Vortex Dynamics of a Thick Sinusoidally Pitching Airfoil at Low Reynolds Numbers in the Presence and Absence of Synthetic Jets

1. Introduction

1.1. The Dynamic Stall on a Pitching Airfoil

The dynamic stall phenomenon on airfoils and lifting surfaces is both a practically important and fundamentally challenging problem that arises in unsteady flow environments, including helicopter rotor blades, rapidly maneuvering aircraft, jet engine compressor blades, wind turbines, and even biological propulsion systems such as fish tails and insect wings [1,2]. For a static airfoil, when the angle of attack reaches a critical value ${\alpha }_s$, the adverse pressure gradient downstream of the suction peak inevitably causes the flow to break away from the surface, leading to static stall, which manifests as a reduction in lift and an increase in drag. Dynamic stall occurs when the effective angle of attack exceeds the static stall angle through a time-varying motion, such as pitching, ramping, or plunging, where the time scales of the airfoil motion are of similar order to or shorter than the time scales of the flow [1,3,4,5]. Early studies on the subject identified the formation of an energetic vortex on the suction surface, referred to as the dynamic stall vortex (DSV), as the predominant feature of the phenomenon [6,7,8,9]. The convection of the DSV along the chord temporarily delays stall onset; however, it is followed by abrupt flow separation and substantial excursions of highly unsteady aerodynamic loads once the vortex is shed into the wake. These impulsive load changes, which can be several times greater than those predicted under static conditions, are not only a major concern for aerodynamic performance but also induce vibrations, accelerate fatigue, and potentially cause structural damage to the airfoil and its supporting components [10]. Dynamic stall is therefore characterized by a highly nonlinear, fluctuating pressure field and pronounced load hysteresis, fundamentally distinct from the response observed under static stall conditions [11,12,13,14]. To simplify the dynamic stall problem, it has become common practice to investigate the canonical case of an airfoil undergoing sinusoidal pitching about its aerodynamic center, which for symmetric airfoils in subsonic flow lies at the quarter-chord point measured from the leading edge, though some studies suggest that pitching and plunging motions are equivalent when effective angles of attack are matched [15]. The sinusoidal pitching motion is generally appropriate for rotorcraft and vertical-axis wind turbines, as the temporal evolution of the effective angle of attack during the phase of the rotation cycle in which dynamic stall occurs is well approximated by a sinusoidal oscillation [16,17,18]. The flow field around a pitching airfoil is illustrated in Figure 1(a), where $(x,y,z)$ denote the global Cartesian coordinates, $\alpha \left(t\right)=\overline{\alpha }+\Delta \alpha sin\left(2\pi f_{\alpha }t\right)$ is the instantaneous angle of attack, $f_{\alpha }$ is the pitching frequency, c is the chord length, and $U_{\infty }$ is the freestream velocity.

Figure 1. Schematic of a pitching-airfoil configuration and a synthetic-jet actuator: (a) global coordinates, chord length c, freestream velocity $U_{\infty }$, and instantaneous angle of attack $\alpha$ for a sinusoidally pitching airfoil, and (b) synthetic-jet actuator and notations, including cavity, diaphragm, orifice diameter d, jet velocity $v_j$, and excitation frequency $f_j$.

Figure 1. Schematic of a pitching-airfoil configuration and a synthetic-jet actuator: (a) global coordinates, chord length c, freestream velocity $U_{\infty }$, and instantaneous angle of attack $\alpha$ for a sinusoidally pitching airfoil, and (b) synthetic-jet actuator and notations, including cavity, diaphragm, orifice diameter d, jet velocity $v_j$, and excitation frequency $f_j$.

For a static airfoil, the flow regime primarily depends on the fixed angle of attack $\alpha =\overline{\alpha }$ and the chord-based Reynolds number $Re=U_{\infty }c/\nu$, where $\nu$ is the kinematic viscosity of the fluid. Depending on these parameters, the boundary layer may remain fully attached, become fully separated, or exhibit partial separation [19]. The angle of attack and Reynolds number have qualitatively similar, yet opposing, effects on the flow state. At low Reynolds numbers, the laminar boundary layer near the leading edge is prone to separation under an adverse pressure gradient. At the critical stall angle ${\alpha }_s$, when reattachment is not possible, a turbulent wake develops on the suction surface after a laminar-to-turbulent transition of the separated shear layer [20]. Prior to stall, however, the separated shear layer sometimes reattaches to the airfoil surface after separation-induced transition, forming a localized recirculating region known as a laminar separation bubble (LSB). LSBs have been documented over a wide range of Reynolds numbers and for both thin and thick airfoils [21,22]. Depending on the bubble-to-chord-length ratio, a bubble is classified as either long or short [23,24]. An increase in angle of attack may cause a short bubble to expand into a long one. An LSB eventually bursts at the critical stall angle ${\alpha }_s$, when reattachment is no longer possible. Although LSBs are detrimental to aerodynamic performance, the drastic loss-of-lift signature of stall is experienced only when the separated shear layer fails to reattach at the critical angle ${\alpha }_s$ [25].

The sinusoidal variation of the angle of attack introduces two additional kinematic parameters to the problem, namely the reduced pitching frequency $k=\pi f_{\alpha }c/U_{\infty }$ and the oscillation amplitude $\Delta \alpha$. Typical flow evolution stages within a dynamic stall cycle include attached flow, stall development, a fully stalled phase, and flow reattachment, with the presence and duration of each stage determined primarily by the kinematic parameters, Reynolds number, and airfoil geometry, among other secondary factors [26,27,28]. The airfoil kinematics in the present study are restricted to a zero mean angle of attack $\overline{\alpha }=0$, representative of conditions encountered in vertical-axis wind turbines. In this configuration, the Reynolds number dictates the static stall angle, while the oscillation amplitude determines the extent of stall penetration and the severity of flow separation [26]. Increasing either the reduced frequency or the oscillation amplitude generally delays the onset of dynamic stall; however, variations in reduced frequency can substantially alter the shape of the hysteresis loop, whereas increasing the oscillation amplitude enlarges the enclosed area [2,28,29,30,31]. The Reynolds number and airfoil geometry also play a significant role in the complex underlying viscous–inviscid interactions. On thick airfoils, or sometimes thin airfoils if the pitch frequency is slow [5,32], a thin layer of reverse flow forms at the trailing edge and gradually progresses upstream, whereas on thin airfoils, flow reversal usually first emerges locally near the leading edge, close to the suction peak [26]. At low Reynolds numbers, the shear layer above the viscous–inviscid interface may become susceptible to instabilities, causing the shear layer to roll up into coherent structures known as shear-layer vortices. A shear-layer vortex (SLV) may shed prematurely or merge with the primary vortex through complex interactions, and its formation may persist even after the primary vortex has been shed [10,33,34,35,36,37]. At low-to-moderate Reynolds numbers, an LSB may form prior to stall, typically near the leading edge of thin airfoils.

