Investigating the Effects of Leading- and Trailing-Edge Shapes of a Flapping Wing on Power Extraction Performance

Abstract

Flapping wings present a promising approach to harnessing energy from fluid flow by leveraging a synchronized pitching and heaving motion of the airfoil. The impact of modifying the leading and trailing edge shapes of a flapping wing on energy harvesting performance is investigated using sinusoidal pitching motion. The pitch angle varies between 80° and 90°. The wing thickness (T1) varies from 8% to 48% of the chord length, with a flat plate chord length of c = 1.0. A promising airfoil profile is achieved by increasing only the leading-edge thickness to 32% of the chord, significantly enhancing energy capture by improving the generation of pushing forces and power. The results show that a wing configuration with a semicircular leading edge and a rectangular trailing edge outperforms the baseline case (a rectangular flat plate) and all other configurations under the same conditions. This configuration shows a notable improvement in power output and efficiency at a pitch angle of 85° and a leading-edge thickness of 32% of the chord. The maximum power output $\left({\mathrm{C}}_{\mathrm{p}\mathrm{t}}\right)$ represents a 16.73% increase over the baseline, while the maximum efficiency (η) reflects a 12.77% improvement. These findings highlight the superior energy extraction performance of the new configuration, emphasizing the dominant role of the leading edge in enhancing energy harvesters compared to the trailing edge.

1. Introduction

Amid the escalating global energy crisis and environmental concerns, the search for sustainable and low-carbon energy solutions has become a major priority for many countries. This initiative aims to enhance energy security, reduce the impacts of climate change, and promote long-term sustainability. As a result, there is a growing emphasis on sustainable replacements for traditional fossil fuels, which are both limited and highly polluting. One promising innovation in this field is oscillating wing energy harvesters, which utilize the oscillatory motion of an airfoil to extract energy from wind or water. The concept of generating power from fluid flow through a flapping motion was first introduced by [1]. This method served as a foundation for advancing research in energy extraction technologies. The first experimental verification of a flapping airfoil prototype was conducted by [2]. The surrounding flow structures, as presented in [3], were analyzed and discussed. Moreover, a study was carried out to examine how variations in the amplitude of heaving and pitching motions influence the two-dimensional aerodynamic behavior of flapping foils [4]. Computational simulations were utilized to analyze the thrust generation and fluid interaction responses of actively controlled flapping wings across a range of free-flow scenarios. This study provided significant insights into the fluid dynamics of flapping foil systems [5]. Wind tunnel experiments were conducted to investigate the effect of synthetic jet control on the stall characteristics of an airfoil. The findings indicated that when the momentum coefficient is relatively low, employing jet control significantly enhances aerodynamic efficiency by suppressing flow separation [6]. Furthermore, investigations into the aerodynamic behavior of an airfoil equipped with a synthetic jet actuator demonstrated that a jet directed normally from the trailing edge can emulate the aerodynamic benefits typically associated with passive Gurney flaps [7]. The use of Gurney flaps was investigated as a means to improve efficiency in different aerodynamic conditions [8]. The intricate interaction between the distance to the ground and power harvesting in flapping foils was investigated. The results indicated that as the foil moves nearer to the ground, power output improves owing to amplified plunging dynamics and increased lift. Nonetheless, placing the foil excessively close to the surface leads to a decline in performance [9,10,11]. To optimize airfoil characteristics and maximize energy extraction, a flapping airfoil system employing a combination of blowing and suction jet control was designed [12]. In another approach, oscillating suction mechanisms were installed on both the upper and lower surfaces of the flapping airfoil to boost its efficiency [13]. Additionally, investigations into wall confinement revealed that single-sided confinement enhances the energy harvesting capability of oscillating foils, whereas double-sided confinement does not yield the same benefit [14].

