A Deeper Insight into Dynamic Stall of Vertical Axis Wind Turbines: Parametric Study of Symmetric Airfoils

Abstract

Vertical axis wind turbines (VAWTs) suffer from dynamic stall (DS) at low tip-speed ratios (λ), where cyclic variations in angle of attack (α) dominate the blade aerodynamics, severely undermining aerodynamic performance and power extraction. The coupled influence of airfoil parameters on DS remains unexplored. To address this gap, a fully coupled parametric study using 126 incompressible URANS simulations is conducted, examining three geometric parameters of symmetric airfoils: maximum thickness (t/c), chordwise position of maximum thickness (x_t/c), and leading-edge (LE) radius index (I). The results show that coupled geometric modification fundamentally alters the stall mechanism, shifting it from abrupt, LE-driven separation toward a gradual, trailing-edge (TE)-controlled process as airfoils transition from thin, forward-x_t/c profiles to thicker configurations with aft x_t/c and reduced I. This transition enhances boundary-layer (BL) stability, delays DS onset, weakens dynamic stall vortex (DSV) formation, and mitigates unsteady aerodynamic loading. Within the investigated design space, the best-performing configuration (NACA0024–4.5/3.5) achieves a 73% increase in turbine power coefficient (C_P) relative to the baseline airfoil (NACA0018–6.0/3.0), mainly through passive control of BL separation and vortex development. These findings highlight the limitations of single-parameter optimization and establish a physics-based, coupled-design framework for mitigating DS-induced performance losses in VAWTs.

1. Introduction

1.1. State of the Art and Research Gaps

The aerodynamic complexity of Darrieus-type VAWTs represents a trade-off associated with their mechanically simple design. The unsteady and highly nonlinear aerodynamics of these turbines mainly arise from large, rapid variations in blade angle of attack ($\alpha$) beyond the static stall limit (${\alpha }_{ss}$), resulting in pronounced unsteady loading under strong relative velocity and rotational effects [1]. Multiple interacting flow phenomena occur simultaneously, including unsteady separation and dynamic stall [2,3], blade–vortex interactions [4], curvature-induced effects [5,6], and Coriolis influences [7], which further amplify this complexity.

At low tip-speed ratios ($\lambda$ < 3), VAWT blades experience $\alpha$ values that exceed ${\alpha }_{ss}$, causing each blade to operate through both pre- and post-stall regimes within a single rotation. This leads to dynamic stall, a transient phenomenon characterized by hysteresis in aerodynamic loads, delayed flow separation, and pronounced lift overshoot associated with the formation of a coherent DSV [8,9]. As this vortex convects and detaches from the airfoil surface, sudden lift loss and strong oscillations occur. The rotational motion of VAWT blades relative to the freestream further complicates unsteady vortex dynamics, leading to vibration, fatigue, noise, and significant aerodynamic losses [10,11]. Mitigating the adverse effects of DS in VAWTs therefore offers a promising pathway toward enhanced power output, thereby improving the competitiveness of VAWTs relative to HAWTs.

In addition to VAWTs [3,12,13,14], DS has been extensively investigated in other contexts, such as HAWTs [15,16,17,18], helicopter rotors [19,20,21], and maneuvering aircraft [22]. Previous studies have established that DS onset is governed by several flow parameters, including reduced frequency (K) [23,24], Reynolds number ($Re$) [25,26], Mach number (M) [27,28], and surface roughness [29]. However, the influence of airfoil shape-defining parameters on DS physics in VAWTs has not been systematically addressed in the literature [30].

Table 1 summarizes key studies examining the role of geometric parameters in DS. These investigations cover a wide range of features, including LE geometry, thickness variation, camber, and adaptive strategies, across different flow regimes and motion types. However, only a limited number of studies have addressed specific airfoil configurations; most studies summarized in Table 1 have focused on varying a single geometric parameter while keeping others fixed. Early experiments by McCroskey [31] established the light–deep DS framework, showing that increasing stall intensity shifts the separation mechanism from gradual TE-type stall in thicker airfoils to abrupt LE-type stall in thinner ones. Sharma and Visbal [8] further demonstrated that increasing $t/c$ modifies DS onset mechanism from LE bubble bursting to TE separation growth, indicating a shift in the dominant separation mode. Bangga et al. [32] reported that increasing $t/c$ in high-solidity VAWTs weakens the DSV and enhances power performance. More recently, J-shaped blade profiles have been shown to enhance aerodynamic stability and reduce flow separation losses in H-Darrieus VAWTs, contributing to improved torque uniformity and wake coherence [33].

Such a single-parameter approach fails to fully resolve the coupled geometric effects that govern DS physics, thereby offering only limited physical insight. Earlier investigations [30,34] suggested that airfoil parameters exert a coupled influence on the BL behavior and power performance of VAWTs across a wide range of $\lambda$. To date, no study has systematically explored the fully coupled impact of these parameters on DS characteristics in VAWTs. Moreover, most existing work on DS has been conducted with a few test cases, mostly thin airfoils at high $Re$ (≈${10}^6$), typical of helicopter and aircraft applications, rather than the moderate-$Re$ regime (∼${10}^5$) and thicker profiles (e.g., NACA0015–0018) relevant to VAWT operation. A comprehensive understanding of DS under these conditions is essential for informing next-generation blade design strategies aimed at mitigating DS-induced performance losses. This study aims to address this gap through a fully coupled parametric analysis of airfoil shape effects on DS characteristics. While our previous work [30] focused on turbine power performance under such geometric variations, the present study investigates the underlying flow physics governing DS in VAWTs. Specifically, BL evolution, laminar separation bubble (LSB) dynamics, and vortex formation are analyzed, and new physics-based indicators are introduced to quantify stall onset and severity. This approach establishes causal links between airfoil geometry, flow physics, aerodynamic loads, and power performance, enabling design-relevant strategies for passive DS mitigation.

Table 1. Overview of airfoil geometry effects on dynamic stall.

Year	Author	#Cases	Parameter	Dedicated Airfoil/Baseline Airfoil/Range [%c]	Method	PA/WT	AOA Range [°]	Re [$\times {10}^5$]/M	K
1976	McCroskey et al. [21]	4	LE shape	Baseline: NACA0012	Exp (WDT)	PA	6–14	25	0.25
1982	McCroskey et al. [31]	8	–	NACA0012, AMES-01, VR-7, NLR-7301, NACA4418, S809, AMES-01, FX-098	Exp (WDT)	PA	5–25	$M=0.07$–0.3	0.025–0.2
1991	Grohsmeyer et al. [35]	3	$r_{LE}$	NACA0012; NACA0012-63; NACA0012-33	Num (ADI)	PA	0–30	40	0.01–0.02
2002	Joo et al. [36]	7	t	12–16	Num (CFD)	PA	5–25	$M=0.283$	0.151
2006	Hamdani et al. [37]	4	t	NACA0012; NACA0012–0016; 0012–0016 hybrid; leading-edge bump	Num (CFD)	PA	5–25	34.5	0.1
2012	Ramesh et al. [38]	4	I	Baseline: SD7003; $I=0.5$–2 [−]	Num (CFD)	PA	0–42.3	0.01–10	0.4
2018	Ouro et al. [39]	2	c	0 and 4	Num (CFD)	WT	$-5$–25	1.35	0.1
2019	Sharma and Visbal [8]	4	t	10–18	Num (FDL3DI)	PA	5–30	2	–
2020	Jain and Saha [40]	5	t	9–21	Num (CFD)	WT	–	1.25	0.16
2021	Bangga et al. [32]	7	t	8–40	Num (CFD)	WT	–	1.1–1.9	0.25
2023	Wang et al. [41]	6	t	12–30	Num (CFD)	PA	6.5–26.5	10	0.1

Note: ADI = alternating direction implicit; WDT = wind tunnel; PA = pitching airfoil; WT = wind turbine.