In general, while the main characteristics of dynamic stall are well established, the mechanisms leading to the formation and detachment of the primary vortex remain an active area of research owing to the complex interplay among flow, kinematic, and geometric parameters [36,37,38]. Two widely accepted explanations of incompressible dynamic stall have been proposed, namely the Van Dommelen–Shen process [39] and the bubble-bursting mechanism [40,41]. The Van Dommelen–Shen interaction, which describes unsteady boundary-layer separation, is speculated to be Reynolds-number independent, beginning with near-wall flow reversal triggered by a strong adverse pressure gradient and continuing with the formation of a zero-vorticity line that progressively lifts away from the wall. The process culminates in streamwise compression of converging forward- and reverse-flow particles along this line, where inertial effects dominate viscous and pressure-gradient terms, ultimately leading to fluid eruption away from the wall and boundary-layer separation. The bubble-bursting mechanism, which is considered an extension of the same process leading to airfoil stall in steady or quasi-steady conditions, links the formation of the DSV to the bursting of an LSB [40,42].

Although the role of bubble bursting in the inception of dynamic stall is well established, stall does not necessarily originate from LSB bursting, and stall may instead be initiated by turbulent separation downstream of the reattachment point [4,13,29,43]. According to Carr et al. [13], in the former mechanism, referred to as leading-edge bubble-bursting stall, the LSB contracts with increasing angle of attack until turbulent reattachment is no longer possible. The latter mechanism is classified as either leading-edge or trailing-edge stall, depending on the location of turbulent separation, with the bubble remaining stable until separation eventually reaches the reattachment point. The mechanism responsible for dynamic stall onset is reported to shift from leading-edge bubble bursting to trailing-edge stall with increasing airfoil thickness [43,44]. A less frequently reported behavior is thin-airfoil stall, where the bubble progressively elongates and may extend over the entire airfoil surface [45].

1.2. Dynamic Stall Control by Synthetic Jets

Flow control is generally defined as any attempt to drive the character or disposition of a flow field toward a more favorable state [46]; accordingly, depending on the context, dynamic stall control may involve delaying detachment, enhancing, or completely suppressing the DSV. Due to the adverse impacts described in Section 1.1, the objective of control on turbine blades is typically to delay or eliminate the formation of the DSV. Flow control devices are broadly classified as passive or active, with the latter requiring an external power source to add momentum to the flow. A synthetic jet actuator (SJA) is an active flow control device, typically consisting of a piezoelectric disk that periodically modulates the fluid volume within a cavity. If the fluid expelled during the momentary ejection is fast enough to escape the subsequent suction phase, the vortex ring or a vortex pair that forms due to the roll up of the separated shear layer at the sharp edges of the aperture propagates away from its origin, ultimately producing a quasi-steady jet from the working fluid itself, adding momentum to the surrounding medium despite the net mass flux remaining zero [47]. Although synthetic jet actuators reduce the mechanical complexity and weight of the flow control system by eliminating the need for external pumps or plumbing, they expand the geometric and operational parameter space and thereby complicate control design. The pertinent actuator parameters are illustrated in Figure 1b, where $f_j$ denotes the diaphragm excitation frequency, d the aperture width or diameter, and $v_j$ the jet velocity.

The control authority of synthetic jets over a static airfoil is well documented. The interaction between the synthetic jet and the crossflow primarily depends on the actuator location, the reduced frequency $f^{+}=f_{j}c/U_{\infty }$, and the momentum coefficient $C_{\mu }$, defined as the ratio of the time-averaged jet momentum to that of the freestream. The momentum coefficient is typically characterized under quiescent conditions and, for an actuator with a circular orifice, is mathematically given by the following relation:

$$C_{\mu }={{\displaystyle \frac{1}{bc{U}_{\infty }^2}}}_{A_j}\overline{v_j^2}dA={{\displaystyle \frac{\pi }{bcT{U}_{\infty }^2}}}_{-\infty }^{+\infty }{\int }_0^T{v}_j^{2}rdtdr$$

(1)

where b is the reference spanwise length, $A_j$ is the effective jet area, r is the radial distance from the symmetry axis, and T is a sufficiently long reference time to ensure convergence. In general, a minimum threshold value of $C_{\mu }$ exists below which actuation is ineffective because the jet cannot penetrate the crossflow [48]. With a large enough momentum coefficient, the separated shear layer, which is generally highly receptive to disturbances, locks into the driving frequency, ultimately forming discrete coherent structures that mix high-momentum freestream fluid into the near-wall region, thereby suppressing flow separation [49]. Forcing is most effective when the reduced frequency targets an instability of the baseline flow, which for a static airfoil corresponds to either the global instability $f^{+}\sim O\left(1\right)$, the natural shear-layer frequency $f^{+}\sim O\left(10\right)$, or the separation bubble frequency lying between these two [50,51]. Both low- and high-frequency forcing have been shown to be effective, though forcing at $f^{+}\sim O\left(1\right)$ results in unsteady reattachment, whereas forcing at $f^{+}\sim O\left(10\right)$ leads to the damping of global flow oscillations. For optimal performance, actuators are often placed near the natural separation location [51,52].

Due to the success of SJAs in suppressing separation on static airfoils, efforts have been made to extend their application to dynamic stall control, with varying degrees of success [53,54,55,56,57]. Parametric analyses suggest that control authority of synthetic jets in dynamic stall depends on the choice of momentum coefficient and jet location [58,59], much as in static stall, though the influence of unsteady forcing on the resulting flow field is not as straightforward. Traub et al. [49] reported that although leading-edge forcing often delayed the onset of DSV formation to higher incidence, or even suppressed it entirely, in some cases it instead promoted its formation, resulting in increased aerodynamic loading. Taylor and Amitay [60] also observed that leading-edge forcing can delay or suppress the DSV formation, although this may occur at the expense of enhanced trailing-edge separation.

1.3. Objectives

The present work is partly motivated by the relative scarcity of studies on dynamic stall in thick airfoils and low-Reynolds-number flows, as also highlighted in some recent studies [61,62]. The vast majority of the literature reviewed in Section 1.1 has focused on thin airfoils at relatively high Reynolds numbers, largely because systematic investigations of the phenomenon originated from studies of retreating blade stall in helicopters during forward flight. However, certain rotary applications, such as vertical-axis wind turbines and water turbines, employ thick airfoils to improve blade stiffness. Vertical-axis wind turbines, in particular, require symmetric airfoils to maintain uniform blade loading as the blades oscillate between positive and negative angles of attack and typically operate in the built environment at low-to-moderate chord-based Reynolds numbers $Re\sim O\left({10}^4\right)$, where dynamic stall is likely to occur with coexisting laminar, transitional, and turbulent flow regions.

Thick symmetric airfoils are generally less aerodynamically efficient than their thinner counterparts and produce lower lift than cambered airfoils; hence, the application of effective flow control strategies is crucial for enhancing their aerodynamic performance. The use of burst modulation for static airfoil control was introduced by Amitay and Glezer [63], and subsequent studies have confirmed its effectiveness. A key advantage of this approach is that it allows the actuator to operate at its optimal excitation frequency, typically much higher than the frequencies required for control, while the modulation frequency targets the relevant flow instabilities. However, the effects of burst modulation in dynamic stall control remain relatively unexplored. A recent study by Rice et al. [64] on a thick NREL S817 airfoil demonstrated that low-frequency modulation $f^{+}\sim O\left(1\right)$ improves several aspects of dynamic stall control while significantly reducing power consumption compared to continuous actuation at $f^{+}\sim O\left(10\right)$. Due to actuator limitations, the high-frequency forcing $f^{+}\sim O\left(10\right)$ in their study was not burst-modulated, and only a single low reduced frequency $f^{+}\sim O\left(1\right)$ was examined. The second objective of the present study is therefore to investigate the role of reduced frequency in burst-modulated synthetic jets for dynamic stall control. To this end, the experimental setup and data processing are described in Section 2, followed by the results and discussion in Section 3, and concluding with the key findings summarized in Section 4. The study is restricted to the spanwise mid-plane, where spanwise non-uniformity is minimal and control authority is greatest, resulting in steadier flow reattachment [65,66].