Studies have shown that the incorporation of flaps on wings influences the aerodynamic performance of both rigid and flexible structures [15]. When a jet flap was mounted at the trailing edge of an aircraft wing, it resulted in a notable increase in the production of lift [16]. In addition, the tandem-foil system concept was proposed, highlighting its potential to enhance the overall efficiency of the system [17]. The aerodynamic performance of tandem configurations with multiple flapping and fixed airfoils has been explored to understand flow interactions and their impact on lift and propulsion. Three triple-airfoil configurations were analyzed, revealing potential for enhanced lift and pitch control [18]. A study demonstrated that airborne wind energy systems can effectively generate power without the use of motors or actuators, leading to enhanced system efficiency [19]. Research on passive foils has primarily aimed at determining the most effective submersion depths for optimal operation [20]. Investigations into rear-edge flaps on flapping airfoils revealed a 27% boost in total power output, along with a 21% increase in aerodynamic performance [21]. The use of flexible wings resulted in a 2% increase in total power compared to a conventional airfoil [22]. Research has shown that the incorporation of trailing-edge jet flaps improves performance [23]. The addition of a front-mounted flap to the foil configuration resulted in a 30% increase in power output and a 23% improvement in overall efficiency [24]. Wind tunnel testing was carried out to assess the impact of discrete co-flow jets on enhancing lift and reducing drag, including a comparative analysis of discrete versus open co-flow jet setups [25]. Another investigation focused on combining a parabolic flap-equipped co-flow jet airfoil, which showed substantial aerodynamic benefits. Specifically, the lift coefficient increased by 32.1%, while the corrected lift-to-drag ratio improved by an impressive 93.8% at low angles of attack [26]. Additional research assessed the aerodynamic gains of airfoils outfitted with basic high-lift devices that utilize co-flow jet systems [27]. A comparative analysis was conducted to evaluate the effects of implementing co−flow jet flaps on both the flap and the primary front section of an aircraft control surface [28]. The aerodynamic characteristics resulting from the combination of Gurney flaps and jet flaps were examined through a series of wind tunnel experiments [29].

Adjustments to the wing motions and their positions optimized wake interactions, resulting in enhanced performance [30]. Performance improvements in energy harvesters can be achieved through targeted structural modifications [31]. Observations indicated that power output was greater during left-swing motion than during linear or right-swing movements [32]. An analysis of a wing system incorporating both leading- and trailing-edge flaps operating under sinusoidal motion showed a 28.24% improvement in power generation when compared to a configuration utilizing a single wing [33]. Research has demonstrated that non-sinusoidal motions enhance energy extraction, generating a higher power output compared to sinusoidal motions [34,35]. The performance of an airfoil can achieve an efficiency of up to 30% when it operates at the optimal frequency and plunge amplitude [36]. The control of thrust generation was investigated during the heaving and pitching motions of the airfoil using synthetic jet control, which led to a notable improvement in the production of thrust [37]. Adjusting the pitching motion of the airfoil to specific frequency and amplitude ranges resulted in an achieved efficiency of 34% [38]. Morphing airfoils gain versatility by seamlessly adjusting their geometry using advanced materials and structural mechanisms. In contrast, a three-part system reaches similar aerodynamic benefits by independently modifying components such as the leading and trailing flaps. Further studies examined airfoils with a droop-nose leading edge combined with a flexible, morphing trailing edge. Among the findings, the morphing trailing edge design led to a 10.25% improvement in aerodynamic efficiency [39]. A recent study investigated how flap bending, flap folding, flap sweeping, and flap twisting influence aerodynamic forces and vortex dynamics across different flight phases [40]. A new kinematic involving a droop-nose leading edge has been proposed to enhance the power output of flapping wing turbines, increasing the extracted power by 18% with a reversed D-path and by 21.9% compared to a vertical path [41]. A semi-passive flapping airfoil device with prescribed pitching motion is investigated in this study, with the influence of pitching amplitude on energy harvesting efficiency examined using an overlapping grid technique [42].