Table 1. Overview of airfoil geometry effects on dynamic stall.

Year	Author	#Cases	Parameter	Dedicated Airfoil/Baseline Airfoil/Range [%c]	Method	PA/WT	AOA Range [°]	Re [$\times {10}^5$]/M	K
1976	McCroskey et al. [21]	4	LE shape	Baseline: NACA0012	Exp (WDT)	PA	6–14	25	0.25
1982	McCroskey et al. [31]	8	–	NACA0012, AMES-01, VR-7, NLR-7301, NACA4418, S809, AMES-01, FX-098	Exp (WDT)	PA	5–25	$M=0.07$–0.3	0.025–0.2
1991	Grohsmeyer et al. [35]	3	$r_{LE}$	NACA0012; NACA0012-63; NACA0012-33	Num (ADI)	PA	0–30	40	0.01–0.02
2002	Joo et al. [36]	7	t	12–16	Num (CFD)	PA	5–25	$M=0.283$	0.151
2006	Hamdani et al. [37]	4	t	NACA0012; NACA0012–0016; 0012–0016 hybrid; leading-edge bump	Num (CFD)	PA	5–25	34.5	0.1
2012	Ramesh et al. [38]	4	I	Baseline: SD7003; $I=0.5$–2 [−]	Num (CFD)	PA	0–42.3	0.01–10	0.4
2018	Ouro et al. [39]	2	c	0 and 4	Num (CFD)	WT	$-5$–25	1.35	0.1
2019	Sharma and Visbal [8]	4	t	10–18	Num (FDL3DI)	PA	5–30	2	–
2020	Jain and Saha [40]	5	t	9–21	Num (CFD)	WT	–	1.25	0.16
2021	Bangga et al. [32]	7	t	8–40	Num (CFD)	WT	–	1.1–1.9	0.25
2023	Wang et al. [41]	6	t	12–30	Num (CFD)	PA	6.5–26.5	10	0.1

Note: ADI = alternating direction implicit; WDT = wind tunnel; PA = pitching airfoil; WT = wind turbine.

1.2. Novelty and Objectives

To address the aforementioned gaps, this study pursues the following objectives

• To establish a detailed physical characterization of DS in VAWTs by investigating the individual and coupled impacts of three key geometric parameters—maximum thickness ($t/c$), chordwise position of maximum thickness ($x_t/c$), and LE radius (expressed via a geometric index, I)—on BL evolution, vortex dynamics, and aerodynamic loads.
• To derive robust physical correlations between geometric parameters and DS indicators using 126 URANS simulations validated against experimental data, focusing on symmetric airfoils modified from the NACA four-digit series.
• To identify an optimal airfoil configuration that passively mitigates DS effects through combined geometric modification, thereby enhancing aerodynamic stability and turbine power performance under DS-prone operating conditions.
  This work demonstrates that airfoil optimization for VAWTs cannot be treated as a set of single-parameter geometric modifications, particularly under DS conditions. This finding has important implications for the development of next-generation morphing blades.

1.3. Paper Outline

This paper is organized as follows. Section 2 outlines the computational setup and parameters. Section 3 introduces the modification ranges of the selected geometric variables and presents the resulting airfoil configurations. Section 4 presents the results, highlighting both individual and coupled impacts of the three geometric parameters on key BL phenomena. Section 5 and Section 6 provide the discussion and conclusions, respectively.

2. Computational Settings and Parameters

2.1. Turbine Characteristics

Table 2. Reference turbine characteristics.

Parameter	Value
Turbine type	Darrieus H-type (lift based)
n	1
d	1 m
$\sigma$	0.06
Airfoil shape	Original NACA0018; designated as NACA0018-6/3; $t/c=18\%$; $x_t/c=30\%$; $I=6.0$
Blade-spoke connection point	$c/2$
c	0.06 m
$U_{\infty }$	9.3 m/s
$\Omega$	46.5 rad/s
$\lambda$	2.5
$R{e}_c$ [$\times {10}^5$]	1.03
TI	5%

summarizes the geometrical and operational specifications of the reference turbine adopted from previous studies on VAWT characterization [42,43], representative of low-solidity Darrieus-type turbines. The corresponding chord-based Re is on the order of ${10}^5$, which characterizes the operating conditions typically encountered in small-to-medium-scale VAWTs operating at moderate wind speeds. Figure 1 shows a schematic of the turbine, representing a simplified version of the experimental setup used by Tescione et al. [44]. To limit the computational cost associated with the 126 transient CFD simulations, the model considers a single-blade configuration, with the shaft and spokes omitted. Previous studies have shown that excluding non-lifting components such as the shaft and spokes has a negligible impact on aerodynamic loads and power prediction in low-solidity VAWTs under fixed $Re$ conditions [45]. Furthermore, Rezaeiha et al. [43] showed that, for fixed $Re$ and normalized solidity, turbine aerodynamic performance, quantified by the power and thrust coefficients ($C_P$ and $C_T$), is nearly independent of blade number. Accordingly, a single-blade configuration ($n=1$) is adopted, retaining the dominant BL physics and DS mechanisms while substantially reducing computational cost.

2.2. Computational Settings

Table 3. Details of computational settings.

Parameter	Specification
Domain (Figure 2a)	$30d\times 30d$ (d: turbine diameter)
Grid (Figure 2b–e)	302,815 quadrilateral cells; 800 cells around the airfoil circumference; max $y^{+}<2.5$
Boundary conditions	Inlet: uniform normal velocity (TI = 5%, turbulence length scale = d); outlet: zero static gauge pressure; side boundaries: symmetry; walls: no-slip
Turbulence model	Four-equation transition SST
CFD approach	Incompressible Unsteady Reynolds-Averaged Navier–Stokes (URANS)
Solver	ANSYS Fluent v2019R2
Discretization order (time and space)	Second order
Pressure–velocity coupling scheme	SIMPLE
Azimuthal increment ($d\theta$)	0.1°
Time step size	$3.75339546\times {10}^{-5}$ s
No. of time steps per turbine revolution	3600
No. of turbine revolutions to reach statistical convergence	20 (results shown for the 21st revolution)
Total time steps	72,000
No. of iterations per time step	20
Scaled residuals	<${10}^{-5}$

summarizes the computational settings employed in this study, while Figure 2 illustrates the computational domain and grid topology. The turbulence modeling approach is based on previous investigations, in which three experimental datasets [46] and two advanced scale-resolving simulations [11,47] were compared against seven typical Reynolds-averaged eddy-viscosity turbulence models. Based on these comparisons, the four-equation transition SST model was identified as the most suitable for URANS simulations of VAWTs operating under complex DS conditions.

2.3. Solution Verification and Validation

The numerical setup, including the computational domain, azimuthal increment, and convergence criteria, follows established best-practice guidelines for CFD simulations of VAWTs. The 2D domain type is selected based on previous findings, showing small systematic differences (<1%) in power and thrust coefficients when compared against 2.5D URANS simulations [42,48,49].

A grid refinement analysis was conducted for the reference turbine to ensure spatial independence. The selected mesh was compared with a uniformly refined grid comprising 1,211,260 cells. Figure 3 presents the variation of lateral force coefficient $C_{F_y}$ during the last turbine revolution for both grids. The mean and maximum absolute deviations between the coarse and fine meshes were 0.01 and 0.04, respectively. To further quantify the discretization error, the Grid Convergence Index (GCI) was calculated following Roache’s method [50]. The computed $GC{I}_{\mathrm{coarse}}=0.0073$, corresponding to 2.45% of the Richardson extrapolated value, confirms the mesh independence of the results. Additional verification details are provided in [51].

