1. Introduction
In recent years, the demand for small, agile and energy-efficient aerial platforms has led to increased interest in micro air vehicles (MAVs). Owing to their compact size and low flight speeds, these platforms typically operate at Reynolds numbers around
$Re \sim 10^5$
(McMichael Reference McMichael1997; Mueller Reference Mueller1999; Shyy et al. Reference Shyy, Lian, Tang, Liu, Trizila, Stanford, Bernal, Cesnik, Friedmann and Ifju2008), where laminar separation bubbles (LSBs) frequently develop on the suction side of aerofoils (Gaster Reference Gaster1967; O’meara & Mueller Reference O’meara and Mueller1987). The presence of an LSB can substantially increase pressure drag, reduce lift and trigger early stall, posing a major challenge to aerodynamic performance and stability (Fitzgerald & Mueller Reference Fitzgerald and Mueller1990; Gad-el Hak Reference Gad-el-Hak2001).
To mitigate these effects, numerous passive flow control strategies have been proposed, including vortex generators, Gurney flaps and riblets (Viswanath Reference Viswanath2002; Wang, Li & Choi Reference Wang, Li and Choi2008; Portal-Porras et al. Reference Portal-Porras, Fernandez-Gamiz, Zulueta, Garcia-Fernandez and Uralde-Guinea2024). While these devices are effective, they remain active throughout the flight envelope, introducing unwanted drag even when flow separation is absent. In contrast, active flow control methods enable targeted flow manipulation, improving efficiency by activating only when needed. Among these, air jet vortex generators (AJVGs) have gained particular attention for their ability to inject jets of air through the aerofoil surface, forming streamwise vortices that transport high-momentum fluid from the free stream into the near-wall region, thereby delaying separation (Wallis Reference Wallis1952).
The effectiveness of AJVGs depends on various design and operational parameters, such as jet pitch and skew angles, orifice geometry, and the velocity ratio (VR) between the jet and the local free-stream velocity (Compton & Johnston Reference Compton and Johnston1992; Henry & Pearcey Reference Henry and Pearcey1994). Pitched and skewed jets have been shown to generate coherent longitudinal vortices, whose circulation scales with VR up to an optimal range (Compton & Johnston Reference Compton and Johnston1992). These vortices induce local thinning of the boundary layer under the downwash region and redistribute momentum spanwise, enhancing flow attachment in a controllable manner.
At low Reynolds numbers, AJVGs have proven especially useful. Crowther (Reference Crowther2006) demonstrated that a linear array of jets on a flapped aerofoil effectively suppressed laminar separation and restored aerodynamic performance. Volino (Reference Volino2003) reported similar benefits on low-pressure turbine blades, highlighting the ability of AJVGs to promote transition and maintain attachment under adverse pressure gradients. These results underline their relevance for MAV applications, where boundary layers are highly susceptible to separation.
Recent studies employing high-resolution particle image velocimetry (PIV) have further clarified the vortex dynamics induced by AJVGs. These investigations consistently report the formation of concentrated vortex cores, characterised by asymmetric circulation, spanwise velocity gradients and elevated turbulence levels. Compton & Johnston (Reference Compton and Johnston1992) observed that skewed jets generate embedded streamwise vortices, producing boundary-layer thinning under downwash and thickening near upwash zones. Khan & Johnston (Reference Khan and Johnston2000) and Crowther (Reference Crowther2006) confirmed enhanced near-wall momentum and increased turbulent kinetic energy downstream of the jet. More recently, Ramaswamy & Schreyer (Reference Ramaswamy and Schreyer2023) showed that elliptical orifices increase vertical penetration and vortex strength, with optimal jet spacing being crucial to maintaining coherence without inducing destructive interactions. Across these studies, AJVG-induced vortices are consistently linked to enhanced wall shear, localised peaks in turbulent energy, and improved boundary-layer recovery.
Although considerable progress has been made in understanding the effects of AJVGs, the three-dimensional structure of the vortices they generate at low Reynolds numbers – particularly in the context of MAV-representative aerofoils – remains insufficiently understood. Most previous experimental studies have relied on planar or stereoscopic PIV, which limit the analysis to two-dimensional sections of inherently three-dimensional flows. As a result, the spatial organisation, coherence and boundary-layer interaction of AJVG-induced vortices have largely been inferred rather than directly observed. Moreover, the effects of these vortices on laminar separation and stall suppression have not been fully characterised under controlled, low-Reynolds-number conditions.
The objective of this study is to fill this gap by performing volumetric flow measurements using the double-pulse Shake-the-Box (STB) technique (Schanz, Gesemann & Schröder Reference Schanz, Gesemann and Schröder2016; Novara, Schanz & Schröder Reference Novara, Schanz and Schröder2023). To the best of the authors’ knowledge, this constitutes the first application of STB to investigate the effects of AJVGs on an LSB in a MAV-relevant aerofoil. The experimental campaign is conducted on an SD7003 aerofoil at Reynolds numbers of 30 000 and 80 000, for angles of attack of
$8^{\circ}$
,
$10^{\circ}$
and
$14^{\circ}$
. The analysis is structured in three parts: first, the three-dimensional vortex structures induced by the jets are characterised in detail; second, the modification of laminar separation in pre-stall conditions is assessed; and finally, the potential for stall suppression and reattachment at higher angles is investigated. The combination of advanced volumetric diagnostics with a systematic analysis of separation metrics offers new insight into the physics of active flow control at low Reynolds numbers.
2. Experimental set-up
2.1. Flow facility and model description
Experiments were conducted in a closed-loop water tunnel featuring a test section of
$0.50 \times 0.50$
m
$^2$
cross-sectional area and a total length of 2.5 m. The tunnel provides a steady flow with a controllable velocity range between 0.01 and 1 m s−1, and a free-stream turbulence intensity below 1 %. A schematic overview of the test facility and measurement configuration is provided in figure 1. The blockage ratio, evaluated at the highest tested angle of attack (
$14^{\circ}$
), was 4.2 %, remaining well below the 6 % threshold generally recommended to limit wall interference effects (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012).
Schematic representation of the experimental configuration. (a) Overview of the flow facility and measurement set-up. (b) Sketches illustrating the side and top views of the test section and measurement volume. Illustrations are not to scale for clarity.