2. Methodology

2.1. Wind Tunnel Facility

The experiments were carried out in a closed-loop, low-speed wind tunnel housed within the Turbulence Research Laboratory at the Department of Mechanical and Industrial Engineering, University of Toronto. The wind tunnel features a 5000 $\mathrm{m}$$\mathrm{m}$-long test section with an octagonal cross-section measuring 1220 $\mathrm{m}$$\mathrm{m}$ in height and 910 $\mathrm{m}$$\mathrm{m}$ in width. The octagonal corners maintain a constant angle but taper in width along the length of the test section to expand the cross-sectional area and offset boundary layer growth. Optical access is provided through clear acrylic panels forming the ceiling and one side wall of the test section. Flow is driven by a six-bladed axial fan powered by a REEVES^® MotoDrive^® 500 series motor, mounted outside the tunnel on a vibration-isolating concrete pad. Flexible couplings connect the fan housing to the tunnel, minimizing vibration transmission to the test section. Freestream velocity is adjustable between $2.5$ $\mathrm{m}$/$\mathrm{s}$ and $18.0$ $\mathrm{m}$/$\mathrm{s}$, measured at the test section inlet with a pitot-static tube, with an uncertainty estimated at less than $\pm 1\%$. Upstream of the test section, flow passes through a conditioning unit composed of seven screens and a 9:1 converging section, which reduces turbulence and promotes flow uniformity. The resulting freestream exhibits low turbulence, with intensity typically below $0.1\%$ . After traversing the test section, the flow is redirected through four 90° bends using a turning vane system, enabling recirculation within the tunnel. The freestream velocity was maintained approximately constant at $U_{\infty }=4.2\mathrm{m}/\mathrm{s}$ throughout all experiments.

2.2. Airfoil Instrumentation and Motion Control

The airfoil model used in this study has a NACA 0018 profile with an open trailing edge, a configuration commonly employed in vertical-axis wind turbines. The model was machined from aluminum, with a chord length of $c=150\mathrm{m}\mathrm{m}$ and a spanwise extent of 600 $\mathrm{m}$$\mathrm{m}$, yielding an aspect ratio of 4. Approximately 83 of the central section is hollow to accommodate synthetic jet actuators and reduce weight. To minimize end effects, the model was tightly fitted with two circular endplates, each having a 300 $\mathrm{m}$$\mathrm{m}$ diameter and sharp leading edges. The endplates, made from polycarbonate rather than acrylic to withstand fatigue loading, were installed flush against the model to eliminate any gaps that could cause flow leakage or generate tip vortices. The model with attached endplates was installed 300 $\mathrm{m}$$\mathrm{m}$ downstream of the test section entrance, where it was connected to a shaft at each end, allowing it to pivot about its aerodynamic center while remaining centered within the span of the wind tunnel. The spacing between each endplate and the nearest sidewall was sufficiently large to prevent interactions with the sidewall boundary layer. One shaft was supported by two SKF 61802-2Z deep groove ball bearings mounted on the transparent sidewall, while the other was coupled to a ClearPath CPM-SDSK-3432S-ELS servo motor capable of controlling the direction and angular position with a resolution of 0.056°. An Arduino UNO board was used to program the prescribed sinusoidal motion and to establish communication with National Instruments LabVIEW, which served as the primary user interface for all other controls.

An insert spanning 480 $\mathrm{m}$$\mathrm{m}$ in width and 100 $\mathrm{m}$$\mathrm{m}$ in length was carefully machined to preserve the NACA 0018 profile when installed within the hollow section of the model. The insert housed an array of ten Murata MZB1001T02 microblowers, flush-mounted beneath the surface and distributed symmetrically along the span with a center-to-center spacing of $b=45\mathrm{m}\mathrm{m}$ between neighboring actuators, with no actuator located on the centerline. Each SJA had an optimal excitation frequency of $25.5$ $\mathrm{k}$$\mathrm{Hz}$ and discharged through an orifice of diameter $d=0.8\mathrm{m}\mathrm{m}$. The driving signal for the microblowers was supplied by a Rigol DG1022Z function generator and amplified by a YAMAHA HTR5470 amplifier to a peak-to-peak voltage of 25 $\mathrm{V}$. Once activated, the issued jets exited at 14% chord from the leading edge at an angle of 80.4° relative to the airfoil surface. The SJA array was characterized outside the tunnel under quiescent conditions using a single-sensor hot-wire probe traversed along the span 5 $\mathrm{m}$$\mathrm{m}$ downstream of the jet exit, sampling at 10 $\mathrm{k}$$\mathrm{Hz}$ for 30 seconds at each position. This test was conducted to verify spanwise symmetry, with the momentum coefficient of each jet measured using Eq. (1) with a reference freestream velocity $U_{\infty }=4.2\mathrm{m}/\mathrm{s}$. An example of the time-averaged jet exit velocity $\overline{v_j}$ and the corresponding momentum coefficients are reported in Figure 2. After the insert was installed in the model, the airfoil surface was coated with matte black paint to minimize reflections. For flow control, the driving signal was burst-modulated with a duty cycle of 50%. The momentum coefficient was held constant at approximately $C_{\mu }\approx 2.5\times {10}^{-3}$ along the centerline, while forcing was applied at $f_j=0$, 28 $\mathrm{Hz}$, and 280 $\mathrm{Hz}$, corresponding to $f^{+}=0$, 1, and 10, respectively, to systematically investigate the effect of modulation frequency on dynamic stall control. An additional frequency at $f_j=18\mathrm{Hz}$ ($f^{+}=0.643$) was applied to examine potential competition between the synthetic jets and pitching motion due to their interaction in time.

2.3. Velocity Measurements

A single-sensor hot-wire probe and a two-dimensional, two-component (2D2C) particle image velocimetry (PIV) system were used in tandem to characterize the shedding behavior and instantaneous velocity fields at the spanwise mid-plane of the airfoil. The assumption of spanwise uniformity, required to minimize perspective errors in 2D2C PIV, is supported by the relatively large aspect ratio of the airfoil, the selection of the mid-plane for measurements, and the spanwise uniformity of the SJA array [65,66]. All measurements were conducted at a chord-based Reynolds number of $Re=40000$. Due to to the relatively low freestream and jet-exit velocities, the Mach number remained very low ($Ma\ll 0.3$), allowing the flow to be treated as incompressible throughout the experimental campaign. The pitching airfoil was oscillated about a zero mean angle of attack ($\overline{\alpha }\approx 0$) at a frequency of $f_{\alpha }=1\mathrm{Hz}$, corresponding to a reduced pitch frequency of $k=0.112$. Two pitching amplitudes, $\Delta \alpha =15.0°$ and $20.0°$, were selected to ensure the occurrence of dynamic stall for both the uncontrolled and controlled cases. All cases considered are summarized in Table 1, and further experimental details are provided in the following paragraphs.