Two distinct dynamic spanwise twisting modes were examined to assess their impact on unsteady aerodynamic forces and flow patterns. The results suggest that applying a backward twist during the downstroke and a forward twist during the upstroke leads to a more favorable distribution of force coefficients [43]. Additionally, spanwise bending control was studied in relation to the unsteady force response of a flat plate under acceleration. The results underscore the importance of both bending direction and acceleration rate in influencing the formation of edge vortices and overall force behavior [44]. Further research into semi-active foils undergoing cosine-based motion revealed a significant increasement in energy harvesting performance [45]. It has been shown through research that efficiencies of over 33% can be reached with sinusoidal motions [46]. The impact of non-sinusoidal motions on performance was explored, revealing that a pitch angle of 75° resulted in an efficiency of 32% [47]. The incorporation of a modified trailing edge featuring a jet flap was studied, leading to a 1.46% improvement in efficiency compared to conventional designs [48]. A recent study explored the use of hybrid airfoil profiles, reporting a power coefficient of 1.16 when operating at a pitch angle of 70° [49]. The implementation of hybrid motion techniques resulted in a substantial 32.50% increase in power output for a single-flap wing compared to one using purely sinusoidal kinematics [50]. In further developments, the application of a hybrid motion path to a dual-flap wing demonstrated a remarkable enhancement in energy harvesting capability, yielding a 34.06% rise in power production and reaching a peak efficiency of 44.21% [51]. The performance of flapping-foil turbines was analyzed, revealing a substantial enhancement in the performance [52]. Efficiency was enhanced by 21.7% with adjustable side flaps positioned on the pressure surface, compared to traditional airfoil designs [53]. Research has confirmed that the integration of flaps enhances performance across various motion types—including sinusoidal [54], non-sinusoidal [47], and hybrid motions [49]—in both single-wing configurations and wings equipped with one [50] or two flaps [33]. These applied studies form the backbone of this research, reinforcing the credibility and practical viability of the proposed approach. In Australia, BioPower Systems Pty Ltd. designed a bio-stream energy device modeled after the oscillatory motion of fish fins, enabling the extraction of energy from fluid flows via horizontal movements. Flow-induced vibration simulations were conducted on three NACA airfoils—0009, 0012, and 0015—in a fully passive mode. The energy conversion efficiencies were recorded as 43.9%, 44.2%, and 36.3%, respectively [55]. This cutting-edge design is capable of generating up to 250 kW of power [56]. Developed by the European Marine Energy Centre Ltd., the Stingray device was the first large-scale tidal stream generator to utilize vertical oscillations for energy extraction. It achieved a peak power output of 150 kW [57]. The dual-wing generator was engineered to mimic the flapping motion of wings for the purpose of energy harvesting. This system show a notably high efficiency in its operation [58].

While previous studies have explored the effects of flapping kinematics, flap attachments, and various motion profiles on energy harvesting, this study explores the impact of modifying the leading- and trailing-edge shapes of a wing on energy harvesting efficiency under varying edge thicknesses and pitch angles while maintaining consistent operational conditions. A computational analysis evaluates how these factors influence performance by examining pushing force, power curves, and flow characteristics. The findings highlight how adjustments to the leading-edge shape can enhance aerodynamic performance and provide a new design guideline that emphasizes the aerodynamic influence of the leading edge in energy harvester applications.

2. Numerical Methodology

Table 1. Parameters employed in the wing-based energy harvesting system.

Description	Symbol	Value
Time	t
Thickness of the flat plate	${\mathrm{t}}_{\mathrm{w}}$	4%c
Thickness	T1	(8%–48%c)
Chord length	c	1.0
Pivot point	$x_p$	c/3
Heaving amplitude	${\mathrm{h}}_{\mathrm{o}}$/c	1.0
Frequency	$f$
Angular frequency	ω = 2πf
Reduced frequency	$f^{*}$ = ${\displaystyle \frac{f\mathrm{c}}{{\mathrm{U}}_{\mathsf{\infty }}}}$	0.14
Reynolds number	$\mathrm{R}\mathrm{e}$ = ${{\displaystyle \frac{\rho {\mathrm{U}}_{\mathsf{\infty }}\mathrm{c}}{\mathsf{\mu }}}}^1$	500,000
Strouhal number	St = ${\displaystyle \frac{2f{\mathrm{h}}_{\mathrm{o}}}{{\mathrm{U}}_{\mathsf{\infty }}}}$	0.28
Phase angle	$\varnothing$	90°

¹ $\rho$ is density, ${\mathrm{U}}_{\mathsf{\infty }}$ is uniform speed, μ is dynamic viscosity.

presents the key parameters and operational settings utilized in this study to enhance power generation in a wing-based energy harvesting system [54].