The aerodynamic model employed is the SD7003 aerofoil, with a chord length of 120 mm and a span of 300 mm. The model was fabricated using high-resolution additive manufacturing, followed by manual polishing to ensure a smooth surface finish and reduce surface roughness. Circular endplates with a diameter of 350 mm were mounted on both sides of the aerofoil to suppress tip vortex formation and approximate two-dimensional flow conditions. This configuration follows the experimental layout used in Kubo, Miyazaki & Kato (Reference Kubo, Miyazaki and Kato1989). Detailed sketches of the test section, including side and top views of the model integration, are also shown in figure 1.
Geometric parameters of the JVG.
(a) Schematic view illustrating the definition of the skew angle (
$\theta$
), pitch angle (
$\phi$
), the free-stream velocity
$U_{\infty}$
and the jet exit velocity
$U_j$
. Not to scale for clarity. (b) Sectional view of the aerofoil showing the internal plenum and the jet array distributed along the span.
2.2. The JVG system
The aerofoil was equipped with an array of 14 jets distributed uniformly along the span. The geometric parameters of the jet vortex generators (JVGs) employed in this study are summarised in table 1. The skew and pitch angles were set to
$60^{\circ}$
and
$30^{\circ}$
(figure 2
a), respectively, as this combination has been shown in previous studies to promote the formation of strong and stable longitudinal vortices (Singh et al. Reference Singh, Peake, Kokkalis, Khodagolian, Coton and Galbraith2006).
Additionally, prior research has reported that vortex strength is strongly correlated with the jet-to-crossflow VR (Henry & Pearcey Reference Henry and Pearcey1994; Oliver Reference Oliver1997). Therefore, to maximise vortex intensity while minimising the required mass flow rate, the smallest reliably manufacturable jet diameter was selected, i.e. 1 mm.
The chordwise location of the jet array was chosen based on the separation point identified in the uncontrolled aerofoil at
$\alpha = 10^{\circ}$
, where
$\alpha$
is the angle of attack. The jets were positioned just upstream of this point, at
$x/c = 0.05$
where
$x$
is the chordwise coordinate measured from the leading edge, and
$c$
is the aerofoil chord length, ensuring vortex formation as close as possible to the onset of separation.
Finally, a spanwise spacing of 12 mm (
$0.1c$
) was adopted following successful implementations in similar active jet control studies. This distance was found to be sufficient to ensure vortex coherence and avoid destructive interference between neighbouring jets (Freestone Reference Freestone1985; Prince et al. Reference Prince, Badalamenti and Regas2017).
To supply the JVGs, a closed-loop pumping system was implemented. Water seeded with tracer particles was extracted directly from the tunnel and injected into the model to ensure proper visualisation of the jet structures within the measurement domain. The flow rate was regulated using an in-line flowmeter installed between the pump and the jet array.
To ensure uniform distribution of the flow across all jets, the aerofoil was internally equipped with a plenum chamber (figure 2 b). This chamber was fed by the pump through the flowmeter and operated under approximately constant pressure. The plenum was directly connected to all jet orifices, allowing simultaneous and steady injection across the entire spanwise array.
3. Methodology
3.1. Shake-the-Box configuration
The volumetric velocity measurements were acquired using a volumetric imaging system composed of four Imager CX3p-12 cameras (4080
$\times$
2984 px, 12-bit) equipped with integrated Scheimpflug mounts and Zeiss 50 mm f/1.4 lenses and a maximum acquisition rate of 54 Hz in double-frame mode at full resolution. Illumination was provided by a dual-cavity Nd:YAG laser (Spitlight PIV 600–100) operating at 100 Hz with a maximum pulse energy of approximately 210 mJ per cavity at 532 nm (
$\approx$
420 mJ total). The laser beam was routed through a 1.8 m articulated arm and shaped into a measurement volume using LaVision® volume optics with 5
$\times$
collimating lenses. A crosswise aperture was installed to enhance the homogeneity of the illumination across the entire volume.
A custom, laterally offset aerofoil-shaped mask was mounted on the lower endplate to fix the illuminated region, ensuring the laser sheet intersected the same measurement volume in the global frame for all angles of attack.
System synchronisation, camera control and data acquisition were handled via a LaVision® PTU X timing unit, interfaced with DaVis® 11 software. The working fluid was seeded with 60
$\unicode{x03BC}$
m polyamide particles (density 1.03 g cm−3) to enable accurate Lagrangian tracking. The water temperature was maintained at
$19.2\,^{\circ}$
C during the experiments with JVGs, and at
$21.5\,^{\circ}$
C for the clean (no JVG) cases used for comparison.
To match Reynolds numbers of 30 000 and 80 000, the free-stream velocities were set to
$U_{\infty } =$
0.24 m s−1 and 0.64 m s−1 for the clean configuration. In the jet-controlled cases, a lower water temperature required slightly higher velocities of
$U_{\infty } =$
0.256 m s−1 and 0.682 m s−1 to maintain the same Reynolds numbers. The measurement volume, illuminated with 150 mJ attenuated laser pulses, extended approximately
$130 \times 120 \times 45$
mm
$^3$
, covering the region from the leading edge up to approximately
$0.8c$
, encompassing the typical location of the LSB and the jet location. In the wall-normal direction, the illuminated volume was constrained to roughly
$0.1c$
using a masking system to restrict illumination to the flow region of interest, enabling higher seeding density while avoiding excessive image density and reducing ghost particle formation during reconstruction.
The cameras were arranged in a cross configuration, with adjacent optical axes forming angles of
$35^{\circ}$
, which improved triangulation accuracy. A numerical aperture f-number of 16 was selected to maintain an adequate depth of field (45 mm) throughout the illuminated domain. The chosen magnification resulted in a digital resolution of
$28.44$
px mm−1, with particle images spanning approximately 2.5 pixels in diameter.
System calibration was carried out by scanning five planar targets spaced 10 mm apart along the optical depth. A polynomial mapping function was used for initial calibration, followed by a volume self-calibration procedure (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006; Wieneke Reference Wieneke2008), which yielded an average disparity of 0.03 pixels and a maximum disparity of 0.08 pixels.