Hot-wire experiments were conducted at a sampling frequency of 10 $\mathrm{k}$$\mathrm{Hz}$ for up to 900 seconds. For the static airfoil cases, measurements were performed at a single downstream location for three angles of attack $\alpha =10°$, 15°, and 20°. For the pitching airfoil, two streamwise locations, $x/c=2.00$ and 3.75, were probed for the uncontrolled case ($f^{+}=0$), while only $x/c=3.75$ was considered again for the controlled cases with $f^{+}=1$ and 10. For the PIV measurements, the flow was seeded using a SAFEX^® 2010F fog generator operating with SAFEX^®-Inside-Nebelfluid, a diethylene glycol–water mixture. Illumination was provided by a Litron Bernoulli neodymium-doped yttrium aluminum garnet (Nd:YAG) laser emitting green light at a wavelength of 532 $\mathrm{n}$$\mathrm{m}$ with a maximum pulse energy of 200 $\mathrm{m}$$\mathrm{J}$/pulse. The initially circular laser beam was routed above the test section ceiling, where it was spread into a thin light sheet using a pair of THORLABS cylindrical lenses with focal lengths of $-13.7$ $\mathrm{m}$$\mathrm{m}$ and 1000 $\mathrm{m}$$\mathrm{m}$. The laser sheet was approximately 1 $\mathrm{m}$$\mathrm{m}$ thick and was carefully aligned at the airfoil centerline. Scattered light from the tracer particles was recorded using a 12-bit complementary metal-oxide-semiconductor (CMOS) JAI SP-5000M-USB camera with a sensor resolution of $2560\mathrm{pixels}\times 2048\mathrm{pixels}$, equipped with an Azure 5022ML12M 50 $\mathrm{m}$$\mathrm{m}$ lens, providing a field of view approximately 175 $\mathrm{m}$$\mathrm{m}$ in length and 140 $\mathrm{m}$$\mathrm{m}$ in height. A schematic of the experimental arrangement, including the airfoil model, SJA array, and the PIV setup, is presented in Figure 3. PIV acquisition was timed by an NI PCI-6232e data acquisition card at a sampling frequency of 12 $\mathrm{Hz}$, with the interframe time delay for an image pair set to $60\mu \mathrm{s}$ to obtain an appropriate in-plane particle displacement.

PIV measurements were performed near the static stall angle at $\alpha =8.5°$, $9.0°$, $9.5°$, and $10.0°$ for the uncontrolled airfoil, and at $\alpha =17.0°$ and $15.0°$ for the controlled cases with $f^{+}=1$ and 10, to examine the flow behavior prior to stall. Image acquisition was phase-locked to 6 different positive angles of attack, with 400 image pairs collected at each angle. The PIV processing was accomplished using the open-source software OpenPIV-Python-GPU, employing a window-deformation iterative multigrid algorithm starting with an initial window size of $64\mathrm{pixels}\times 64\mathrm{pixels}$, followed by two iterations at $32\mathrm{pixels}\times 32\mathrm{pixels}$, and concluding with two iterations at a final window size of $16\mathrm{pixels}\times 16\mathrm{pixels}$, yielding a physical resolution of $0.55°$ $\mathrm{m}$$\mathrm{m}$. Median validation with a neighborhood kernel size of three and a threshold of two was applied at every iteration to identify and replace outliers. MATLAB and Python were used for data post-processing, while visualization was performed using Origin^®.

3. Results and Discussion

As briefly outlined in Section 1, dynamic stall and its control with SJAs involve the formation and shedding of both small- and large-scale vortical structures. A variety of coherent-structure identification methods exist to locate, extract, and visualize flow features across different spatial and temporal scales [36]. In the present study, the Galilean-invariant ${\Gamma }_2$ criterion [67] is the primary structure-identification tool for instantaneous realizations. Given the velocity field $\mathbf{v}(x,y)=(u,v)$ and the in-plane position vector $\mathbf{r}=(x,y)$, ${\Gamma }_2$ is defined locally at the reference point ${\mathbf{r}}_{\mathbf{i}}=(x_i,y_i)$ according to:

$${\Gamma }_2\left(\mathbf{v}\right)={{\displaystyle \frac{1}{A_i}}}_{A_i}{\displaystyle \frac{(\mathbf{r}-{\mathbf{r}}_{\mathbf{i}})\times (\mathbf{v}-\langle \mathbf{v}\rangle )}{\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{i}}\right|·\left|\mathbf{v}-\langle \mathbf{v}\rangle \right|}}dA,\langle \mathbf{v}\rangle ={{\displaystyle \frac{1}{A_i}}}_{A_i}\mathbf{v}dA$$

(2)

where $A_i$ is the area of interest around the reference point $(x_i,y_i)$, here taken as a square window here taken as a square window with side length equal to 5% of the chord. By definition, ${\Gamma }_2$ is a dimensionless scalar bounded by $-1\le {\Gamma }_2\le 1$, with vortex cores identified by the local extrema of ${\Gamma }_2$ and the sense of rotation given by the sign of the extrema. In dynamic stall, the instantaneous flow field contains valuable information that is often lost in the mean field and includes vortical structures that are not necessarily phase coherent. As gradient-based approaches are sensitive to numerical differentiation noise, especially in instantaneous realizations, the ${\Gamma }_2$ criterion has been widely employed instead [36,56,60]. In what follows, we continue to use the nomenclature introduced in Section 1, while all other symbols are defined upon first use. In agreement with the terminology used so far and common turbulence conventions, $\overline{\left(·\right)}$, $\left|·\right|$, and $\langle ·\rangle$ are used to denote time-averaging, magnitude (norm), and conditional or ensemble averaging, respectively. All plots are normalized by the freestream velocity $U_{\infty }$ and the airfoil chord length c. The SJA location is indicated by a black triangle, and upward and downward arrows represent the airfoil pitching direction, corresponding to the upstroke and downstroke phases of the motion cycle.

3.1. An Overview of the Static Airfoil

In this section, the static airfoil is examined first to establish a reference frame for the unsteady events occurring in the pitching configuration. Global flow features are extracted from the mean streamlines shown in Figure 4, complemented by instantaneous velocity fields overlaid with ${\Gamma }_2$ contours in Figure 5. For all cases considered here, the boundary layer near the leading edge is initially attached and laminar, followed by the onset of transition. Laminar-to-turbulent transition in shear flows, including separated shear layers on bluff bodies [68], boundary layers [69] , and laminar separation bubbles [70], exhibits similar characteristics. A laminar separation bubble is observed in Figure 4(a) and Figure 4(c), whereas Figure 4(b) and Figure 5(d) show a separated shear layer on a bluff body and a fully attached boundary layer, respectively. For all of these cases, the roll-up process initially redistributes the contained vorticity into elongated spanwise vortices, forming the so-called cat’s-eye pattern, and is subsequently followed by the emergence of fully rolled-up discrete spanwise vortices. The described evolution is commonly explained by the continued growth of the Kelvin–Helmholtz instability wave [70,71].

Beyond the initial roll-up phase, the discrete spanwise vortices enhance the entrainment of high-momentum fluid from the outer flow into the near-wall region. If conditions are met, the separated shear layer can reattach onto the airfoil surface in a mean sense, leading to the formation of an LSB, as observed for the $f^{+}=0$ case at $\alpha =8.5°$. A comparison between Figure 4(a) and Figure 5(b) indicates that a further increase in angle of attack to $\alpha =9.5°$ for the uncontrolled airfoil leads to the failure of shear layer reattachment, resulting in massively separated flow in the mean sense. This regime change is evidenced by the onset of reverse flow over the aft portion of the suction side, along with a marked reduction in streamwise velocity near the leading edge, indicating the collapse of the suction peak. Previous studies on the NACA 0018 airfoil at $Re\approx 4\times {10}^4$ similarly identify $\alpha =10.0°$ as representative of post-stall conditions [72,73,74,75], in agreement with the present observations.