Figure 1 shows a schematic representation of a two-dimensional rectangular flat plate, featuring a thickness-to-chord ratio (${\mathrm{t}}_{\mathrm{w}}$) of 4%c. According to the best conditions for maximizing power output and efficiency, as identified in [54], the system operates with a reduced frequency of $f^{*}$ = 0.14, a pitch angle of ${\mathsf{\theta }}_{\mathrm{o}}$ = 75°, a pivot location at $x_p$ = c/3, and a heave amplitude of ${\mathrm{h}}_{\mathrm{o}}$ = 1.0c, ensuring meaningful comparisons. The baseline geometry is a simple rectangular flat plate, providing a clear reference point for evaluating performance improvements. The modified geometries introduce variations such as rounded and rectangular leading edges, along with rounded, rectangular, or sharp trailing edges, to systematically investigate their effects on the generation of pushing force and overall energy harvesting performance. The thickness values (T1) varied from 8% to 48% of the chord length to assess their impact, while maintaining a constant flat plate thickness ${\mathrm{t}}_{\mathrm{w}}$ of 4% for consistency across all cases.

Figure 2 shows various configurations with different leading- and trailing-edge shapes, highlighting the design variations considered in this study.

Figure 3 presents a schematic depicting various configurations, where ${\mathsf{\theta }}_{\mathrm{o}}$ represents the pitch angle of the wing. The following variables are defined: h(t) denotes the vertical displacement of the plate at any given time, while ${\mathrm{h}}_0$ indicates the amplitude of the heaving motion.

$$\mathrm{h}\left(\mathrm{t}\right)={\mathrm{h}}_0\sin (\mathsf{\omega } \mathrm{t}+\varnothing )$$

(1)

Equation (2) shows sinusoidal pitching motion for the wing [54]. For the instantaneous pitch angle, $\mathsf{\theta }$(t), and the maximum pitch angle, θ_o,

$$\mathsf{\theta }\left(\mathrm{t}\right)={\mathsf{\theta }}_{\mathrm{o}} \mathrm{s}\mathrm{i}\mathrm{n} (\mathrm{\omega } \mathrm{t})$$

(2)

The instantaneous power is denoted by P, while $\overline{\mathrm{P}}$ represents the average power.

$$\mathrm{P}=\mathrm{Y}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}(\mathrm{t})}{\mathrm{d}\mathrm{t}}}+\mathrm{M}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}$$

(3)

$$\overline{\mathrm{P}}={\displaystyle \frac{1}{\mathrm{T}}}\underset{0}{\overset{\mathrm{T}}{\int }}\mathrm{P} \mathrm{d}\mathrm{t}$$

(4)

The function Y(t) denotes the force applied in the heaving direction, whereas M(t) refers to the moment generated due to pitching motion.

$${\mathrm{C}}_{\mathrm{p}\mathrm{t}}={\displaystyle \frac{\mathrm{P}}{\frac{1}{2}\rho {\mathrm{U}}_{\mathsf{\infty }}^3\mathrm{s} \mathrm{c}}}={\displaystyle \frac{2}{\rho {\mathrm{U}}_{\mathsf{\infty }}^3\mathrm{s} \mathrm{c}}}[\mathrm{Y}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}+\mathrm{M}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}]$$

(5)

$${\mathrm{C}}_{\mathrm{p}\mathrm{t}}={\mathrm{C}}_{\mathrm{p}\mathrm{l}}+{\mathrm{C}}_{\mathrm{p}\mathrm{m}}={\displaystyle \frac{1}{{\mathrm{U}}_{\mathsf{\infty }}}}[{\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}+{\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right){\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}]$$