The CFD framework used in this study was previously validated against multiple experimental datasets involving three VAWTs with different geometric and operational characteristics, including blade number, blade profile, solidity, tip-speed ratio, reduced frequency, and Reynolds number [3,44,52]. This variety of experimental conditions, including both flow characteristics and turbine quantities, supports the reliability of the employed URANS approach. For brevity, full validation details are not repeated here; the reader is referred to [46] for a comprehensive discussion.

3. Airfoil Shapes

Figure 4 shows a schematic of the symmetric NACA four-digit airfoil, highlighting three modified geometric parameters: relative maximum thickness $t/c$, its chordwise position $x_t/c$, and the leading-edge radius $r_{LE}$, expressed through a nondimensional geometric index denoted as I. The specific values for each parameter are summarized in

Table 4. Modification values for the studied parameters.

Parameter	Values
$t/c$ [%]	10, 12, 15, 18, 21, 24
$x_t/c$ [%]	20, 22.5, 25, 27.5, 30, 35, 40
I [−]	4.5, 6.0, 7.5

.

To prevent highly undesirable airfoil shapes from an aerodynamic standpoint, $x_t/c$ is limited to values ≤ 40. Furthermore, the selected range of I ensures that the resulting profiles maintain a physically realistic LE curvature, avoiding excessively sharp or overly blunt geometries. Applying these constraints, a total of 126 distinct airfoil geometries were generated (Figure 5). The full set of equations governing airfoil modification is provided in [30] and is not repeated here.

Symmetrical test cases are designated in the format NACA00$t/c$–$I/x_t/c$, where the leading zeros indicate zero camber; $t/c$ specifies the maximum thickness as a percentage of the chord (%c); I denotes the LE radius index (with one decimal precision); and $x_t/c$ represents the chordwise location of maximum thickness, expressed in tenths of chord. According to this naming scheme, the baseline airfoil, NACA0018–6.0/3.0, is a symmetric profile with maximum thickness of 18%c, LE radius index of $I=6.0$, and a thickness peak located at 30%c.

4. Results

This section examines the combined influence of the three airfoil parameters on turbine aerodynamics under DS-prone conditions at $\lambda$ = 2.5, where instantaneous $\alpha$ exceeds the static stall threshold (${\alpha }_{ss}$; Figure 6). Such conditions are typical of small-scale urban VAWTs [3], where deep DS is known to limit aerodynamic efficiency. The angle of attack was obtained from CFD simulations following [42]. The analysis focuses on the first half of the final revolution, where key BL phenomena occur: formation and bursting of laminar separation bubble, DSV emergence, and other pre-stall viscous effects [11]. Before addressing the coupled impact, single-parameter sensitivity analysis is first performed to isolate individual aerodynamic contributions. The analysis was confined to configurations within the vicinity of the optimal performance envelope identified in [30]. Reference values for optimal maximum thickness and its chordwise location ($t_{\mathrm{opt}}/c$ and $x_{t,\mathrm{opt}}/c$) were taken from the best-performing NACA0024–4.5/3.5 [30]. Variations in $t/c$ and $x_t/c$ were first examined at fixed $I_{\mathrm{base}}=6.0$. Subsequently, with $t_{\mathrm{opt}}/c$ and $x_{t,\mathrm{opt}}/c$ held constant, the effect of I is assessed by comparing $I=4.5$ and 7.5 against the baseline case ($I_{\mathrm{base}}=6.0$).

4.1. Laminar Separation Bubble (LSB)

LSBs form when the laminar BL detaches under a strong adverse pressure gradient (APG), typically at low-to-moderate $Re$ [53,54]. As important precursors to DS, they contribute to lift loss, vibration, noise, and overall performance degradation [55,56]. LSBs are identified by pressure plateaus followed by sharp recovery, localized suction in surface pressure contour (CoP), and a negative skin friction coefficient ($C_f$) near the LE. Their spatiotemporal evolution is characterized here by tracking laminar separation ($x_s$) and turbulent reattachment ($x_r$) points using Prandtl’s $C_f=0$ criterion.

4.1.1. Impact of Maximum Thickness (t/c)

Figure 7 and Figure 8 show spatiotemporal $C_f$ and CoP contours and trajectories of $x_s$ and $x_r$ on the suction side of NACA00t–6.0/3.5, revealing three distinct regimes based on $t/c$.

Thin airfoils ($t/c$ = 10–12%): Sharp LEs promote strong flow acceleration and intense favorable pressure gradients (FPG), followed by severe APG, shifting the maximum suction peak ($S{P}_{max}$) beyond static-stall conditions (${\theta }_{SP,max}>{\theta }_{SP,ss}$). This delayed $S{P}_{max}$ is characteristic of DS and induces transient lift overshoot via DSV formation. The strong APG triggers early separation both temporally (${\theta }_{\mathrm{sep}}$) and spatially ($X_{\mathrm{sep}}$), while the upstream migration rate of the separation point $x_s$ ($U_m$) remains limited due to concentrated short bubbles (reflected as small mean bubble length $L_b$), which stabilize the bubble spatially through enhanced reversed-flow adherence [57] (Figure 8a,b and Figure 9a,d).

With increasing $\theta$, APG intensification triggers short-bubble bursting into fully detached shear layers or elongated attached bubbles [53]. According to Tani’s criterion [54], short bubbles induce minimal pressure deviation from inviscid profiles, whereas long bubbles cause pronounced pressure distortion.

Although bubble bursting can be identified by sudden $SP$ drops [8], high-fidelity simulations indicate that wall-shear-stress behavior provides a more reliable indicator of LSB dynamics [58]. In this work, LSB bursting is characterized by a pronounced negative $C_f$ spike within the turbulent region. Prior to bursting, the local minimum skin friction ($C_{f,min}$) declines toward a critical threshold ($C_{f,\mathrm{crit}}$) at ${\theta }_{C_f,\mathrm{crit}}$, corresponding to the most negative $C_f$ spike (see Figure 10a and Section 4.3). The $C_{f,min}$ reduction rate, denoted as $R_{f,min}$, is defined as the temporal rate at which $C_{f,min}$ approaches $C_{f,\mathrm{crit}}$ and is adopted as a proxy for the growth rate of shear-layer instability up to ${\theta }_{C_f,\mathrm{crit}}$. This condition coincides with maximum opposing wall shear, peak shear-layer amplification, and reduced momentum transfer [57,59], signaling incipient LSB breakdown. At $t/c=10\%$, the LSB rapidly evolves toward bursting at ${\theta }_{C_f,\mathrm{crit}}={47}^{\circ }$, consistent with a high $R_{f,min}$, before abruptly transitioning into a long bubble. For $t/c=12\%$, a lower $R_{f,min}$ and a less negative $C_{f,\mathrm{crit}}$ at higher ${\theta }_{C_f,\mathrm{crit}}$ indicate more gradual shear-layer growth and delayed bursting (Figure 8a,b and Figure 10a).

Moderately thick airfoils ($t/c$ = 15–18%): A larger LE radius ($r_{LE}$) produces a smoother FPG-to-APG transition, delaying ${\theta }_{\mathrm{sep}}$, and shifting $X_{\mathrm{sep}}$ downstream. Longer bubbles form with weaker circulation, reducing surface-flow adhesion and accelerating upstream migration of $x_s$. The monotonic decrease in $R_{f,min}$ and $\left|C_{(f,\mathrm{crit})}\right|$ indicates weaker instability growth, accompanied by a delayed ${\theta }_{C_f,\mathrm{crit}}$. Under these conditions, classical LSB bursting and long-bubble formation are suppressed, and short bubbles persist until the TES reaches $x_r$ (Figure 8a–b, Figure 9a and Figure 10a).