For each configuration, 250 double-frame recordings were acquired to compute statistically converged time-averaged velocity fields while keeping computational cost reasonable. All presented results, including derived quantities such as vorticity, Reynolds stresses, shape factor, and separation and reattachment locations, were obtained from these time-averaged fields. This averaging approach provides a clearer assessment of the flow modifications induced by the JVGs, enabling direct and consistent comparison between the controlled and uncontrolled configurations. Additionally, the dominant spanwise vortices were observed to be steady and coherent, supporting the use of averaged quantities for flow characterisation. The exposure time was set to
$40\,\unicode{x03BC} \textrm {s}$
, and the time between laser pulses (
$\Delta t$
) ranged from
$400\,\unicode{x03BC} \textrm {s}$
to
$800\,\unicode{x03BC} \textrm {s}$
, resulting in particle displacements of 4–10 pixels. The particle image density was maintained at approximately 0.01 ppp. Due to the larger calibration volume compared with the measurement domain, seeding density had to be kept low to avoid compromising calibration accuracy. Image preprocessing involved three main steps. First, a local intensity normalisation was applied using a moving average filter with a window size of 150 pixels, aiming to homogenise the brightness of particle images across the field of view. Next, a constant background offset of 80 counts was subtracted from all pixels to suppress residual background illumination. Finally, the intensity of all pixels was scaled by a factor of two to optimise bit depth usage and enable finer thresholding through virtually increased signal-to-noise ratio (SNR). The resulting images exhibited uniform particle intensities and negligible background noise, thereby improving the robustness of the particle detection and reconstruction process.
Particle tracking was conducted using the STB algorithm with four iterative passes. A two-dimensional detection threshold of 40 counts and a triangulation error of 1 voxel were employed. For the computation of Eulerian statistics, particle tracks were binned onto a grid using subvolumes of
$52 \times 52 \times 52$
voxels with 75 % overlap, producing a final grid resolution of 13 voxels, i.e. 0.457 mm or 0.38 % of the chord. Each instantaneous snapshot contained on average
$\approx 33$
track entries per bin, which over the full ensemble of 250 recordings results in a median of
$\approx 1.0\times 10^{4}$
entries per bin. This level is sufficient to capture reliable flow trends, but remains below the
$10^{5}$
–
$10^{6}$
entries per bin typically required for fully converged second-order statistics (Schröder et al. Reference Schröder, Schanz, Novara, Philipp, Geisler, Knopp, Schroll and Willert2018). Consequently, second-order statistics are reported here as indicative trends, while a fully converged evaluation would require a larger number of entries per bin.
The angle of attack was initially adjusted mechanically and later refined using iterative particle reconstruction (IPR) by tracking reference markers on the aerofoil surface. Table 2 reports the nominal and measured angles of attack for both the clean and jet-controlled configurations. Deviations from the nominal values are within
$0.3^{\circ}$
in all cases. Additionally, the relative difference between both configurations is
$0.1^{\circ}$
, indicating consistent alignment and negligible influence on the comparative results. Nominal values are used throughout the discussion and figures for clarity.
Nominal and measured angles of attack for clean and JVG configurations.
AoA = angle of attack.
$\Delta _{\textit{clean}}$
= deviation between the measured AoA and the nominal AoA in the clean case.
$\Delta _{\textit{JVG}}$
= deviation between the measured AoA and the nominal AoA in the JVG case.
$\Delta _{\textit{rel}}$
= difference in measured AoA between the clean and JVG cases.
3.2. Identification of separation bubble parameters
Figure 3 illustrates the three main parameters used to characterise the LSB: the separation point (
$x_s$
), transition point (
$x_{\textit{tr}}$
) and reattachment point (
$x_r$
). The separation point (
$x_s$
) was determined from STB particle trajectories by identifying a distinct deviation in particle paths and a local drop in velocity magnitude. This method proved particularly advantageous in the present set-up, where near-wall resolution was insufficient to accurately estimate the wall-normal velocity gradient, precluding the use of classical criteria such as those proposed by Schlichting & Kestin (Reference Schlichting and Kestin1961).
Schematic representation of the LSB parameters, including the separation point (
$x_s$
), transition point (
$x_{\textit{tr}}$
) and reattachment point (
$x_r$
). The contour shows the normalised streamwise velocity field,
$U/U_\infty$
, where
$U$
is the streamwise velocity component (
$U_x$
), overlaid with streamlines illustrating the flow topology. This figure is intended as an illustrative sketch of the LSB structure for
$\alpha = 10^{\circ}$
and
$Re = 30{\,}000$
clean case, and the marker positions may not exactly coincide with the values reported in the results.
The reattachment point (
$x_r$
) was extracted from the time-averaged velocity field by detecting the streamwise location at which the flow resumed a positive wall-parallel direction, indicating full boundary-layer recovery.
In contrast to previous studies in which the transition onset (
$x_{\textit{tr}}$
) was identified through the exponential growth of the normalised Reynolds shear stress
$(-\overline {u'w'}/U_\infty ^2)$
(Burgmann & Schröder Reference Burgmann and Schröder2008; Nati et al. Reference Nati, De Kat, Scarano and Van Oudheusden2015), where
$u'$
and
$w'$
denote the fluctuations of the streamwise and wall-normal velocity components, respectively, the present work adopts a different criterion based on the behaviour of the boundary-layer shape factor,
$H = \delta ^{\ast }/\theta$
, where
$\delta ^{\ast }$
is the displacement thickness and
$\theta$
is the momentum thickness. This change was motivated by the fact that, in the jet-controlled cases, the presence of actuation significantly altered the Reynolds stress distribution, making the exponential growth region difficult to identify reliably. Instead, the transition point was defined as the chordwise location where the shape factor
$H(x)$
reaches its maximum. This method was initially proposed by McAuliffe & Yaras (Reference McAuliffe and Yaras2005) and later validated through a comparative study by Burgmann & Schröder (Reference Burgmann and Schröder2008), who showed that both approaches yield nearly identical transition locations.
The bubble thickness (
$\delta _b$
) was defined as the vertical distance between the aerofoil surface and the point of maximum velocity at the chordwise location of the transition point,
$x_{\textit{tr}}$
.
3.3. Momentum coefficient and jet flow rate
The momentum coefficient (
$C_\mu$
) is widely used in the literature as a non-dimensional parameter to quantify the energy input associated with active flow control using jet-based actuators (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000; Shun & Ahmed Reference Shun and Ahmed2011; Liu et al. Reference Liu, Wei, Wan, Lin, Li and Tang2022). It represents the ratio of injected momentum to the free-stream dynamic pressure over the reference surface area and is defined as
\begin{equation} C_\mu = \frac {2 \dot {m} U_{\!j}}{\rho U_\infty ^2 A}, \end{equation}
where
$\dot {m}$
is the mass flow rate,
$U_j$
is the jet velocity,
$\rho$
is the fluid density,
$U_\infty$
is the free-stream velocity, and
$A$
is the reference area of the aerofoil.