(6)

$${\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right)={\displaystyle \frac{\mathrm{Y}\left(\mathrm{t}\right)}{\frac{1}{2}\rho {\mathrm{U}}_{\mathsf{\infty }}^2 c s}}$$

(7)

$${\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right)={\displaystyle \frac{\mathrm{M}\left(\mathrm{t}\right)}{\frac{1}{2}\rho {\mathrm{U}}_{\mathsf{\infty }}^2 c s^2}}$$

(8)

${\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right)$ denotes the instantaneous pushing coefficient, ${\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right)$ corresponds to the instantaneous moment coefficient, and s indicates the span length of the flat plate. Given that this study focuses on a two-dimensional simulation, s is set to 1. The average power coefficient, denoted as $\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}}$, is determined from ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$.

$$\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{T}}{\mathrm{C}}_{\mathrm{p}\mathrm{t}}\mathrm{d}\mathrm{t}=\overline{\mathrm{P}}/({\displaystyle \frac{1}{2}}\rho {\mathrm{U}}_{\mathsf{\infty }}^3\mathrm{c})$$

(9)

$$\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{t}}}=\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{l}}}+\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{m}}}={\displaystyle \frac{1}{\mathrm{T} {\mathrm{U}}_{\mathsf{\infty }}}}[{\int }_0^T{\mathrm{C}}_{\mathrm{L}}\left(\mathrm{t}\right)({\displaystyle \frac{\mathrm{d}\mathrm{h}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}})\mathrm{d}\mathrm{t}+{\int }_0^T{\mathrm{C}}_{\mathrm{M}}\left(\mathrm{t}\right)({\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}})\mathrm{d}\mathrm{t}]$$

(10)

${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$ and $\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{l}}}$ correspond to the pushing powers, while ${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$ and $\overline{{\mathrm{C}}_{\mathrm{p}\mathrm{m}}}$ represent the moment powers. The efficiency η [54] is computed using the following formula:

$$\mathsf{\eta }={\displaystyle \frac{\mathrm{P}}{\frac{1}{2}\rho {\mathrm{U}}_{\mathsf{\infty }}^3\mathrm{s} \mathrm{d}}}={\mathrm{C}}_{\mathrm{p}\mathrm{t}}(\mathrm{c}/\mathrm{d})$$

(11)

Here, d denotes the maximum vertical displacement achieved by the motion of the flapping wing.

This study utilized the overset mesh method [59] in conjunction with ANSYS Fluent 21 R1 [60]. The simulations were based on the Reynolds-averaged Navier–Stokes equations, employing the finite volume approach and a k−ω SST turbulence model low analysis [59,61]. A second-order upwind scheme was used for spatial discretization to accurately capture flow features, and a second-order implicit scheme was applied for temporal discretization to ensure numerical stability in the unsteady simulations. A fixed time step was employed.

$${\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{i}}}=0}$$

(12)

$${\displaystyle \frac{\partial }{\partial \mathrm{t}}}(\rho {\mathrm{u}}_{\mathrm{i}})+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}(\rho {{\mathrm{u}}_{\mathrm{i}}\mathrm{u}}_{\mathrm{j}})=-{\displaystyle \frac{\partial \mathrm{P}}{\partial {\mathrm{x}}_{\mathrm{i}}}}+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}(\mathsf{\mu }({\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}+{\displaystyle \frac{\partial {\mathrm{u}}_{\mathrm{j}}}{\partial {\mathrm{x}}_{\mathrm{i}}}}-{\displaystyle \frac{2}{3}}{\mathsf{\delta }}_{\mathrm{i}\mathrm{j}}{\displaystyle \frac{\partial \mathrm{u}\mathrm{l}}{\partial \mathrm{x}\mathrm{l}}})+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}(-\rho \overline{{\mathrm{u}}_{\mathrm{i}}^{\mathrm{’}}{\mathrm{u}}_{\mathrm{j}}^{\mathrm{’}}}))$$

(13)

Reynolds stress is represented by −$\rho \overline{{\mathrm{u}}_{\mathrm{i}}^{\mathrm{’}}{\mathrm{u}}_{\mathrm{j}}^{\mathrm{’}}}$. A dynamic mesh approach is implemented, and the simulations make use of a pressure–velocity coupling algorithm to resolve the flow field. The motion of the plates is specified using user-defined functions. The computational mesh illustrated in Figure 4 spans a length of 70c and a width of 50c. The boundary conditions are depicted in Figure 4a, while Figure 4b provides an overview of the moving plates and the geometric modifications explored in this study.