Thick airfoils ($t/c$ = 21–24%): Thick profiles show more predictable $S{P}_{max}$ timing (${\theta }_{SP,max}$) with minimal offset from ${\theta }_{SP,ss}$, although their aerodynamic response remains nonlinear. Despite downstream displacement of $X_{\mathrm{sep}}$ due to improved BL stability, ${\theta }_{\mathrm{sep}}$ occurs earlier than expected from the thickness trend, indicating spatial–temporal decoupling of separation onset. This behavior suggests the dominance of unsteady effects, including rapid $\alpha$ variations and rotational forcing, over geometric scaling. Similar thickness-dependent changes in DS behavior have been observed by [8], and the role of unsteady forcing is well established in DS theory [60]. At $t/c=24\%$, both $U_m$ and $L_b$ reach their maximum values, indicating a diffuse bubble. Nevertheless, the minimal $R_{f,min}$, least negative $C_{f,\mathrm{crit}}$, and most delayed ${\theta }_{C_f,\mathrm{crit}}$ signify the slowest bubble evolution and most suppressed shear-layer growth. Consequently, the thickest airfoil exhibits enhanced temporal coherence and aerodynamic stability, consistent with its superior power performance at $I=6.0$ (Figure 8a,b, Figure 9a,d and Figure 10a).

Overall, the $C_f$-based indicators effectively capture LSB onset, evolution, and bursting. Since bursting immediately precedes DSV formation [8,58,61], the azimuthal occurrence of the critical $C_{f,min}$ (${\theta }_{C_f,\mathrm{crit}}$) provides a consistent precursor for DS onset across all tested airfoils. The analysis further indicates that $x_t/c$ governs the spatial–temporal synchronization of laminar separation onset. While synchronization is preserved for $t/c\le 18\%$ at $x_t/c=35\%$, decoupling emerges in thick profiles; shifting $x_t/c$ to 40% restores synchronization across the full thickness range, confirming that $x_t/c$, rather than thickness alone, controls the spatiotemporal coupling of separation onset.

4.1.2. Impact of Chordwise Position of Maximum Thickness ($x_t/c$)

Figure 8c,d illustrate the trajectories of $x_s$ and $x_r$ for varying $x_t/c$, while Figure 11 shows the corresponding spatiotemporal $C_f$ and CoP contours on the suction side over $0^{\circ }\le \theta \le {180}^{\circ }$. Variations in $SP$ and $C_{f,min}$ are presented in Figure 9b and Figure 10b. The analysis reveals four distinct flow regimes.

Forward positions ($x_t/c=20$–25%): High LE bluntness intensifies local flow acceleration and APG severity, which advances ${\theta }_{SP,max}$ while remaining well above ${\theta }_{SP,ss}$ (Figure 9d). Consequently, $X_{\mathrm{sep}}$ and ${\theta }_{\mathrm{sep}}$ occur early, with larger $x_t/c$ shifting $X_{\mathrm{sep}}$ downstream and delaying ${\theta }_{\mathrm{sep}}$. However, the resulting short LSBs (small $L_b$) sustain coherent vortical structures that resist upstream migration of $x_s$. In contrast, LSBs rapidly approach $C_{f,\mathrm{crit}}$, yielding peak $R_{f,min}$, while classical bubble bursting remains suppressed due to moderated shear-layer instability. Instead, residual instabilities arising from LSB–TES interaction induces localized $C_p$ deviations, signaling DSV emergence and convection.

Intermediate positions ($x_t/c=27.5$–30%): $X_{\mathrm{sep}}$ and ${\theta }_{\mathrm{sep}}$ progressively delay with improved synchronization, yielding more predictable separation dynamics. Extended FPG regions due to refined LE shape, reduce SPGR, producing lower $S{P}_{max}$ at higher ${\theta }_{SP,max}$, narrowing the offset from ${\theta }_{SP,ss}$. Higher $U_m$ reflects weakened vortical coherence within longer bubbles, reducing resistance to upstream migration. LSBs evolve more gradually, with milder $R_{f,min}$ and lower $\left|C_{(f,\mathrm{crit})}\right|$ occurring at higher ${\theta }_{C_f,\mathrm{crit}}$, indicating weaker shear-layer growth. Correspondingly, LSB–TES interaction effects on $C_p$ remain minor.

Transitional position ($x_t/c=35\%$): A transitional regime emerges with stronger APG, exhibiting a small offset between ${\theta }_{SP,max}$ and ${\theta }_{SP,ss}$. Both $X_{\mathrm{sep}}$ and ${\theta }_{\mathrm{sep}}$ delay consistently, while $U_m$ increases even as $L_b$ decreases, indicating a decoupling between bubble morphology and unsteady migration dynamics. This configuration yields the highest $C_P$ among all 42 cases at $I=6.0$, suggesting unique aerodynamic optimization potential. Shear-layer instability becomes marginally stronger, reflected in higher $R_{f,min}$ and larger $\left|C_{(f,\mathrm{crit})}\right|$, yet exhibiting the most delayed ${\theta }_{C_f,\mathrm{crit}}$. Although bubble bursting remains suppressed, LSB–TES merging induces $C_p$ deviations during the early downstroke, reflected as a hotspot streak in CoP contour.

Aft position ($x_t/c=40\%$): Further rearward shift sharpens the LE and destabilizes the flow, inducing an abrupt FPG-to-APG transition with increased $S{P}_{max}$ and delayed ${\theta }_{SP,max}$ well beyond ${\theta }_{SP,ss}$, thereby reinforcing DS signatures, including stronger DSV. Consequently, $X_{\mathrm{sep}}$ shifts upstream despite delayed ${\theta }_{\mathrm{sep}}$, indicating a pronounced spatial–temporal decoupling. The resulting short, unstable bubble shows intensified vortical activity, causing $U_m$ to collapse due to increased resistance to upstream migration. The stronger APG increases $R_{f,min}$ and $\left|C_{(f,\mathrm{crit})}\right|$, advancing ${\theta }_{C_f,\mathrm{crit}}$ and indicating intensified shear-layer instability. This is evident from pronounced $C_p$ deviations at lower $\theta$, following LSB–TES interaction.

These findings indicate distinct evolutionary regimes: from rapid instability growth at forward $x_t/c$, through progressively damped dynamics at intermediate positions, to a concentrated yet destabilized LSB at aft $x_t/c$, where sensitivity to geometric constraints is increased.

4.1.3. Impact of Leading-Edge Radius Index (I)

To examine the transitional behavior of NACA0024–6.0/3.5, two additional cases with $I=4.5$ and 7.5 were analyzed. Figure 8e,f present the azimuthal trajectories of $x_s$ and $x_r$, with the corresponding $C_f$ and CoP contours in Figure 12. Variations in $SP$ and $C_{f,min}$ are presented in Figure 9c and Figure 10c. The results confirm that LSB behavior in thick profiles is highly sensitive to I, as variations in I substantially alter $r_{LE}$ and amplify nose-curvature effects. These effects, confined near the LE, intensify and localize the APG, making BL stability strongly dependent on I [62].

Reducing I to 4.5 favorably sharpens the LE, producing gentler pressure gradients with reduced SPGR and $S{P}_{max}$ occurring at higher ${\theta }_{SP,max}$ with minimal offset from ${\theta }_{SP,ss}$. This introduces partial spatiotemporal decoupling between $X_{\mathrm{sep}}$ and ${\theta }_{\mathrm{sep}}$; however, despite increased $L_b$ and decreased $U_m$, separation dynamics re-synchronize relative to the comparison case NACA0024–6.0/3.0, eliminating the morphological–dynamical decoupling observed in NACA0024–6.0/3.5. The LSB exhibits milder temporal evolution with reduced instability growth, reflected by smaller $R_{f,min}$ and $\left|C_{(f,\mathrm{crit})}\right|$, and delayed ${\theta }_{C_f,\mathrm{crit}}$, promoting smoother LSB–TES interaction with negligible pressure disturbances (Figure 8f and Figure 12). Notably, NACA0024–4.5/3.5 achieves the highest $C_P$ among all 126 configurations, indicating that transitional geometries can reach peak efficiency when appropriately refined, which is achieved here by reducing $r_{LE}$.