Forcing at either $f^{+}=1$ or $f^{+}=10$ selectively amplifies the instabilities, thereby promoting earlier transition to turbulence. The resulting turbulent boundary layers exhibit increased resistance to flow separation, leading to a substantial delay in static stall to higher angles of attack for both $f^{+}=1$ and $f^{+}=10$. As the angle of attack increases beyond the baseline static stall angle, the laminar portion of the boundary layer initially remains confined within the favorable pressure gradient region near the airfoil leading edge, thereby preventing laminar separation. Concurrently, the stagnation point moves down the pressure side, while the suction peak progressively shifts upstream and eventually moves ahead of the SJA location. At sufficiently high angles of attack, the laminar boundary layer upstream of the SJAs is once again exposed to an adverse pressure gradient, rendering separation unavoidable. If the entrainment of synthetic jet structures within the separated shear layer is sufficient to allow reattachment, an LSB can form in the mean sense, analogous to the baseline $f^{+}=0$ case.

Reattachment is more likely under low-frequency forcing, owing to the larger synthetic jet structures and enhanced entrainment. The formation of an LSB prior to stall is evident for the $f^{+}=1$ case, as shown in Figure 4(c). Under high-frequency forcing $f^{+}=10$; however, flow control is less effective and the response becomes more sensitive to small variations in angle of attack near the static stall condition. Consequently, prior to stall, only a fully attached boundary layer, as shown in Figure 4(d), could be captured for this case. The similarities and differences between LSBs formed with and without forcing are also of interest. At low Reynolds numbers, relatively long separation bubbles are known to form on the airfoil [76]. In the unforced case $f^{+}=0$, the LSB is longer than in the forced case $f^{+}=1$, due not only to disturbances introduced by the synthetic jets, but also to the increased adverse pressure gradient downstream, which promotes spanwise instabilities, leading to earlier transition and reattachment [77]. The solid blue lines in Figure 4 and Figure 5, extracted from the onset of the reverse flow on the airfoil surface, denote the dividing streamline and indicate the extent of the LSBs. In both cases, the separated shear layer undergoes periodic roll-up into spanwise vortices near the point of maximum bubble height, followed by breakdown into smaller-scale structures downstream of reattachment.

For the Reynolds number considered in this study, flow separation in the early post-stall range of angles of attack is highly intermittent in instantaneous realizations. Similar intermittency was reported by Aniffa and Mandal [78] for a NACA 0012 airfoil at a chord-based Reynolds number of $Re=50000$ and a stall angle of ${\alpha }_s=10.0°$, conditions comparable to those of the present NACA 0018 case with $Re=40000$ and ${\alpha }_s\approx 9.0°$. In their work, proper orthogonal decomposition (POD) was used to examine intermittency. Here, the intermittency factor $\gamma$ and the reverse-flow area A are used instead. The intermittency factor $\gamma$ is defined as the fraction of instances during which reverse flow is present at a given location, such that $\gamma (x,y)=P(u<0)$, where P denotes the probability operator. Accordingly, $\gamma =0.5°$ contour, commonly referred to as the 50% forward-fraction line, identifies locations where the probability of reverse flow is 50%. This line is shown in Figure 4 and Figure 5 to indicate the extent of flow separation for $\alpha =9.5°$ case. The global reverse-flow area A may be defined on the Cartesian grid shown in Figure 1(a), and is mathematically given by:

$$A\left(u\right)={\int }_{u<0}dA={\int }_{u(x,y)<0}dxdy$$

(3)

The reverse-flow area is evaluated for three uncontrolled cases, $\alpha =9.0°$, $9.5°$, and $10.0°$, with the corresponding probability density function (PDF) ${\varphi }_A$ shown in Figure 6. The skewness $S_A$ and flatness $K_A$ of each distribution are reported in the top-right corner of each panel. The dashed blue and green lines denote the reverse-flow area computed from the mean flow field $A\left(\overline{u}\right)$ and the mean reverse-flow area $\overline{A\left(u\right)}$ respectively. Because the reverse-flow area is not a linear operator owing to the temporally varying topology of the reverse-flow region, in general $A\left(\overline{u}\right)\ne \overline{A\left(u\right)}$. Nevertheless, as the massively separated flow becomes less intermittent, the two quantities become increasingly comparable, as evident for $\alpha =10.0°$ case in Figure 6(c). Despite the flow appearing completely separated in the mean sense for all three cases, $\alpha =9.0°$ case is highly intermittent, which motivated the selection of $\alpha =9.5°$ as the representative post-stall condition in Figure 4 and Figure 5. Statistics of the reverse-flow area for the three cases are reported in Table 2. The maximum, and consequently the mean, reverse-flow area increases considerably from $\alpha =9.0°$ to $9.5°$, and remains relatively unchanged from $\alpha =9.5°$ to $10.0°$. Overall, for $\alpha =9.0°$ the mean flow field is no longer a suitable representation of the flow state, and conditional averaging is therefore required. The conditioning criterion used here is $A<\overline{A}$, which partitions the dataset approximately in half such that $P(A<\overline{A})\approx 50\%$ for all three cases, as reported in Table 2. This criterion effectively classifies the flow into two distinct states corresponding to reattached and massively separated flow, as shown in Figure 7.

Around the static stall angle, irrespective of reattachment, the vortical structures in the airfoil wake lack coherence, resulting in broad peaks in the velocity spectra centered about a dominant frequency [73,79]. At sufficiently high post-stall angles of attack, however, the wake exhibits characteristics similar to those of a bluff body [80,81], with the shedding frequency decreasing as the angle of attack is further increased [82]. These patterns are evident in the premultiplied power spectra ${\Phi }_{uu}$ presented in Figure 8 for post-stall angles of $\alpha =10$, 15, and 20. Overall, for a pitching airfoil with continuously varying angle of attack, any of the above forcing frequencies may be employed within an open-loop control strategy. The low reduced frequencies considered here, $f^{+}=1$ and 0.643, are selected to correspond to the shedding frequencies at the midpoint and upper limit of the range $10°\le \alpha \le 20°$, respectively.