Due to the lack of direct reliable measurements of the jet exit velocity (
$U_j$
), the momentum coefficient was estimated using the measured volumetric flow rate
$Q$
and the total exit area of the jets
$A_j$
. Assuming uniform and steady injection, the standard definition simplifies to
\begin{equation} C_\mu = \frac {2 Q^2}{U_\infty ^2 A A_{\!j}}. \end{equation}
All experiments in the present study were conducted at a constant momentum coefficient of
$C_\mu = 0.04$
, a value selected to ensure effective flow control while remaining within the range typically used in similar investigations. The corresponding volumetric flow rates required to achieve this
$C_\mu$
were
$59.02\,\textrm{L}\,\textrm{h}^{-1}$
and
$157.41\,\textrm{L}\,\textrm{h}^{-1}$
for the Reynolds numbers of 30 000 and 80 000, respectively.
4. Results
4.1. The JVG vortex structure
Understanding the structure of the vortices generated by JVGs is essential for interpreting their effect on boundary-layer re-energisation. While previous studies have largely relied on planar or numerical data, the present work provides, for the first time, a volumetric experimental characterisation of these vortex structures at low Reynolds numbers. This allows for a direct assessment of their spatial organisation and dynamics, laying the foundation for understanding their role in flow separation and stall delay.
(a) Visualisation of the vorticity field generated by the JVGs through normalised spanwise vorticity contours,
$\omega _x^{\ast } =\omega _x c / U_\infty$
in multiple
$y$
-
$z$
planes, where
$\omega_x$
is the streamwise vorticity component. (b) Three-dimensional view of the vortical structures induced by the JVGs, illustrated by isosurfaces of Q-criterion. The light grey dots indicate the approximate jet locations, shown for visualisation purposes. Case:
$\alpha = 10^{\circ}$
and
$Re = 80{\,}000$
.
Figure 4 illustrates the three-dimensional organisation of the vortical structures induced by the JVGs. Figure 4(a) displays the normalised spanwise vorticity
$\omega _x^{\ast }$
on several
$y$
-
$z$
planes, uniformly distributed along the chordwise direction at intervals of
$x/c = 0.1$
. At
$x/c = 0.1$
, the presence of a pair of counter-rotating vortices is observed. The vortex with positive sign (clockwise) exhibits a compact and well-defined circular core, with high vorticity magnitude. In contrast, the negative-sign vortex (counter-clockwise) forms closer to the surface, showing a more diffuse structure and significantly lower vorticity levels. As the vortices convect downstream, the counter-clockwise vortex dissipates rapidly and is no longer visible beyond
$x/c = 0.3$
. Apart from this difference within each vortex pair, there is also a mild variation along the span: the spanwise asymmetry in the counter-rotating vortices is attributed to slight manufacturing imperfections in the jet orifices, which result in variations in vortex strength and trajectory between jets.
The asymmetry of the vortex pairs is a direct consequence of the jet’s skew angle, which introduces a spanwise momentum bias that promotes the formation of a dominant vortex on one side and a weaker opposite-sign vortex on the other. In the configuration shown, the jets are skewed to the left (as seen in figure 2), resulting in a stronger, clockwise-rotating vortex forming on the leeward side of the jet. Conversely, the weaker, counter-clockwise vortex forms closer to the suction surface and on the windward side of the jet. This positioning exposes it directly to the high shear region of the free stream, where it is subjected to stronger strain rates and velocity gradients. As a result, it dissipates more rapidly than its counterpart, which remains partially shielded by the jet core and the induced upwash. Prior studies have shown that this asymmetry increases with larger skew angles (Toth et al. Reference Toth, Nichols, Vukasinovic, Glezer, DeFore, Chun, Butler and Harris2024). In contrast, symmetric pitch-only jets typically generate two equally strong counter-rotating vortices (Johnston Reference Johnston1999; Toth et al. Reference Toth, Nichols, Vukasinovic, Glezer, DeFore, Chun, Butler and Harris2024).
Figure 4(b) shows isosurfaces of the Q-criterion, highlighting the dominant streamwise vortices generated by the jets. Due to the applied threshold, only the strongest (clockwise) vortices are visible. These structures originate near
$x/c = 0.05$
, i.e. the chordwise position of the jets, and propagate downstream at a certain distance from the surface. As they travel, they gradually weaken and show a slight lateral deviation aligned with the jet’s skew orientation, rather than following the exact direction of the external flow.
To further illustrate the three-dimensional development of the dominant vortex, figure 5 presents a volumetric rendering of the vortex core using Q-contours along with streamlines seeded near the jet exit. This visualisation indicates the presence of a coherent streamwise vortex originating at the jet location (
$x/c \approx 0.05$
). In figure 5(a), the vortex emerges from the orifice and follows a curved trajectory, maintaining a certain distance from the surface that increases slightly as it convects downstream. The streamlines highlight the helical motion induced by the vortex and their smooth and continuous paths indicate a well-organised rotational structure.
Figure 6 presents the
$U_y$
$U_z$
velocity vector field in the
$y$
–
$z$
plane at
$x/c = 0.1$
, where
$U_y$
and
$U_z$
denote the spanwise and wall-normal velocity components, respectively, overlaid with scalar fields of key flow quantities. Figure 6(a) shows the normalised streamwise vorticity
$\omega _x^{\ast } = \omega _x c / U_\infty$
, which reveals the presence of coherent vortical structures generated by two adjacent jets. The flow induced by each jet forms a pair of counter-rotating vortices, hereafter referred to as vortex pairs A and B. Within each pair, the dominant vortex rotates clockwise (CW) and appears as a region of positive
$\omega _x^{\ast }$
, shown in red. The weaker counter-clockwise (CCW) vortex is identified by a localised region of negative
$\omega _x^{\ast }$
, shown in blue. These structures are anchored near the surface, with the CW vortex located slightly farther from the wall than its CCW counterpart.
Visualisation of a streamwise vortex generated by a single JVG. Panel (a) shows a side view from the positive
$y$
side, while (b) presents an oblique perspective view. The vortex core is identified using isosurfaces of Q-criterion, while the helical trajectory induced by the vortex is illustrated through streamlines. The streamlines are computed from the bin-averaged velocity field and do not represent individual particle trajectories. Case:
$\alpha = 10^{\circ}$
and
$Re = 80{\,}000$
.
