex centres ${\boldsymbol{x}_c} ( t )$ are identified from stereo-PIV snapshots via a robust and accurate method based on peak helicity (Skinner et al. Reference Skinner, Green and Zare-Behtash2020), given by the following equation:

(3.2) \begin{equation} {\boldsymbol{x}_c}\left ( t \right ) = \left ( y_c,z_c \right ) = \mathop {\arg \max }\limits _{y,z} {H_x}, \quad \mbox{where} \quad {H_x}=\left | u\left ( t \right )-{U_\infty }\right | \boldsymbol{\cdot }\left | {\omega _x}\left ( t \right ) \right |. \end{equation}

Following Edstrand et al. (Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016) and Dghim, Ferchichi & Fellouah (Reference Dghim, Ferchichi and Fellouah2020), the triple decomposition (Hussain & Reynolds Reference Hussain and Reynolds1970) is employed to extract the coherent wandering motion of the wingtip vortex. First, each stereo-PIV snapshot is shifted by ${\boldsymbol{x}_c} ( t )$ into a new reference frame $ ( \boldsymbol{x}-\boldsymbol{x}_c ( t ) )$ , yielding the wandering-corrected velocity field $\hat {\boldsymbol{u}} ( \boldsymbol{x}-{\boldsymbol{x}_c} ( t ),t )$ . The coherent wandering velocity field $\tilde {\boldsymbol{u}} ( \boldsymbol{x},t )$ is then derived as

(3.3) \begin{equation} \tilde {\boldsymbol{u}}\left ( \boldsymbol{x},t \right )= \left [ \boldsymbol{u}\left ( \boldsymbol{x},t \right )-\overline {\boldsymbol{u}}\left ( \boldsymbol{x} \right ) \right ]-\big [ \hat {\boldsymbol{u}}\left ( \boldsymbol{x},t \right )-\overline {\hat {\boldsymbol{u}}}\left ( \boldsymbol{x} \right ) \big ], \end{equation}

where the first right-hand term contains the coherent wandering motion and the background turbulence, while the second right-hand term removes the turbulence.

Figure 6.

Self-similar wandering-corrected circulation profiles at four angles of attack: (I) inner core region; (II) buffer region; (III) logarithmic region; (IV), outer region.

Figure 6 illustrates the core structure of the wingtip vortex with the wandering motion corrected. The wandering-corrected wingtip vortex core structure is analysed through the radial distribution of the normalised circulation ${\overline {\hat {\varGamma }}}/{\overline {\hat {\varGamma }}_c}$ , plotted against the similarity variable $\eta = r /{\overline {\hat {r}}_c}$ . Here, circulation $\overline {\hat {\varGamma }}$ is computed via line integration of wandering-corrected tangential velocity along a circular path of radius $r$ centred at the vortex core. The core radius $\overline {\hat {r}}_c$ is defined as the radius where the tangential velocity reaches its maximum and the core circulation is given by $\overline {\hat {\varGamma }}_c = \overline {\hat {\varGamma }} (\overline {\hat {r}}_c )$ .

The normalised circulation ${\overline {\hat {\varGamma }}}/{\overline {\hat {\varGamma }}_c}$ collapses onto a single curve in figure 6, indicating a self-similar behaviour across different $\alpha$ . To characterise this self-similar behaviour, two empirical curve-fitting models are applied. First, for the inner region ( $\eta \lt 1.4$ ), Hoffmann & Joubert (Reference Hoffmann and Joubert1963) proposed the following model:

(3.4) \begin{equation} \left \{ \begin{array}{lllll} \overline {\hat {\varGamma }}\left (\eta \right )/{\overline {\hat {\varGamma }}_c} = \hspace {-0.75em}\quad A\eta ^2 \qquad \qquad \enspace \mbox{for}\enspace \eta \lt 0.4, \\[11pt] \overline {\hat {\varGamma }}\left (\eta \right )/{\overline {\hat {\varGamma }}_c} = \hspace {-0.75em}\quad 1+ B\log \left (\eta \right ) \!\! \quad \mbox{for}\enspace 0.5\lt \eta \lt 1.4. \end{array} \right . \end{equation}

In the present study, the curve-fitting constants $A$ and $B$ are determined as $1.717$ and $2.163$ , respectively. Second, Ramaprian & Zheng (Reference Ramaprian and Zheng1997) demonstrated that ${\overline {\hat {\varGamma }}}/{\overline {\hat {\varGamma }}_c}$ follows a sixth-order polynomial for $\eta \lt 1.2$ , given by

(3.5) \begin{equation} {\overline {\hat {\varGamma }}\left (\eta \right )}/{\overline {\hat {\varGamma }}_c} = a_0 \eta ^2+a_1 \eta ^4+a_2 \eta ^6. \end{equation}

Herein, $a_0$ , $a_1$ and $a_2$ take values of $1.799$ , $-1.085$ and $0.284$ , respectively. As shown in figure 6, the normalised circulation profiles are in excellent agreement with these empirical fitting curves. Furthermore, these five curve-fitting constants agree well with the published results (Birch et al. Reference Birch, Lee, Mokhtarian and Kafyeke2004; Gerontakos & Lee Reference Gerontakos and Lee2006). These results suggest that the core structure of the wingtip vortices remains unaffected by separation bubble development across all tested angles of attack.