3.2. The Dynamic Stall Cycle Without Synthetic Jets

As discussed in Section 1, a pitching airfoil exhibits flow features fundamentally distinct from those of the static case. To further highlight these differences, the frequency spectra for two cases with $\Delta \alpha =15°$ and $\Delta \alpha =20°$ at two downstream locations are presented in Figure 9. Based on the discussion in Section 3.1, these cases correspond to stall depths of $\Delta {\alpha }_s=\Delta \alpha -{\alpha }_s\approx 6°$ and 11°, respectively. In contrast to the spectra shown earlier in Figure 8(b) and Figure 8(c), the pitching-airfoil spectra do not exhibit a sharp peak but instead display a broadband peak over a range close to the shedding frequency of the static airfoil. As expected, a distinct peak appears at the fundamental pitching frequency ($f_{\alpha }=1\mathrm{Hz}$) due to the imposed airfoil motion. In addition, several higher-order harmonics remain relatively energetic, extending into the higher-frequency range and interacting with the broadband peak region, indicative of nonlinear coupling between the imposed pitching motion and vortex shedding. The energy content of some harmonics, most notably the second harmonic, increases with pitching amplitude as $\Delta \alpha$ rises from 15° to 20°. Furthermore, the energy around the broadband peak intensifies at the downstream measurement location, suggesting progressive loss of vortex coherence. Beyond the broadband peak, the spectra exhibit an approximate $-2/3$ slope, consistent with inertial-subrange scaling in turbulent flows. To quantitatively compare the spectra across different cases and measurement locations, the energy content of the nth harmonic is defined as:

$$E_n={\displaystyle \frac{{\displaystyle {\int }_{(n-\frac{1}{2})f_{\alpha }}^{(n+\frac{1}{2})f_{\alpha }}{\Phi }_{uu}df}}{{\displaystyle {\int }_0^{+\infty }{\Phi }_{uu}df}}}$$

(4)

The harmonic energy content for the first through fourth harmonics is reported in Table 3, again reinforcing that the energy content of the second harmonic increases significantly as $\Delta \alpha$ rises from 15° to 20°, independent of measurement location. Table 3 further shows that the second and fourth harmonic energies remain approximately unchanged, whereas the first and third harmonic energies decrease at the farther downstream location. Overall, the multiple peaks observed in the pitching-airfoil spectra indicate the shedding of several energetic coherent structures, rather than the single dominant DSV typically observed for high-Reynolds-number pitching airfoils. These structures are not necessarily phase coherent and therefore may not appear in the phase-averaged flow field, as will be shown in the PIV results discussed next.

The characteristic features of dynamic stall are largely similar for the $\Delta \alpha =15°$ and 20° cases, and most events can be described using either case. At the beginning of the pitch-up motion, the boundary layer remains laminar and fully attached to the airfoil surface. As the airfoil continues to pitch up toward the static stall angle, an adverse pressure gradient develops downstream of the leading edge, while a thin region of reversed flow propagates upstream from the trailing edge [83], causing the upper portion of the boundary layer to behave as a local free shear layer. Shortly after its formation, this shear layer becomes susceptible to Kelvin–Helmholtz instability [37,84], which progressively spreads upstream and generates small-scale shear-layer vortices. As the rolled-up vortices are not phase coherent, these structures are illustrated at two representative instances in Figure 10, with the zero mean streamwise velocity contour ($\overline{u}=0$) denoted by solid blue lines to highlight the extent of reverse flow. Evidently, the boundary layer over the rear half of the airfoil thickens due to flow reversal, while the onset of shear-layer roll-up progressively shifts toward the leading-edge region.

Beyond the static stall angle, the boundary layer becomes susceptible to laminar separation at the leading edge, as well as to trailing-edge separation as the adverse pressure gradient rapidly intensifies while the airfoil decelerates toward its maximum angle of attack. The onset of dynamic stall, and the precise mechanism responsible for it, cannot be definitively identified with the temporal resolution available here; however, both trailing-edge separation and leading-edge bubble bursting have been reported under similar kinematic and flow conditions [43,84]. In the present work, near stall onset, flow reversal at the trailing edge is intermittent and exhibits cyclic variations; consequently, the phase-averaged velocity field is again not particularly instructive. Instantaneous velocity fields and conditional averages are presented instead in Figure 11, with the 50% forward-fraction line ($\gamma =50\%$) highlighted by solid green lines. Comparing Figure 11 with Figure 10, it is evident that the extent of reverse flow has decreased considerably at the present stage. In the instances shown in Figure 11(a) and Figure 11(b), interaction with the reverse flow leads to a sudden growth of shear-layer vortices over the aft portion of the airfoil, highlighted by dashed circular outlines. These structures are associated with a local ejection of streamlines away from the wall, causing the transverse velocity to switch from negative to positive values. To isolate these events, conditional averaging is performed using the criterion $v(x_i,y_i)>0$ at the reference points $(x_i,y_i)/c=(0.53,-0.04)$ and $(0.48,-0.03)$ for $\Delta \alpha =15°$ and 20° cases, as shown in Figure 11(a) and Figure 11(b). The reference points are located immediately above the upstream extent of the forward-fraction line, where the conditioning criterion partitions the datasets such that $P(v(x_i,y_i)>0)\approx 30\%$. The defining feature of dynamic stall, namely a large energetic vortex (DSV) in the mean sense, has not yet emerged in either case.

For both the $\Delta \alpha =15°$ and 20° cases, the DSV first appears near the end of the upstroke, as shown for two representative instances in Figure 12. From a topological standpoint, the DSV here is classified as a leading-edge vortex (LEV), a structure commonly encountered in studies of flapping flight and biomimetic flyers, which continuously grows in size until its eventual detachment from the feeding leading-edge shear layer. The signature of the LEV was extracted from the mean flow using the ${\Gamma }_2\left(\overline{\mathbf{v}}\right)=-0.6$ contour, highlighted by solid gray lines in Figure 12. The threshold value of $|{\Gamma }_2|=2/\pi \approx 0.6$ was selected following Graftieaux et al. [67]. In addition to the LEV, the remnants of boundary-layer shear-layer vortices roll up into another large structure over the aft portion of the airfoil, consistent with the observations of Visbal and Garmann [38]. Using the ${\Gamma }_2\left(\overline{\mathbf{v}}\right)=-0.6$ criterion, this large shear-layer vortex could only be identified for the $\Delta \alpha =20°$ case, as it is not strongly phase coherent for $\Delta \alpha =15°$ case. At higher Reynolds numbers $Re\sim O\left({10}^5\right)$, under similar stall depth ($\Delta {\alpha }_s\approx 10°$) and reduced frequency ($k\approx 0.1$) examined by Mueller-Vahl et al. [61], a lift-generating phase-coherent aft structure may be observed, whereas dynamic stall in the low-Reynolds-number cases considered here is dominated by LEVs (compare Figure 12(b) with Fig. 5(c) therein).

Prior to detachment, the LEV is bounded by two saddle points; one located where the vorticity-feeding leading-edge shear layer separates, and the other at the LEV reattachment point. Two principal LEV detachment mechanisms have been proposed in the literature, illustrated conceptually in Figure 13; each causes distinct topological changes in the flow and therefore determines the maximum achievable circulation. The first mechanism, shown in Figure 13(a), is referred to as bluff-body detachment, as it is analogous to the vortex-shedding process behind bluff bodies. This mechanism is initiated when the rear half-saddle of the LEV moves beyond the trailing edge and transforms into a full saddle in the wake, allowing the upstream-directed boundary layer to carry positive vorticity toward the leading edge and forming a new node associated with the trailing-edge vortex (TEV). In this case, the chord length acts as the characteristic length scale governing the frequency of LEV formation and the limiting circulation. The second mechanism, known as the boundary-layer eruption mechanism or, alternatively, vortex-induced separation, is a viscous–inviscid interaction inherent to any vortex–wall configuration and independent of any geometric length scale, in which the adverse pressure gradient imposed by the LEV on the upstream-directed boundary layer causes it to separate from the airfoil surface, forming spike-like eruptions containing positive vorticity that eventually pinch off the feeding shear layer from the LEV [36,38,85]. The topological changes caused by this eruption, shown in Figure 13(b), induce a new node at the leading edge associated with a secondary LEV.