The
$U_y$
-
$U_z$
velocity vector fields in the
$y$
-
$z$
plane at
$x/c = 0.1$
, illustrating the flow induced by two adjacent JVGs. Each vector field is overlaid with scalar contours of a different flow quantity: (a) normalised streamwise vorticity
$\omega _x^{\ast } = \omega _x c / U_\infty$
, (b) normalised streamwise velocity
$U_x^{\ast } = U_x/U_\infty$
, where
$U_x$
is the streamwise velocity component (c) normalised spanwise velocity
$U_y^{\ast } = U_y/U_\infty$
and (d) normalised vertical velocity
$U_z^{\ast } = U_z/U_\infty$
. In (a), the letters A and B identify the pair of counter-rotating vortices generated by each jet. Case:
$\alpha = 10^{\circ}$
and
$Re = 80{\,}000$
.
Notably, the locations of maximum
$\omega _x^{\ast }$
do not coincide with the apparent vortex cores seen in the velocity field. This offset has been reported in previous studies (Toth et al. Reference Toth, Nichols, Vukasinovic, Glezer, DeFore, Chun, Butler and Harris2024) and is attributed to residual jet-induced shear distorting the local vorticity distribution. As the shear weakens farther downstream, the displacement between the velocity and vorticity centres tends to diminish.
The
$U_y$
-
$U_z$
velocity vector fields in the
$y$
-
$z$
plane at
$x/c = 0.1$
, overlaid with (a) the normalised average kinetic energy contour,
$\textit{AKE}^{\ast }= \textit{AKE} / U_{\infty}^2$
, and (b) the normalised turbulent kinetic energy contour,
$\textit{TKE}^{\ast }= \textit{TKE} / U_{\infty}^2$
, where AKE and TKE denote the average and turbulent kinetic energies, respectively. Case:
$\alpha = 10^{\circ}$
and
$Re = 80{\,}000$
.
The
$(U_y, U_z)$
velocity vectors and the
$U_z^{\ast }$
contours in figure 6(d), reveal the upward motion produced by the jet, corresponding to the vertical momentum injection. This upward motion generates a counter-rotating vortex pair: to its right, the dominant clockwise (CW) vortex, and to its left, a weaker counter-clockwise (CCW) vortex. At this streamwise location (
$x/c = 0.1$
), the CW vortices remain coherent and produce well-defined rotational patterns in the velocity field. In contrast, the CCW vortices have diffused significantly and do not generate clearly identifiable velocity structures, although their footprints persist in the
$\omega _x^{\ast }$
field (figure 6
a).
The CW vortices induce a characteristic downwash toward the wall (i.e. a vertical component of velocity directed downward), observable as a blue region of negative
$U_z^{\ast }$
to the right of their cores (figure 6
d), and a spanwise sweep along the surface, visible as a red region of positive
$U_y^{\ast }$
directly beneath the vortex cores (figure 6
c). These flow components – downward momentum transfer and near-wall lateral advection – will later be shown to play a central role in the re-energisation of the separated boundary layer.
4.2. Mechanism of boundary-layer re-energisation
Moreover, the set of visualisations presented in figures 6 and 7 provides insight into the mechanism by which the boundary layer is re-energised. The
$U_x^{\ast }$
contours (figure 6
b) show a local increase in streamwise velocity near the core of each dominant CW vortex. This indicates that high-momentum fluid from the free stream is advected into the vortex, enhancing the local axial momentum. A similar, though weaker, peak is observed near the expected location of the CCW vortices, particularly in vortex pair A, around
$y/c \approx 0.04$
, suggesting that some entrainment persists despite the lower coherence of the structure at this downstream position.
While the local velocity peaks reflect entrainment of high-momentum fluid near the vortex cores, the
$U_x^{\ast }$
field in figure 6(b) also exhibits a low-velocity region near the wall, corresponding to the separated boundary layer. However, this region becomes narrower or locally suppressed between approximately
$y/c = -0.09$
and
$-0.10$
, and again near
$y/c = -0.01$
to
$0.01$
, both located beneath the CW vortices. In the same spanwise intervals, the
$U_y^{\ast }$
field in figure 6(c) displays local maxima, indicating enhanced spanwise velocity directed along the positive
$y$
-axis, consistent with the rotational sense of the CW vortices. This pattern results from the downwash induced by the vortices (see figure 6
d), which transports high-momentum fluid from the free stream toward the wall. Upon reaching the low-momentum near-wall region, this flow is redirected laterally due to the velocity gradient across the boundary layer. This lateral swiping motion contributes to the redistribution of momentum within the separated shear layer, locally reducing or suppressing flow separation. A similar, though less pronounced, lateral motion is also observed near the inferred location of the secondary CCW vortices, particularly around
$y/c \approx 0.04$
and
$y/c \approx -0.05$
. Although these structures are weaker and less coherent, the associated spanwise velocity peaks suggest that they also contribute to the redistribution of momentum in the near-wall region.
Three-dimensional visualisation of the interaction between jet-induced vortices and the boundary layer, using isosurfaces of normalised velocity components. All panels display the
$U_t^{\ast } = U_t / U_\infty = 0$
surface (blue) and Q-criterion isosurfaces (red, threshold = 4 % of
$Q_{max }$
), where
$U_t$
is the tangential velocity component along the aerofoil surface and
$Q_{max}$
is the maximum value of the Q-criterion used for thresholding, combined with (a) no additional component, (b)
$U_z^{\ast } = U_z / U_\infty = -0.27$
(green) and (c)
$U_y^{\ast } = U_y / U_\infty = 0.18$
(purple). Case:
$\alpha = 10^{\circ}$
and
$Re = 30{\,}000$
.
The distribution of
$\textit{AKE}^{\ast }$
, defined as
$\textit{AKE}^{\ast } = (1/2)|\boldsymbol{V}_{\!\textit{a}v\textit{g}}|^2 / U_{\infty }^{2}$
, where
$V_{avg}$
is the time-averaged velocity magnitude, is shown in figure 7(a). Localised maxima of
$\textit{AKE}^{\ast }$
appear at the vortex cores, consistent with the presence of high-speed fluid in these regions. In contrast, reduced values of
$\textit{AKE}^{\ast }$
are observed near the wall, corresponding to the low-momentum region associated with the separated boundary layer. The velocity vectors overlaid on the
$\textit{AKE}^{\ast }$
field suggest that fluid with high
$\textit{AKE}^{\ast }$
is deflected downward by the vortex-induced motion and directed toward the wall, consistent with the downwash pattern previously observed in
$U_z^{\ast }$
. This motion brings high-momentum fluid into regions of low average velocity, thereby increasing the
$\textit{AKE}$
of the near-wall region and supporting the re-energisation of the boundary layer.