Increasing I to 7.5 enlarges $r_{LE}$, creating a blunter LE that destabilizes the flow. This intensifies APG with higher $S{P}_{max}$ and delayed ${\theta }_{SP,max}$ well beyond ${\theta }_{SP,ss}$, as geometric saturation at the LE limits local curvature and APG redistribution [63]. Relative to NACA0024–6.0/3.5 and the comparison case NACA0024–6.0/3.0, separation dynamics destabilizes with delayed ${\theta }_{\mathrm{sep}}$ but upstream-shifted $X_{\mathrm{sep}}$, indicating pronounced spatial–temporal decoupling. Nevertheless, LSB morphology and migration remain synchronized, as both $L_b$ and $U_m$ decrease, reflecting intensified shear-layer activity. The enhanced bubble evolution is evidenced by the highest $R_{f,min}$ and $\left|C_{(f,\mathrm{crit})}\right|$ and earliest ${\theta }_{C_f,\mathrm{crit}}$, yet the instability buildup remains insufficient to trigger long-bubble formation. Instead, early LSB–TES merging occurs, causing the most pronounced $C_p$ deviation (Figure 8f and Figure 12).

These transitions illustrate the sensitivity of separation dynamics to LE geometry. While typical bursting is suppressed across all I values, the LSB response is distinctly non-monotonic, evolving from mild and stable behavior at $I=4.5$, through intensified nonlinear dynamics at $I=6.0$ (a critical transition), to destabilized conditions at $I=7.5$.

4.2. Dynamic-Stall Onset Criterion

Accurate identification of DS onset is essential for understanding and mitigating unsteady aerodynamic phenomena. Commonly employed indicators include deviations in normal force, lift and drag coefficients ($C_N$, $C_l$ and $C_d$) as well as early drops in the pitching moment coefficient ($C_M$) [21,64]. In addition, pressure-based signatures associated with the formation and growth of the DSV have been widely reported [65]. Recent approaches also include LE-based criteria extended to mixed and TE stall cases [66], and the boundary enstrophy flux (BEF) [67], a vorticity-based parameter that peaks at the moment of maximum net wall shear. This behavior is physically analogous to the $C_{f,\mathrm{crit}}$ criterion discussed earlier (Section 4.1.1), as both reflect conditions of maximum opposing wall shear and have been shown to correlate with DS onset.

Figure 13a compares different DS onset criteria for NACA00t–6.0/3.5, while Figure 13b–f present ${\theta }_{DS}$ obtained using different criteria for NACA0012–6.0/3.5, selected as a representative thin-airfoil case. Lift overshoot reliably detects DS in thin airfoils with strong DSVs but fails in thick profiles where DSVs and lift jumps are suppressed. In contrast, the $C_f$-based criterion (${\theta }_{C_f,\mathrm{crit}}$) shows close temporal alignment with conventional indicators, particularly the $C_d$-deviation criterion, even when lift overshoot disappears. Its agreement in thin airfoils supports the consistency of the method, while its persistence in thick airfoils—where other criteria may fail—indicates its potential as a consistent DS onset indicator across a wide range of airfoil geometries.

4.3. Formation and Evolution of the Dynamic Stall Vortex

Dynamic stall in the present configurations develops through a strong coupling between BL instability, LSB evolution, and vortex-induced load response. When present, the DSV emerges following LSB destabilization or TES–LSB interaction, appearing as a coherent suction footprint in spatiotemporal CoP contours and localized nonlinearity in $C_p$ distributions. Its formation, growth, and convection directly govern the magnitude, timing, and smoothness of lift and drag variations. The coupled effects of airfoil parameters on both flow physics and aerodynamic loads are discussed below.

4.3.1. Impact of Maximum Thickness ($t/c$)

Figure 14 shows $C_p$ and $C_f$ distributions for selected $t/c$ and $\Delta \theta$, with observations categorized as follows (also see Figure 7, Figure 8b, Figure 9a and Figure 10a). Figure 15 presents lift and drag coefficients ($C_l$ and $C_d$) as functions of $\theta$ and $\alpha$ under various geometric modifications. For clarity, a subset of values have been omitted.

Thin airfoils ($t/c$ = 10–12%): Strong LE acceleration rapidly contracts the LSB, producing concentrated vorticity and an early, sharp negative $C_f$ spike ($C_{f,\mathrm{crit}}$) at low ${\theta }_{C_f,\mathrm{crit}}$, signaling early DS onset. At $t/c=10\%$, the LSB bursts abruptly, leading to $SP$ collapse and $C_p$ deviation as the short bubble evolves into a large-scale, surface-attached DSV, distinct from wake-type vortices that detach immediately upon bursting [68]. The DSV grows while convecting downstream, leaving a pronounced diagonal hotspot streak in CoP contour with its fully developed form (FDV) observed at ${\theta }_{\mathrm{fdv}}=74.6^{\circ }$. It also appears as a strong negative $C_f$ band indicating intense flow reversal with limited TES interaction. At $t/c=12\%$, slightly gentler pressure gradients yield a lower $\left|C_{(f,\mathrm{crit})}\right|$ at higher ${\theta }_{C_f,\mathrm{crit}}$, delaying stall onset and DSV emergence. Despite weaker local pressure peaks, the FDV develops at higher ${\theta }_{\mathrm{fdv}}=81.9^{\circ }$, inducing stronger $C_{p,\mathrm{fdv}}$, likely due to the higher effective flow speed. Moreover, enhanced interaction between the developing DSV and TES is observed prior to full vortex formation. These flow features directly translate into pronounced lift and drag overshoots. Early, strong DSV formation induces abrupt $C_l$ and $C_d$ jumps; at $t/c=12\%$ overshoot intensity remains high despite delayed ${\theta }_{DS}$ and ${\theta }_{C_l,max}$, reflecting stronger vortex convection at higher azimuths (Figure 15a–d).

Moderately thick airfoils ($t/c$ = 15–18%): LE acceleration weakens and bubble instability diminishes, reflected by lower $\left|C_{(f,\mathrm{crit})}\right|$ occurrence at higher ${\theta }_{C_f,\mathrm{crit}}$. The LSB becomes less prone to long-bubble formation, and DS originates from TES–LSB merging rather than abrupt bubble bursting. As a result, DSVs are weaker, less coherent, and delayed. For $t/c=15\%$, the FDV appears at higher ${\theta }_{\mathrm{fdv}}$ and induces significantly weaker suction compared to thinner airfoils. At $t/c=18\%$, DSV signatures are further attenuated, with only marginal pressure deviations visible in $C_p$ distributions, indicating a considerably weakened DSV. Correspondingly, lift and drag responses become smoother, with delayed ${\theta }_{DS}$ and ${\theta }_{C_l,max}$, reduced overshoot amplitudes, and diminished peak values (Figure 15a–d).

Thick airfoils ($t/c=24\%$): LE acceleration becomes insufficient to sustain strong shear-layer instabilities, effectively suppressing DSV formation, with the maximum induced suction shifted to the early downstroke ($\theta \approx {118}^{\circ }$). As a result, lift overshoot disappears and unsteady lift augmentation is governed by successive shedding of weaker trailing-edge vortices (TEVs), consistent with the accelerated-flow hypothesis [69], which attributes lift overshoot mainly to intense LE acceleration. Consequently, unsteady lift enhancement shifts from a sharp DSV-induced overshoot to gradual TEV-driven fluctuations, drag jumps are moderated, and the $C_l$–$\alpha$ slope decreases due to earlier TES onset and enhanced BL deceleration near the TE (Figure 15a–d). Overall, increasing $t/c$ stabilizes the BL and shifts the stall mechanism from abrupt LE-driven behavior with strong DSV to a milder, TE-controlled regime with suppressed DSV effects.