To place the above discussion into context, instantaneous velocity fields and phase-averaged flow fields are presented in Figure 14, where the 5 velocity-magnitude contour ($\left|\overline{\mathbf{v}}\right|/U_{\infty }=5$) is highlighted by solid green lines to better visualize the nodes and saddles in the mean flow. For both cases, the mean streamline topology at this stage is consistent with the bluff-body detachment mechanism illustrated in Figure 13(a), exhibiting two nodes and a single full saddle in the wake immediately above the TEV. Boundary-layer eruptions, however, may only be identified from instantaneous realizations, as the mechanism is not phase-locked and is therefore washed out by phase averaging. In Figure 14(a), a boundary-layer eruption has resulted in the formation of a secondary LEV, as illustrated earlier in Figure 13(b). Several large vortices, however, have already detached from the feeding shear layer, with the rearmost structure having reached the trailing edge. Consequently, the bluff-body detachment mechanism is also active, as reverse flow has propagated upstream to the secondary LEV and is displacing the large-scale vortical structures away from the airfoil surface. In Figure 14(b), a large vortex has detached from the feeding shear layer before reaching the trailing edge; consequently, the bluff-body detachment mechanism is absent and no TEV is observed at this instant. Overall, for both the $\Delta \alpha =15°$ and 20° cases, the DSV exhibits a multi-scale character, comprising multiple large shear-layer vortices that arise due to low-Reynolds-number effects and the strong viscous response of the boundary layer. Furthermore, the instantaneous and mean flow fields suggest that the LEV detachment mechanism at the present chord-based Reynolds number $Re=40000$ lies within a transitional range in which both bluff-body detachment and boundary-layer eruption may occur simultaneously, consistent with Widmann and Tropea [86], who observed this transitional behavior for a ramping flat plate at $Re=10000$ and 35000 . The breakdown of the LEV into multiple large-scale structures excites the low-order harmonics of the fundamental pitching frequency, as observed in the spectra shown in Figure 9.

The primary distinction between the $\Delta \alpha =15°$ and 20° cases emerges after the detachment of the primary mean LEV. For $\Delta \alpha =15°$, the flow transitions almost immediately into a massively separated state. For $\Delta \alpha =20$, however, the reverse flow intensifies due to a stronger adverse pressure gradient, leading to large cycle-to-cycle variations, before the flow eventually reaches a massively separated state similar to that observed for the $\Delta \alpha =15$ case. These variations arise from distinct topological changes associated with the two LEV detachment mechanisms. Conditional averages based on the reverse-flow area A, introduced in Section 3.1, are therefore presented in Figure 15. The conditioning criterion $A<\overline{A}$ partitions the dataset approximately in half, such that $P(A<\overline{A})\approx 50\%$. To better isolate the nodes and saddles in the averaged flow fields, the 5% and 8% velocity-magnitude contours ($\left|\langle \mathbf{v}\rangle \right|/U_{\infty }=5\%$ and 8%) are highlighted by solid green lines. As described earlier in Figure 13(b), the boundary-layer eruption mechanism generates additional LEVs which, if they persist, can prevent reverse flow from propagating into the leading-edge region and thereby significantly reduce the reverse-flow area. The flow topology in Figure 15(a) corresponds to such instances, with the rear reattachment point of the secondary LEV forming a half-saddle on the airfoil surface and a full saddle located just above the TEV node, indicating complete detachment of the TEV from the airfoil. Examples of studies reporting the formation and shedding of a secondary LEV immediately following the detachment of the primary LEV through boundary-layer eruption include those of Mulleners and Raffel [36] (see Fig. 3(k)) and Widmann and Tropea [86] (compare Figure 15(a) with Fig. 7(a) therein). If bluff-body detachment is dominant, the reverse-flow area increases considerably, yielding the flow topology shown in Figure 15(b). In this case, the TEV remains attached to the airfoil, as evidenced by a half-saddle on the surface, while the leading-edge shear layer is completely separated, forming a train of nodes and saddles (compare Figure 15(b) with Fig. 7(b) in Widmann and Tropea [86]).

Another distinction between the $\Delta \alpha =15°$ and 20° cases is the strength of the TEV, reported at the vortex eye for both cases in Table 4. Evidently, from the ${\Gamma }_2$ values, although the TEVs are relatively weak and of comparable strength at the phases shown in Figure 14, the TEV in the $\Delta \alpha =20$ case continues to grow into a much larger and stronger coherent structure owing to the stronger adverse pressure gradient. The substantial increase in the energy of the second harmonic from the $\Delta \alpha =15°$ case to the $\Delta \alpha =20°$ case in Figure 9 is attributed to this enhanced TEV strength. During the remainder of the cycle, as the airfoil pitches down, the boundary layer progressively reattaches from the leading edge toward the trailing edge before eventually relaminarizing.

3.3. The Dynamic Stall Cycle with Synthetic Jets

In the presence of synthetic jets, dynamic stall is completely suppressed for $\Delta \alpha =15°$, with neither an LEV nor a TEV forming. For $\Delta \alpha =20°$, however, all cases still experience dynamic stall, as the maximum angle of attack exceeds the control limit identified in Section 3.1. The wake frequency spectra, shown for the $f^{+}=10$ and $f^{+}=1$ cases in Figure 16, are quite similar to that of the baseline flow in Figure 8, exhibiting several low-order harmonic peaks and a broadband region, although the energy of the second harmonic appears to be notably reduced. For a fair comparison between cases, the harmonic energy content for the first through fourth harmonics is reported in Table 5, indicating comparable harmonic energy between the $f^{+}=10$ and $f^{+}=1$ cases while also confirming that the second-harmonic energy is reduced for the controlled cases compared with the baseline $\Delta \alpha =20°$ case reported in Table 3. Overall, while synthetic jets appear to mitigate the severity of dynamic stall, strong vortex shedding persists.

The early stages of upstroke with the addition of synthetic jets are similar to the baseline flow, with the boundary layer initially fully attached to the surface. Particularly, under a favorable pressure gradient, the synthetic jet structures initially remain close to the airfoil surface, as was demonstrated in a recent study by Rice and Amitay [87]. Under an adverse pressure gradient, however, the synthetic jet structures immediately lift off from the surface and dissipate more quickly in space. For the pitching airfoil, unlike the static case, the adverse pressure gradient is not steady but instead rapidly builds up as the airfoil suddenly comes to rest. Consequently, the boundary-layer response to these unsteady conditions may depend on the immediate location of the synthetic jet on the airfoil surface.