Additional insight is provided by the
$\textit{TKE}^{\ast }$
field shown in figure 7(b), computed as
${\textit{TKE}^{\ast }} = ( {1}/{2})( \overline {u'^2} + \overline {v'^2} + \overline {w'^2})/ U_{\infty }^2$
, where
$v'$
is the fluctuation of the spanwise velocity component, representing the energy associated with velocity fluctuations, highlights regions of enhanced mixing. In this case, near-wall
$\mathrm{TKE}^{\ast }$
peaks occur directly beneath the vortex core and, as also noted by Khan & Johnston (Reference Khan and Johnston2000), are particularly pronounced in the upwash region, indicating that the interaction between the vortex and the boundary layer promotes localised turbulent activity.
Figure 8 provides a three-dimensional visualisation of the re-energisation mechanism induced by the jet-generated vortices and their interaction with the boundary layer. In figure 8(a), the dominant vortices generated by the jets are visualised using Q-criterion isosurfaces, along with surfaces where the tangential velocity component
$U_t^{\ast }$
vanishes, which have been used to identify the separated region on the suction side of the aerofoil. The flow exhibits a highly three-dimensional structure, with significant variation along the spanwise direction. It can be observed that the vortices interact directly with the boundary layer, substantially reducing, and in some cases locally suppressing, separation beneath the jets. In contrast, separation is only partially alleviated in the regions between jets, where the influence of the vortices is weaker. It is also noticeable that not all jets exhibit the same effect on the boundary layer, which is attributed to slight manufacturing imperfections in the jet orifices, leading to variations in vortex strength and behaviour across the span, as also observed in the vortex structures shown in figure 4. Figure 8(b,c) further illustrate the momentum exchange mechanism previously described. Figure 8(b) adds isosurfaces of the vertical velocity component
$U_z^{\ast }$
to the Q-criterion and
$U_t^{\ast }=0$
surfaces. In agreement with the two-dimensional planar results shown in figure 6, a strong downwash induced by the dominant clockwise vortex is visible to the right of each vortex core as identified by the Q-criterion. Notably, the
$U_z^{\ast }$
isosurface exhibits a conical shape that widens downstream, in contrast to the Q-criterion surface which progressively weakens. This implies that although the rotational intensity of the vortex decreases as it convects downstream (as indicated by the decreasing
$Q$
value), the physical size of the vortex grows, increasing its spatial footprint and thereby enhancing the downwash effect on the near-wall region. Figure 8(c) shows isosurfaces of the spanwise velocity component
$U_y^{\ast }$
, which illustrate how high-momentum fluid, driven downward by the vortex, impinges on the boundary layer and is subsequently redirected laterally along the span. These regions of elevated
$U_y^{\ast }$
coincide with the areas where separation is suppressed, as seen in figure 8(a) (blue isosurface), suggesting that the momentum transfer into the near-wall region occurs through a spanwise sweeping motion. Furthermore, as with the
$U_z^{\ast }$
isosurfaces, the spanwise motion persists even as the vortex weakens, i.e. as the Q-criterion value decreases. However, the shape of the
$U_y^{\ast }$
isosurfaces remains relatively constant, or even diminishes downstream. This behaviour is attributed to the increasing distance between the vortex core and the surface, which reduces the vortex’s ability to interact with and energise the near-wall region, resulting in a decrease in spanwise velocity near the surface.
4.3. Separation behaviour in pre-stall conditions
This section presents a comparative analysis of boundary-layer separation for the clean and JVG-configured cases at two angles of attack,
$\alpha = 8^{\circ}$
and
$\alpha = 10^{\circ}$
, and for Reynolds numbers
$Re = 30{\,}000$
and
$Re = 80{\,}000$
. Under these conditions, the clean configuration is known to develop a well-defined LSB on the suction side of the aerofoil, as extensively documented in the literature (Burgmann, Dannemann & Schröder Reference Burgmann, Dannemann and Schröder2008; Burgmann & Schröder Reference Burgmann and Schröder2008; Zhang, Hain & Kähler Reference Zhang, Hain and Kähler2008; Nati et al. Reference Nati, De Kat, Scarano and Van Oudheusden2015). These cases were therefore selected to assess the capability of skewed JVGs to reduce separation and mitigate the extent of the LSB under pre-stall conditions.
As discussed in the previous section, the flow in the JVG cases exhibits highly three-dimensional features due to the presence of streamwise vortices. To ensure a consistent and meaningful comparison with the clean configuration, data for the JVG cases were extracted from a fixed spanwise plane located at
$y/c = -0.03$
, corresponding to the midpoint between two adjacent jets. This location avoids regions of strong jet interference and offers a representative view of the local boundary-layer development. Although the flow is not spanwise invariant, preliminary inspection of neighbouring planes confirmed that the trends observed at this location are qualitatively consistent across the central region of the span.
Comparison of normalised streamwise velocity profiles,
$U^{\ast } = U / U_\infty$
, between the uncontrolled and jet-controlled configurations. The profiles are extracted from the
$y/c = -0.03$
spanwise plane.
Figure 9 provides a visual representation of the effectiveness of JVGs in mitigating the separated flow region, shown through contours of normalised velocity,
$U^{\ast }$
, at
$Re = 30{\,}000$
and angles of attack
$\alpha = 8^{\circ}$
and
$10^{\circ}$
.
In the clean configuration, the development of a LSB is visible on the suction side of the aerofoil, characterised by an extended region of reversed or low-speed flow near the surface. As the angle of attack increases from
$8^{\circ}$
to
$10^{\circ}$
, both the length and thickness of this low-velocity region increase, indicating stronger separation and delayed reattachment.
In contrast, the cases with JVGs exhibit a marked reduction in the extent of the separated region. The low-speed zone is significantly suppressed both in the wall-normal direction and along the chordwise coordinate, indicating a re-energised boundary layer with enhanced resistance to the adverse pressure gradient. Moreover, while the clean configuration shows a substantial increase in the size of the separated region from
$8^{\circ}$
to
$10^{\circ}$
, particularly in its thickness, the JVG cases display negligible variation between the two angles of attack.