4.3.2. Impact of Chordwise Position of Maximum Thickness ($x_t/c$)

Across all $x_t/c$ values, long-bubble formation is suppressed, and DS initiates mainly through TES–LSB interaction rather than classical bubble bursting. Note that $C_p$ and $C_f$ distributions are omitted for brevity; however, the relevant trends are evident from the preceding figures (Figure 8d, Figure 9b, Figure 10b and Figure 11), while the corresponding lift and drag coefficients are presented in Figure 15e–h. The evolution of DSVs and the associated aerodynamic loads exhibit four distinct regimes.

Forward positions ($x_t/c=20$–25%): At $x_t/c=20\%$, residual shear-layer instabilities following TES–LSB merging generate weak but discernible DSV, appearing as suction imprint in CoP contours and modest lift overshoots accompanied by pronounced drag rise. Increasing $x_t/c$ to 22.5% (excluded from the plotted set) weakens these instabilities, delays ${\theta }_{DS}$, and reduces DSV-induced suction and load excursions. At $x_t/c=25\%$, the DSV remains less pronounced during convection; however, the FDV induces stronger localized suction than at $x_t/c=22.5\%$, leading to a higher $C_{d,max}$ despite the absence of a clear lift overshoot.

Intermediate positions ($x_t/c=27.5$–30%): Further downstream shift of $x_t/c$ intensifies DSV suppression. $C_p$ disturbances become minimal, FDV occurrence is displaced to higher ${\theta }_{\mathrm{fdv}}$, and unsteady lift augmentation is increasingly governed by TEV-induced suction rather than coherent DSV formation.

Transitional position ($x_t/c=35\%$): A transitional regime emerges at $x_t/c=35\%$, characterized by increased $\left|C_{(f,\mathrm{crit})}\right|$ despite reduced DSV coherence. FDV signatures appear during the early downstroke, indicating a shift toward geometry-driven instability while maintaining limited vortex strength.

Aft position ($x_t/c=40\%$): Flow instabilities intensify, yielding earlier but weaker DSV-induced suction. TEV effects amplify, intensifying pre-stall load fluctuations and leading to earlier, more severe lift stall. Drag behavior, however, remains comparable to $x_t/c=35\%$ but with a lower peak value, indicating lift–drag decoupling: enhanced TEV-induced suction amplifies lift fluctuations without a proportional drag overshoot. For $x_t/c\le 30\%$, TES is weakly sensitive to geometry, while aft shifts to $x_t/c\ge 35\%$ drive upstream TES onset and earlier degradation of the $C_l$–$\alpha$ slope.

Overall, increasing $x_t/c$ from 20% to 40% drives a gradual transition from DSV-dominated stall to TEV-controlled unsteady lift generation, with a transitional regime at $x_t/c=35\%$.

4.3.3. Impact of Leading-Edge Radius Index (I)

Varying I for the NACA0024–6.0/3.5 does not promote long-bubble formation; DS onset remains governed by TES–LSB interaction. However, the I strongly influences instability growth rate, DSV strength, and load severity. For brevity, $C_p$ and $C_f$ distributions are omitted; relevant trends can be observed in Figure 8f, Figure 9c, Figure 10c and Figure 12; the corresponding lift and drag coefficients are shown in Figure 15i–l.

Reducing I to 4.5 substantially weakens LE acceleration, suppressing shear-layer growth and DSV formation. Consequently, the blade passes ${\alpha }_{max}$ smoothly without vortex-induced lift overshoot, $C_p$ distributions remain nearly linear prior to stall, and unsteady loads are dominated by mild successive TEV activity. This results in delayed ${\theta }_{DS}$, reduced drag overshoot and $C_{d,max}$, and a significantly smoother $C_l$ response despite earlier ${\theta }_{C_l,max}$. Moreover, the weakened APG shifts TES point downstream, suppressing suction and reducing the $C_l$–$\alpha$ slope at higher $\theta$.

Conversely, increasing I to 7.5 intensifies LE acceleration and bubble vorticity, producing a stronger, earlier-forming DSV with more pronounced nonlinear pressure behavior. The entire DSV lifecycle—formation, growth, and convection—occurs during the upstroke, leading to pronounced TEV-suction nonlinearity, sharp drag rise, intensified lift fluctuations with earlier ${\theta }_{C_l,max}$, and the most severe stall behavior observed. The steeper APG accelerates BL momentum loss and promotes earlier TES, causing a sharper decline in the $C_l$–$\alpha$ slope.

For $t/c\ge$$18\%$, where TEVs become the primary contributor to unsteady lift generation, DS onset no longer aligns with distinct features in the lift curve. As reported in [11], URANS solvers exhibit inherent limitations in accurately predicting TEV dynamics, making correlations between stall onset and TEVs unreliable. Higher-fidelity simulations are required to capture these mechanisms with sufficient accuracy.

4.4. Combined Influence of Airfoil Shape Parameters

Figure 16 maps the DS governing indicators in $t/c$–$x_t/c$ space for three LE radius indices (I), identifying configurations associated with either enhanced or mitigated DS behavior under high-wind operating conditions.

Suction peak growth rate (SPGR; Figure 16a–c) quantifies APG amplification and the transition from stable LSBs to vortex-dominated flow. High-SPGR indicates abrupt destabilization and early DS onset, whereas low SPGR promotes extended flow attachment. At $I=4.5$, a distinct low-SPGR region emerges for thick airfoils $t/c=21$–24% with $x_t/c=25$–35%, where DSV formation is strongly suppressed. In contrast, thin airfoils ($t/c=10$–12%) exhibit a persistent high-SPGR ridge that is largely insensitive to $x_t/c$, reflecting inherent susceptibility to abrupt DS driven by strong LE acceleration. Increasing I raises SPGR across the optimal region, reflecting intensified vortex activity and reduced suppression effectiveness. The critical zone persists with diminished values, implying reduced but not eliminated DS suppression. Similar trends are observed for $S{P}_{max}$, $R_{f,min}$, and $|C_{f,crit}|$.

Bubble stability index (BSI; Figure 16d–f) provides a comparative measure of BL stability by combining separation timing (${\theta }_{\mathrm{sep}}$), location ($X_{\mathrm{sep}}$), and bubble extent ($L_b$). Maximum stability occurs at $I=4.5$ for thick airfoils with mid-to-aft $x_t/c$ ($t/c=21$–24%, $x_t/c=27.5$–40%), whereas thin airfoils with aft $x_t/c$ and thick airfoils with forward $x_t/c$ remain prone to early vortex growth. Increasing I reduces BSI in the optimal region due to earlier separation and more compact bubbles, increasing vortex activity and DS susceptibility.

Dynamic stall onset, identified by ${\theta }_{DS}={\theta }_{C_f,\mathrm{crit}}$ (Figure 16g–i), is most delayed for thick airfoils with aft $x_t/c$ at low $I=4.5$, while thin airfoils experience consistently early DS, largely regardless of $x_t/c$. Increasing I slightly advances ${\theta }_{DS}$ in thick profiles and marginally delays it in thin ones, without altering their inherent susceptibility to DS.

Fully developed DSV (FDV), characterized by its timing (${\theta }_{\mathrm{fdv}}$; Figure 16j–l) and suction strength ($C_{p,\mathrm{fdv}}$; Figure 16m–o), forms earliest and strongest in thin airfoils. Moderately thick airfoils $t/c=15$–18% with $x_t/c=25$–35% exhibit delayed and weakened vortex development, while thick sections with aft $x_t/c$ show pronounced attenuation. Increasing I preserves contour topology but advances FDV formation and amplifies vortex induced suction.