For clarity, consider the mean spanwise-vorticity contours for $f^{+}=10$ and $f^{+}=1$ cases shown in Figure 17, where the boundaries of the large synthetic-jet structures are extracted from the mean flow field using the criterion ${\Gamma }_2\left(\overline{\mathbf{v}}\right)=-0.6$, as before. The spanwise vorticity is particularly useful here for visualizing the smaller vortical structures near the leading edge and is computed from the velocity field using numerical differentiation:

$$\Omega ={\displaystyle \frac{\partial v}{\partial x}}-{\displaystyle \frac{\partial u}{\partial y}}$$

(5)

At the high reduced frequency $f^{+}=10$, the synthetic jet structures are closely spaced along the wall; when the adverse pressure gradient rapidly intensifies, these structures lift off and form larger, yet still relatively closely spaced, clusters. In contrast, at the low reduced frequency $f^{+}=1$, while the structures are larger in size, they are also more widely spaced. Consequently, during the rapid increase in the adverse pressure gradient, regions in the immediate vicinity of the synthetic jets within the boundary layer experience excess momentum, whereas adjacent regions experience a relative momentum deficit. This spatial imbalance leads to distortion of the boundary layer and the associated coherent structures, as evident in Figure 17(b). To characterize the extent of interaction between the two competing sources of unsteadiness, namely the synthetic jets and the airfoil pitching motion, the dimensionless frequency disparity ratio is defined as the ratio of the SJA forcing frequency to the airfoil pitching frequency:

$$F^{+}={\displaystyle \frac{f_j}{f_{\alpha }}}={\displaystyle \frac{\pi f^{+}}{k}}$$

(6)

The interaction between the unsteady effects is negligible for $f^{+}=10$ ($F^{+}=280$) and remains mild for $f^{+}=1$ ($F^{+}=28$). At critically low disparity ratios, phase synchronization between pitching and actuation is required to ensure proper spatiotemporal momentum injection [88].

The presence of synthetic jet structures in the transitional boundary layer enhances entrainment, suppressing flow reversal at the trailing edge. As a result, unlike the baseline case, the boundary layer remains attached as long as the adverse pressure gradient does not reach the laminar segment of the boundary layer upstream of the SJA location. This condition is inevitably reached for $\Delta \alpha =20°$ regardless of $f^{+}$ value, leading to leading-edge stall and the formation an energetic LEV. To further demonstrate the effects of disparity ratio near stall onset, this stage is examined for $f^{+}=1$ ($F^{+}=28$) and $f^{+}=0.643$ ($F^{+}=18$) in Figure 18. For $f^{+}=10$ ($F^{+}=280$), not shown in Figure 18, the synthetic-jet structures grow convectively, and the boundary-layer thickness increases smoothly, similar to that observed in Figure 17(a). For $f^{+}=1$, by the time the LEV forms, the synthetic-jet structures shown in Figure 17(b) have largely dissipated and merged into larger clusters within the boundary layer. In contrast, for $f^{+}=0.643$, an abrupt distortion of the boundary layer is observed near the mid-chord location, forming a single large coherent structure that is evident in Figure 18(b) and is not present in the $f^{+}=10$ and $f^{+}=1$ cases.

Once the LEV forms, regardless of $f^{+}$, the SJAs supply additional vorticity to the LEV, and the rear reattachment point moves rapidly downstream. Similar to the baseline $\Delta \alpha =15°$ and 20° cases, both boundary-layer eruption and bluff-body detachment mechanisms are observed. Instantaneous velocity fields and phase-averaged flow fields are presented in Figure 19, where the 8% velocity-magnitude contour ($\left|\overline{\mathbf{v}}\right|/U_{\infty }=8\%$) is highlighted by solid green lines to emphasize the nodes and saddles in the mean flow. Figure 19(a) and Figure 19(c) are intended to illustrate boundary-layer eruption. The rear reattachment point in this case remains on the airfoil surface, with a half-saddle evident very close to the trailing edge in Figure 19(c), and hence the bluff-body detachment mechanism is not active. The instance shown in Figure 19(a) exhibits both a secondary LEV and a large eruption-induced vortex, consistent with the schematic in Figure 13(b), indicating detachment governed by the boundary-layer eruption mechanism. Figure 19(b) and Figure 19(d) illustrate the bluff-body detachment mechanism, evident from the presence of the TEV and a full saddle point in the wake. The TEV in these cases remains relatively weak, similar to that observed in the baseline flow at $\Delta \alpha =15°$. Overall, while synthetic jets do not fully suppress dynamic stall for $\Delta \alpha =20°$, they significantly attenuate one of the two large mean coherent structures, namely the TEV. Following LEV detachment, the flow transitions to a separated state before progressively reattaching from the leading edge toward the trailing edge, with low-frequency forcing ($f^{+}=1$ or $0.643$) being more effective in suppressing trailing-edge separation, as evident from the streamline patterns and coherent structures in Figure 20.

4. Summary and Conclusions

Static stall and dynamic stall under sinusoidal pitching at a fixed reduced pitch frequency $k=0.112$, along with their control using burst-modulated synthetic jets, were investigated experimentally for a NACA 0018 airfoil at a chord-based Reynolds number of $Re=40000$. A range of static angles of attack and two pitching amplitudes ($\Delta \alpha =15°$ and 20°) were examined to assess the influence of stall depth. Phase-locked PIV measurements and wake velocity spectra were used to characterize the underlying vortex dynamics and the effects of actuation at different reduced frequencies.

For the baseline (uncontrolled) static airfoil, the flow was highly intermittent in the immediate post-stall regime, alternating between a laminar separation bubble (LSB) and a massively separated state. A slight increase in stall depth significantly reduced this intermittency, and the wake spectra became increasingly characteristic of bluff-body shedding at larger stall depths. The controlled cases were highly sensitive near the critical angle of attack, transitioning rapidly to a massively separated state without exhibiting the intermittency observed in the baseline flow. For the baseline (uncontrolled) pitching cases, dynamic stall was governed by the formation and evolution of a leading-edge vortex (LEV), which formed near the end of the upstroke and grew into a multi-scale structure rather than a single coherent vortex. LEV detachment occurred through both boundary-layer eruption and bluff-body detachment mechanisms, leading to the formation of a trailing-edge vortex (TEV), which was particularly larger at the higher pitching amplitude. The wake spectra exhibited several low-order harmonics and a broadband region near the static airfoil shedding frequency, indicating nonlinear coupling between the imposed pitching motion and vortex shedding.

The introduction of synthetic jets significantly modified the boundary-layer development and vortex dynamics. For $\Delta \alpha =15°$, dynamic stall was completely suppressed for all forcing frequencies considered, with the boundary layer remaining attached throughout the cycle. For $\Delta \alpha =20°$, dynamic stall still persisted, with an LEV forming and detaching via the same mechanisms observed in the baseline cases. Nevertheless, flow control mitigated the strength and coherence of the TEV, leading to a weaker wake signature and a reduction in second-harmonic content in the spectra. The control effectiveness depended on the forcing frequency. Both low- and high-frequency forcing delayed stall onset by promoting earlier transition and enhancing momentum exchange within the boundary layer. However, the resulting controlled flow structures differed markedly. High-frequency forcing ($f^{+}\sim O\left(10\right)$) produced closely spaced, rapidly dissipating vortical structures, yielding a relatively uniform boundary-layer response. In contrast, low-frequency forcing ($f^{+}\sim O\left(1\right)$) generated larger, more widely spaced structures, which introduced spatial non-uniformities in the momentum distribution under rapidly varying adverse pressure gradients.

Overall, low-frequency forcing appears more effective in promoting flow reattachment, whereas high-frequency forcing may still be advantageous in high-speed rotary applications, as it is decoupled from the motion frequency and therefore does not require phase synchronization. The results demonstrate that burst-modulated synthetic jets can effectively delay or suppress dynamic stall at low Reynolds numbers, with actuator placement near the leading edge being the critical factor; consequently, complete suppression of separation is limited by the maximum angle of attack. Although lift, drag, and pitching moment are not measured directly in the present study, the observed delay in separation, attenuation of large coherent structures (particularly the TEV), and weaker wake signatures are consistent with a less severe dynamic-stall response. Direct confirmation of load alleviation, which would require dedicated pressure, force, and moment measurements, is planned for future studies.