(a) Separation point (
$x_s$
), transition point (
$x_{\textit{tr}}$
), reattachment point (
$x_r$
), (b) bubble length (
$l_b$
), where l is the physical bubble length, and (c) bubble thickness (
$\delta _b$
) for both the uncontrolled and jet-controlled configurations. Results are shown for angles of attack
$\alpha = 8^{\circ}$
and
$10^{\circ}$
, and Reynolds numbers
$Re = 30{\,}000$
and
$80{\,}000$
. All values are extracted from the spanwise plane at
$y/c = -0.03$
.
Figure 10 shows the main parameters characterising the LSB for both the clean and JVG-configured cases. These values provide a quantitative view of how the presence of the jets modifies the separation behaviour across the different flow conditions.
Focusing on the plots in figure 10(a), which display the separation point
$x_s$
, transition point
$x_{\textit{tr}}$
and reattachment point
$x_r$
for each configuration, a consistent trend is observed across all four tested cases. The introduction of JVGs leads to a clear upstream shift of all three characteristic points, indicating that the LSB is displaced closer to the leading edge in the presence of control. Moreover, it is noteworthy that the separation point
$x_s$
also shifts upstream in the JVG-controlled cases, particularly at
$\alpha = 8^{\circ}$
.
To further analyse this effect, figure 10(b) shows the bubble length, defined as the distance between reattachment and separation points. At
$Re = 30{\,}000$
, the presence of JVGs leads to a significant reduction in bubble length, consistent with a more compact separation region. However, at
$Re = 80{\,}000$
, the bubble length slightly increases in the controlled case, which may appear counterintuitive given the expectation of a more energised boundary layer.
This apparent discrepancy can be clarified by examining the bubble thickness, plotted in figure 10(c). In all controlled cases, the bubble thickness is consistently reduced, indicating a substantial mitigation of the separated region in the wall-normal direction. Furthermore, both in the clean and controlled configurations, a nearly linear correlation is observed between bubble length and thickness, consistent with previous observations reported in Burgmann & Schröder (Reference Burgmann and Schröder2008). While the number of test cases is limited, the JVG-controlled configurations exhibit noticeably reduced variation in bubble length
$l_b$
and thickness
$\delta _b$
in response to changes in Reynolds number and angle of attack, compared with the clean cases.
In addition to the discrete metrics presented, a more comprehensive assessment of the boundary-layer behaviour is provided by the shape factor
$H = \delta ^{\ast }/\theta$
. While most previous studies have focused on parameters such as momentum or displacement thickness (Liu et al. Reference Liu, Wei, Wan, Lin, Li and Tang2022),
$H$
offers a more compact and sensitive indicator of the flow state. Physically, a high shape factor corresponds to a velocity profile with a relatively large displacement thickness and a thin momentum core – typical of low-momentum boundary layers prone to separation. In contrast, lower values indicate fuller, more energised profiles associated with attached or turbulent flow.
Beyond its sensitivity,
$H$
enables a clearer and continuous identification of the LSB and provides a convenient metric to quantify the impact of the JVGs on its extent and stability across varying flow conditions.
Chordwise distribution of the shape factor
$H = \delta ^{\ast }/\theta$
in the spanwise plane at
$y/c = -0.03$
. Results are shown for angles of attack
$\alpha = 8^{\circ}$
and
$10^{\circ}$
, and Reynolds numbers
$Re = 30{\,}000$
and
$80{\,}000$
, comparing the controlled (orange) and uncontrolled (blue) cases.
Figure 11 presents the distribution of
$H(x)$
for all tested cases. The plots reveal a consistent reduction in the peak shape factor when JVGs are active, indicating improved boundary-layer performance. This reduction is most significant for the low-Reynolds-number, high-angle-of-attack condition (
$Re = 30{\,}000$
,
$\alpha = 10^{\circ}$
), where the maximum value of
$H$
decreases by 45.6 %. Conversely, under the most favourable condition (
$Re = 80{\,}000$
,
$\alpha = 10^{\circ}$
), the increase in
$H$
within the separated region becomes negligible, suggesting that separation is nearly suppressed in the controlled case.
Additionally, the previously observed phenomenon of upstream shift of the separation bubble is also evident in these plots. The rise in
$H$
begins earlier in the JVG cases, indicating an earlier onset of separation. However, the peak and subsequent decay of
$H$
also occur sooner, implying earlier transition and reattachment locations.
The results indicate that the role of the JVGs in pre-stall conditions is not necessarily to prevent separation altogether, but rather to stabilise the separation bubble, reducing its intensity and sensitivity to external conditions. Similar behaviour has been described in studies on active flow control, where actuation reshapes the separated region into a shorter and more stable structure rather than eliminating it entirely (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000; Scholz et al. Reference Scholz, Casper, Ortmanns, Kähler and Radespiel2008; Salunkhe et al. Reference Salunkhe, Tang, Zheng and Wu2016). The upstream shift of
$x_s$
observed here, although not explicitly reported in these previous works, may be interpreted as part of a control mechanism that induces an earlier but weaker separation, immediately followed by rapid transition and reattachment, consistent with the formation of energised, transitional shear layers. The combined analysis of
$x_s$
,
$x_{\textit{tr}}$
and
$x_r$
locations, along with the evolution of bubble length, thickness and the shape factor
$H(x)$
, reveals a consistent trend across the tested conditions: the JVGs do not eliminate the LSB, but rather reshape it into a thinner, more stable and less sensitive structure.
4.4. Stall suppression
This section examines the effect of JVGs on flow separation at an angle of attack
$\alpha = 14^{\circ}$
, a condition under which the clean configuration exhibits full stall. The analysis focuses on the controlled case, where flow attachment is partially maintained due to the action of the skewed jets. The objective is to characterise the boundary-layer behaviour in this delayed-stall scenario, and to evaluate how the JVGs modify the structure and extent of separation under conditions that would otherwise result in fully separated flow. As in the previous cases, the analysis is based on data extracted from the spanwise plane located at
$y/c = -0.03$
, corresponding to the midpoint between two adjacent jets.
Figure 12 shows the normalised velocity contour,
$U^{\ast }$
, for the case
$\alpha = 14^{\circ}$
and
$Re = 30{\,}000$
. As seen in figure 12(a), the clean configuration is in full stall, characterised by boundary-layer separation near the leading edge and sustained detachment along the entire suction side, resulting in a large, low-momentum recirculation region. In contrast, the case with JVGs (figure 12
b) exhibits an attached velocity profile, with a small separation bubble forming near the leading edge, although visibly larger than in the previously analysed angles of attack, i.e.