The results demonstrates that a robust favorable regime consistently emerges at thick airfoils $t/c=21$–24% with $x_t/c=27.5$–35%, most pronounced at $I=4.5$. This region combines minimal SPGR, maximal BSI, delayed ${\theta }_{DS}$, and moderated $C_{p,\mathrm{fdv}}$, reflecting stabilized LSB dynamics, delayed DS onset, and reduced DSV strength. These findings highlight the importance of coupled geometric optimization when assessing DS behavior in early-stage VAWT airfoil design.

4.5. Stall Classification

Stall behavior across the $t/c$–$x_t/c$ space is classified using two complementary perspectives: a mechanism-based framework as described by McCroskey [31] and a severity-based assessment. While the dominant stall mechanism remains qualitatively similar across $I=4.5$, 6.0, 7.5, local flow features and stall severity vary significantly.

4.5.1. Mechanism-Based Classification

According to the classification framework proposed by McCroskey [31], four stall types are identified based on the dominant BL separation mechanism: leading-edge stall (LEs), initiated by LSB bursting and DSV formation or abrupt forward TES propagation; thin-airfoil stall (TAs), characterized by laminar LE separation and rearward shift of turbulent reattachment; mixed stall (MXDs), involving simultaneous LE and TE separation, merging near mid-chord, or bidirectional splitting from mid-chord; and trailing-edge stall (TEs), driven by gradual upstream migration of turbulent separation from the TE. Figure 17a shows the resulting stall classification map across $t/c$–$x_t/c$ space for the three I values. Detailed flow visualizations for each stall type observed in this study are provided in Appendix B to maintain focus on design-relevant parametric trends, while representative flow physics for each stall type are documented therein.

Thin airfoils ($t/c=10$–12%) exhibit strong sensitivity to $x_t/c$. For $t/c=10\%$ and $x_t/c=20$–25%, rapid TES–LSB interaction triggers bubble breakdown and DSV formation, classified as TE-triggered LEs. At $x_t/c=27.5\%$, transient interaction occurs near mid-chord before DSV dominance, defining transitional thin-airfoil stall (TTAs). For $x_t/c\ge 30\%$, LSB evolution becomes independent of TES, resulting in TA-dominated behavior governed primarily by LE dynamics. For $t/c=12\%$ and $x_t/c=20$–27.5%, an intermediate mode appears in which TES propagates upstream neither abruptly nor gradually, producing lift overshoot without LE vortex formation. This behavior is consistent with abrupt TEs (aTEs) as defined by McCroskey [31]. At $x_t/c=30\%$, TES–LSB interaction occurs; despite subsequent DSV dominance, the case is classified as transitional thin-airfoil stall (TTAs). With further aft $x_t/c\ge 35\%$, the stall mechanism progressively transitions to TAs as TES influence diminishes.

Moderately thick and thick airfoils ($t/c=15$–24%) are consistently dominated by upstream-propagating TES across all $x_t/c$. Flow topology and surface measurements confirm gradual upstream TES propagation without independent LSB bursting, classifying these cases as TE-type stalls.

This classification shows a systematic transition of DS in VAWTs: from LE-driven mechanisms dominated by LSB bursting and DSV formation in thin airfoils to TE-controlled mechanisms governed by upstream TES propagation in thicker profiles. This transition emerges only through coupled geometric modifications rather than single-parameter variation.

4.5.2. Severity-Based Classification (Light–Deep DS Spectrum)

According to McCroskey [31], light stall occurs when ${\alpha }_{max}$ slightly exceeds ${\alpha }_{ss}$, with the viscous layer remaining comparable to the airfoil thickness, whereas deep stall corresponds to substantial exceedance, accompanied by viscous-layer growth to chord scale. Mulleners and Raffel [64] provided a temporal interpretation, showing that light stall arises when the downstroke begins before full DSV development, while deep stall is characterized by DSV dominance during the upstroke phase.

To quantify this continuum, a stall severity index (SSI) is introduced, combining the timing (${\theta }_{\mathrm{fdv}}$) and suction strength ($C_{p,\mathrm{fdv}}$) of the fully developed DSV. SSI is defined as a relative severity metric within the present parametric space, normalized with respect to the best-performing configuration, which represents the lowest stall severity among the examined cases rather than stall-free behavior. This formulation prevents weak, early vortices from being misclassified as deep stall while appropriately weighting strong, early-forming DSVs, enabling a consistent comparison of stall severity across airfoil configurations.

For $I=4.5$ (Figure 17b), deep stall (SSI $>0.55$–0.60) is confined to thin airfoils ($t/c\le 12\%$), whereas a low-severity region (SSI $\le 0.40$) appears for $t/c=18$–24% and $x_t/c=30$–40%. Near-zero SSI values therefore indicate very light stall relative to other configurations, rather than complete stall suppression. Increasing I (Figure 17c,d) enhances SSI across the domain, indicating intensified vortex activity, while the underlying stall mechanisms remain unchanged.

These trends are consistent with the physical indicators discussed in Section 4.5. For example, thin airfoils with forward $x_t/c$ exhibit high SPGR and strong APG, promoting early LSB bursting and deep DS via LE separation. Increasing $t/c$ and shifting $x_t/c$ aft reduce SPGR, enhance bubble stability, and transition stall behavior toward gradual TE-dominated mechanisms with reduced severity and smoother load response, consistent with McCroskey’s observation that TE-type stall is generally associated with improved aerodynamic robustness.

4.6. Correlation Between BL Behavior, DS Mitigation, and Turbine Power Gain

Table 5. Turbine power coefficient and key boundary-layer parameters for the baseline and optimal airfoils during the last turbine revolution.

Airfoil	Configuration	Rank	${\mathbf{C}}_{\mathbf{P}}$ [$\times {10}^{-4}$]	SPGR	${\mathit{\theta }}_{\mathbf{sep}}$ [°]	${\mathit{X}}_{\mathbf{sep}}$ [%c]	${\mathit{L}}_{\mathit{b}}$	BSI	$|{\mathit{C}}_{\mathit{f},\mathbf{crit}}|$	${\mathit{\theta }}_{\mathit{DS}}$ [°]	SSI
Base.	NACA0018–6.0/3.0	18	559	0.458	26.9	0.31	0.11	0.587	0.0425	76	0.1403
Opt.	NACA0024–4.5/3.5	1	969	0.278	26.8	0.44	0.135	0.749	0.0247	97.7	0.0001
$\Delta$ [%]	–	–	+73.3	$-39.3$	$-0.37$	+41.9	+22.7	+27.6	$-41.9$	+28.6	$-99.9$

Note: SPGR = suction peak growth rate; BSI = bubble stability index; SSI = stall severity index.

compares key BL parameters for the baseline (NACA0018–6.0/3.0) and optimal (NACA0024–4.5/3.5) airfoils (Figure 18a). The optimized configuration yields a 73% increase in turbine $C_P$, associated with modified BL dynamics that alleviate DS effects. This improvement is preceded by a pronounced reduction in SPGR, accompanied by an extended FPG region and a downstream shift of $X_{\mathrm{sep}}$, while ${\theta }_{\mathrm{sep}}$ remains nearly unchanged. The optimized airfoil exhibits a more diffuse bubble with reduced vortex concentration and enhanced BSI. Consequently, instability accumulation prior to bursting is significantly lowered, reflected by reduced $\left|C_{(f,\mathrm{crit})}\right|$, resulting in pronounced attenuation of DSV strength.