Figure 7.

Streamwise velocity contours of the leading POD mode for the wingtip vortex with positive $\overline {\omega }_x$ . Magenta circles denote the locations of the wingtip vortex cores. Solid and dashed black contour lines represent $\tilde {\omega }_x^{\phi (1)}=0.0075$ and $-0.0075$ within each circle, respectively. The percentage in the upper right corner of each panel represents the energy ratio of the corresponding POD mode to the total fluctuation energy.

To gain insight into the coupling between wingtip vortices and separation bubbles, POD is applied to the coherent wandering velocity fields $\tilde {\boldsymbol{u}} (\boldsymbol{x},t )$ at each cross-section $y$ – $z$ plane (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016; Dghim et al. Reference Dghim, Ferchichi and Fellouah2020), expressed as

(3.6) \begin{equation} \tilde {\boldsymbol{u}} \left ( \boldsymbol{x},t \right )=\sum \limits _k^N{a_k{\boldsymbol{\tilde {u}}^{\phi \left ( k \right )}}}. \end{equation}

Here, $a_k$ is the time coefficient of the $k$ th mode $\tilde {\boldsymbol{u}}^{\phi ( k )}$ and $N$ denotes the total number of modes. Figure 7 presents the streamwise velocity component of the leading POD mode $\tilde {u}^{\phi ( 1 )}$ at $x/c = 0.74$ and $1.52$ . This dominant POD mode accounts for more than half of the total fluctuation energy of the wingtip vortex at a given plane, as quantified by the percentage values in each panel of figure 7. The cross-sectional positions of the wingtip vortex cores are indicated by magenta circles, with radii corresponding to the local vortex core radius $\hat {r}_c$ . Owing to the plate symmetry, this study is restricted to the wingtip vortices with positive streamwise vorticity $\overline {\omega }_x$ . Within each magenta circle, two regions of oppositely signed $\tilde {u}^{\phi ( 1 )}$ are clearly observed. The formation of this velocity distribution is consistent with prior studies and is associated with the helical displacement of the wingtip vortices (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016; Dghim et al. Reference Dghim, Ben Miloud, Ferchichi and Fellouah2021).

To illustrate the implications of the $\tilde {u}^{\phi ( 1 )}$ distribution on the wandering motion of the wingtip vortex, the corresponding modal streamwise vorticity $\tilde {\omega }_x^{\phi ( 1 )}$ within each magenta circle is shown in figure 7. The distribution of $\tilde {\omega }_x^{\phi ( 1 )}$ is represented by solid and dashed black contour lines, corresponding to $\tilde {\omega }^{\phi ( 1 )}=0.0075$ and $-0.0075$ , respectively. For each vortex core, regions of oppositely signed $\tilde {\omega }_x^{\phi ( 1 )}$ are symmetrically positioned on either side. When the time coefficient $a_1$ assumes a positive value, vorticity intensifies on the side with positive $\tilde {\omega }_x^{\phi ( 1 )}$ and weakens on the side with negative $\tilde {\omega }_x^{\phi ( 1 )}$ , given that the wingtip vortex has positive $\overline {\omega }_x$ as described earlier. As a result, the vortex cores undergo systematic displacement along the axis connecting these two regions, wandering from the negative to the positive $\tilde {\omega }_x^{\phi ( 1 )}$ side.

Notably, on the left side of the vortex cores (i.e. $ ( z-\overline {z}_c )/c\lt 0$ ), $\tilde {u}^{\phi ( 1 )}$ exhibits distinct behaviour depending on separation bubble development. As shown in figure 7(a), $\tilde {u}^{\phi ( 1 )}$ is relatively uniformly along the spanwise direction for $\alpha = 4^\circ$ . In the absence of a separation bubble, this $\tilde {u}^{\phi ( 1 )}$ distribution is attributed to attached boundary layer instability. Correspondingly, the wingtip vortex displays a vertical wandering tendency, as inferred from the $\tilde {\omega }_x^{\phi ( 1 )}$ regions within the magenta circle. In contrast, at higher angles of attack, $\tilde {u}^{\phi ( 1 )}$ becomes concentrated in the region ${ ( z-\overline {z}_c )}/c\lt -0.2$ and the wingtip vortices exhibit a spanwise wandering tendency. Specifically, when $a_1$ is positive, the localised positive $\tilde {u}^{\phi ( 1 )}$ on the left side of each vortex core indicates the contraction of the separation bubble. This bubble contraction is accompanied by the leftward wandering of wingtip vortices.

Similar $\tilde {u}^{\phi ( 1 )}$ behaviour is observed in the wake. For $\alpha \leqslant 6^\circ$ , the $\tilde {u}^{\phi ( 1 )}$ distribution outside the wingtip vortex cores originates from the wake shear layers and the wingtip vortices display a tendency of vertical wandering (figures 7 e and 7 f). As $\alpha$ increases to $10^\circ$ , a concentrated $\tilde {u}^{\phi ( 1 )}$ region induced by the separation bubble appears at the left side of the wingtip vortex core, accompanied by a pronounced spanwise wandering tendency of the wingtip vortex (figure 7 h). In summary, the results demonstrate that the oscillation of separation bubbles strengthens with increasing $\alpha$ and promotes the spanwise wandering of the wingtip vortices.