$8^{\circ}$
and
$10^{\circ}$
, and progressively reattaching downstream. This indicates that the control strategy effectively delays the onset of large-scale separation, maintaining flow attachment over most of the chord.
Comparison of normalised streamwise velocity contours,
$U^{\ast }$
, between the (a) uncontrolled and (b) jet-controlled cases at
$\alpha = 14^{\circ}$
and
$Re = 30{\,}000$
. The data are extracted in the spanwise plane at
$y/c = -0.03$
. The velocity field illustrates the stall delay induced by the JVGs.
(a) Separation point (
$x_s$
), transition point (
$x_{\textit{tr}}$
), reattachment point (
$x_r$
), (b) bubble length (
$l_b$
) and (c) bubble thickness (
$\delta _b$
) for the jet-controlled cases at
$\alpha = 14^{\circ}$
and Reynolds numbers
$Re = 30{\,}000$
and
$80{\,}000$
. No comparison is shown with the uncontrolled configuration, as the flow is fully stalled in that case. All values are extracted from the spanwise plane at
$y/c = -0.03$
.
Figure 13 presents the main separation bubble parameters for the controlled case at
$\alpha = 14^{\circ}$
and two Reynolds numbers. As observed, the bubble exhibits the same trends previously reported for uncontrolled LSBs (Burgmann & Schröder Reference Burgmann and Schröder2008): as the Reynolds number increases, the separation point shifts slightly upstream. Similarly, the upstream movement of the transition and reattachment points is attributed to the enhanced shear across the separated shear layer. This steeper velocity gradient promotes faster amplification of Kelvin–Helmholtz instabilities, which accelerates the transition to turbulence and ultimately facilitates earlier reattachment through enhanced momentum mixing. Moreover, the bubble length and thickness, as previously discussed in the pre-stall analysis, maintain an approximately linear relationship and both decrease with increasing Reynolds number.
Chordwise distribution of the shape factor
$H = \delta ^{\ast }/\theta$
in the spanwise plane at
$y/c = -0.03$
for the jet-controlled cases at
$\alpha = 14^{\circ}$
and Reynolds numbers
$Re = 30{\,}000$
and
$80{\,}000$
. No comparison is provided with the uncontrolled configuration, as the clean case is fully stalled under these conditions.
Figure 14 shows the distribution of the shape factor
$H(x)$
for the controlled case at
$\alpha = 14^{\circ}$
and both Reynolds numbers. As previously indicated in the bubble characteristics plot (figure 13), the separation bubble at
$Re = 30{\,}000$
is considerably longer than at
$Re = 80{\,}000$
. However, the
$H$
factor reveals a different aspect of the flow: the peak value of
$H$
is significantly higher for
$Re = 80{\,}000$
, indicating a much stronger recirculation zone concentrated near the leading edge. Despite its intensity, the separated region dissipates rapidly downstream, as reflected by the steep drop in
$H$
, showing the compact nature of the bubble at higher Reynolds number. In fact, in both cases, the
$H$
factor eventually decreases to values close to 2, consistent with a fully attached and likely turbulent boundary layer downstream of reattachment. This behaviour aligns with previous findings on active stall mitigation (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000; Scholz et al. Reference Scholz, Casper, Ortmanns, Kähler and Radespiel2008), where actuation strategies similarly promoted early reattachment and partial flow recovery under conditions of severe separation.
5. Conclusions
This study presents an experimental investigation of active flow control using skewed JVGs on an SD7003 aerofoil at low Reynolds numbers. Volumetric measurements obtained via STB provided the first experimental characterisation of the three-dimensional vortex structures generated by JVGs in this regime. The jets produced asymmetric counter-rotating streamwise vortices, with the dominant structure entraining high-momentum fluid from the free stream and convecting it downward toward the wall. This fluid then swept spanwise along the surface, displacing low-momentum recirculating regions and replacing them with higher-momentum flow, forming the physical basis for the observed flow control effects.
In pre-stall conditions, the JVGs did not eliminate laminar separation but reconfigured the separation bubble into a thinner and more stable structure. The separation, transition and reattachment points consistently shifted upstream. At
$Re = 30{\,}000$
, the bubble length was significantly reduced, while at
$Re = 80{\,}000$
it increased slightly. However, in both cases, the bubble thickness decreased markedly, and the shape factor
$H$
exhibited a noticeably lower peak followed by faster recovery, indicating clear re-energisation of the boundary layer. Therefore, rather than preventing separation entirely, the JVGs reshape the LSB into a more compact and robust form, reducing its sensitivity to flow conditions and improving flow stability.
Under stall conditions (
$\alpha = 14^{\circ}$
), the clean case exhibited full separation with no reattachment. In contrast, the JVG-equipped configuration showed partial reattachment, forming a short separation bubble near the leading edge. This bubble shortened with increasing Reynolds number, consistent with the behaviour of a conventional LSB. These results indicate that JVGs can transform massive separation into a shorter and more stable flow configuration, effectively contributing to stall delay in low-Reynolds-number regimes.
Funding
This research was support by the Government of the Basque Country, Elkartek KK-2024/00069, KK-2024/00117 and ELKARTEK25/28 DBaskIN; and CIEMAT, Energía eólica offshore para el ensayo y el desarrollo energético de energías renovables e hidrógeno verde; the work of U.F.G. was partially supported by the Government of the Basque Country, ITSAS-REM (IT1514-22).
Declaration of interests
The authors declare no conflicts of interest.
Authors contributions
Equal contributions to the conceptualisation of the study were made by Y.P., X.U-G., I.U., U.F-G. and A.B. Data curation was carried out by Y.P., X.U-G., I.U. and U.F-G. Formal analysis was performed by Y.P., X.U-G. and U.F-G., while the investigation was conducted by Y.P. and X.U-G. Software development was undertaken by Y.P., and visualisation was prepared by X.U-G. Methodology was developed by I.U. Project administration was coordinated by I.U., U.F-G. and A.B. Resources were provided by A.B. Funding acquisition was managed by U.F-G. and A.B., and supervision was carried out by U.F-G. The original draft of the manuscript was written jointly by all authors, and all authors contributed equally to reviewing and editing the final version.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Open access