Spatiotemporal $C_f$ and CoP contours (Figure 18b–e) confirm these findings: the hotspot streak associated with DSV suction nearly disappears in CoP contour for the optimal case, highlighting a substantial reduction in vortex-induced suction. Stall behavior shifts toward a lighter regime characterized by successive TEV formation and shedding rather than a large-scale vortex. This produces a smoother lift evolution with delayed peak, yielding a higher $C_{l,max}$ and alleviated post-stall fluctuations. (Figure 19a). The optimized airfoil also generates a higher mean $C_m$ with a postponed peak, indicating a delayed onset of moment stall, followed by gradual decline and limited post-stall oscillations due to reduced DS intensity (Figure 19c). Despite a lower $C_l$ curve slope due to promoted TES, and the reduced $C_m$ peak, drag overshoot and $C_{d,max}$ are noticeably reduced (Figure 19b). DS alleviation also maintains higher $C_m$ in the turbine downwind quartile, contributing to $C_P$ enhancement.

Figure 20 illustrates contours of the dimensionless instantaneous tangential velocity ($V_{tan,n}$) and streamlines at selected $\theta$ for baseline and optimized airfoils. Although both configurations exhibit TE-type stall, the baseline airfoil develops a large-scale TE vortex earlier in the cycle. This vortex rapidly expands to dominate the suction surface, promoting strong vortex shedding and pronounced post-stall load fluctuations in $C_l$, $C_d$, and $C_m$. In contrast, the optimized airfoil shows more stable TES evolution with successive, weaker TEV formation and shedding, leading to milder separation and significantly reduced load fluctuations.

5. Discussion

The present study focuses on operation at a low tip-speed ratio ($\lambda =2.5$), a regime in which VAWTs are known to experience pronounced dynamic stall (DS) [8]. Although the present study focuses on a single operating condition ($\lambda =2.5$), previous investigations by the authors have shown that as the TSR increases, the effective $t/c$ in DS regime tends to decrease and its chordwise location moves toward the LE, while the LE radius index remains unchanged [34]. Additionally, higher TSR values lead to a more limited range of variations in $\alpha$, resulting in comparatively lighter DS and improved aerodynamic load behavior. Therefore, the quantitative trends discussed here for $\lambda =2.5$ are consistent with the expected physical mechanisms at higher TSR values, even though extending the study to additional TSR values falls outside the defined scope of the current work, which focuses on coupled geometric effects under a representative VAWT operating condition.

The results demonstrate the clear advantage of a coupled modification strategy—simultaneous variation of airfoil parameters ($t/c$, $x_t/c$, and I)—over single-parameter approaches. This integrated framework effectively mitigates DS, enhances aerodynamic stability, and improves turbine power performance, thereby fulfilling the primary objective of the study.

Future work should extend this framework to asymmetric airfoil parameters, particularly camber and its chordwise position, which may further influence DS behavior. In DS regimes, blade-wake interaction is significant. While this study uses a single-blade model, Rezaeiha et al. [43] showed that, for a given $\sigma$ and $\lambda$, the scale of load fluctuations due to DS and blade-wake interactions is larger and less frequent for turbines with fewer blades due to the larger blade chord length. This phenomenon can alter the local $\alpha$, thereby affecting the onset and progression of DS. Our model, while computationally simplified and focused on fundamental stall physics, does not capture these complex interactions. Dedicated studies could examine multi-blade effects to provide a more comprehensive understanding for practical VAWT designs.

All simulations are performed at a fixed Reynolds number, reduced frequency, and turbulence intensity. Although previous studies report limited sensitivity of $\alpha$ to these parameters [42], their potential role in DS mitigation warrants further investigation.

It is therefore essential to clarify the modeling assumptions and numerical limitations underlying the present analysis. Given the known limitations of URANS in resolving post-stall flow physics, the present analysis deliberately emphasizes the pre-stall and upstroke phases, where DS onset mechanisms—LSB formation, instability growth, bursting dynamics, and early-stage DSV development—can be reliably characterized. Mitigating DS at its onset is of primary importance, as delaying stall initiation or reducing its severity during upstroke directly translates to improved power output and reduced load hysteresis over the turbine revolution. Accordingly, the proposed indicators ($C_{f,\mathrm{crit}}$, ${\theta }_{C_f,\mathrm{crit}}$, BSI, and SSI) assess early-stage DS characteristics, where URANS fidelity remains acceptable for comparative parametric analysis.

URANS limitations must be considered when interpreting the proposed indicators. Although the approach captures the overall DS cycle with reasonable accuracy, it cannot resolve small-scale unsteady vortex dynamics. Hybrid RANS/LES studies [11] show that URANS may predict early LSB bursting and DSV shedding, inaccurate secondary vortex formation, and excessive reverse flow during downstroke, causing discrepancies in drag prediction and load hysteresis. These deficiencies mainly affect post-stall flow evolution, where complex DSV–TEV interactions dominate. Thus, while the present indicators remain reliable for relative comparisons within a consistent numerical framework, their quantitative use in post-stall regimes requires validation using higher-fidelity approaches such as DES or LES.

6. Conclusions

In this study, incompressible URANS simulations—extensively validated against experimental data—were employed to perform a fully coupled parametric analysis of three airfoil shape-defining parameters ($t/c$, $x_t/c$, and I) to understand and mitigate dynamic stall in VAWTs. The main conclusions can be summarized as follows:

• Dynamic stall behavior is strongly governed by leading-edge flow acceleration, which controls suction peak growth rate, adverse pressure gradient development, and laminar separation bubble bursting. While leading-edge radius plays an important role, its effect becomes most pronounced when varied jointly with airfoil thickness and its chordwise position, demonstrating that effective DS mitigation requires coordinated multi-parameter design rather than isolated geometric modification.
• A critical threshold of leading-edge acceleration is identified for LSB bursting into a strong DSV. Below this threshold, the stall mechanism transitions from abrupt, LE-driven behavior in thin airfoils with forward $x_t/c$ to gradual, trailing-edge-controlled stall in thicker, aft-$x_t/c$ configurations, where TEVs replace DSVs as the dominant source of unsteady lift. This transition stabilizes the boundary layer, delays stall onset, suppresses DSV formation, and substantially reduces stall severity.
• The coupled modification reveals a consistently favorable regime at $t/c=21$–24%, $x_t/c=27.5$–35%, and $I=4.5$, characterized by reduced SPGR, enhanced bubble stability, delayed DS onset, and weakened vortex activity. Within this regime, the NACA0024–4.5/3.5 airfoil achieves a 73% increase in turbine power coefficient ($C_P$) relative to the baseline NACA0018–6.0/3.0, representing the best-performing configuration within the investigated design space.
• The leading-edge radius index primarily influences stall severity without fundamentally altering the dominant stall mechanism. Reducing I weakens leading-edge acceleration and adverse pressure gradients, delaying DSV initiation and promoting a transition from deep to light stall, whereas increasing I intensifies leading-edge acceleration and leads to earlier and more severe stall behavior.
• Finally, the azimuth of the most negative skin-friction spike within the laminar separation bubble (${\theta }_{C_f,\mathrm{crit}}$) is shown to provide a reliable indicator of dynamic stall onset within the present modeling framework, remaining effective even for thick airfoils where lift-based criteria fail due to suppressed DSV activity.

Overall, the results demonstrate that fully coupled airfoil modification enables effective passive control of boundary-layer events, producing smoother aerodynamic loading and more stable energy conversion—outcomes which cannot be obtained through single-parameter optimization. These findings establish a clear physical link between airfoil geometry, boundary-layer stability, and dynamic stall mechanism and severity, and provide a solid physical foundation for the development of next-generation VAWT blade concepts, including morphing and adaptive designs, particularly for offshore applications operating at low $\lambda$ where dynamic stall is most critical.