Studies were conducted to assess both grid and time independence, utilizing coarse, medium, and fine meshes. The impact of different mesh resolutions and time intervals on ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ values are summarized in

Table 2. Independence analysis of mesh and time steps.

Mesh Type	Element Count		Time Step/Cycle	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$ (%) Variation with Grid	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$ (%) Variation with Time Step	η (%)
	Wing	Background
Coarse	0.6 × ${10}^5$	0.3 × ${10}^5$	2000	0.891			34.53
Medium	1.2 × ${10}^5$	0.6 × ${10}^5$	500	0.904			35.03
			2000	0.887	0.44	1.88	34.37
			4000	0.883		0.45	34.22
Fine	2.6 × ${10}^5$	1.2 × ${10}^5$	2000	0.886	0.11		34.34

. The simulations were configured with boundary conditions that included a Re = 1100, $f^{*}$ = 0.14, and a pitch angle fixed at 75°. To assess the independence of time-step selection, simulations were performed using a medium-resolution mesh with time steps of 500, 2000, and 4000. The results show that the medium mesh provides sufficient accuracy, with only slight variations observed between different time steps.

The accuracy of the model was verified through a comparison with the results reported in [62]. As shown in Figure 5, this comparison revealed a strong correlation between the outcomes. A medium-resolution grid combined with 2000 time steps was chosen to balance computational efficiency with the accuracy of the results.

3. Results and Discussion

The selected parameters were as follows: the pivot point was located at $x_p$ = c/3, $\varnothing$ = 90°, $f^{*}$ = 0.14, and ${\mathrm{h}}_{\mathrm{o}}=\mathrm{c}$. These conditions are the same as those presented in [54]. The output power of six different configurations was compared to that of the baseline case—a rectangular flat plate with a thickness of 4% of the chord (4%c)—which achieved a power output coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$) of 1.04. Among the modified geometries, case 2 produced a lower ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ of 0.855, indicating reduced output power. In contrast, case 3 delivered the highest performance, reaching a ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ of 1.032. Although slightly below the baseline value, case 3 stands out as the best performing among the six cases studied, highlighting its potential for improving energy harvesting performance through geometric optimization.

To ensure convergence to a periodic steady-state solution, numerical simulations were carried out over seven complete flapping cycles. Power coefficients and efficiency data were subsequently extracted exclusively from the seventh cycle, by which time all transient effects had sufficiently dissipated, ensuring an accurate evaluation of the wing’s performance. Figure 6 shows the time histories of lift and drag coefficients, highlighting the stability and reliability of the simulation results. This approach provides a more comprehensive understanding of both the transient and periodic aerodynamic behaviors inherent in flapping-wing dynamics.

3.1. Impact of the Leading-Edge Thickness on the Output Power

This section explores the effects of edge modifications and pitch angles to emphasize the critical role of leading-edge geometry in enhancing energy harvesting. Figure 7 shows an analysis of power output for the most favorable case identified in this study (case 3), focusing on the influence of leading-edge thicknesses (T1 = 0.28, 0.32, and 0.36) and pitch angles (${\mathsf{\theta }}_{\mathrm{O}}$ = 80°, 85°, and 90°) and on power output. As shown in Figure 7, the output power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$) generally increases with thickness up to T1 = 0.32c, after which it declines, indicating the best thickness for maximizing energy extraction.

The highest power output was achieved at a pitch angle of 85° with a thickness of T1 = 0.32c, yielding a peak ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ of 1.214. The 32%c thickness in case 3 results in a relatively thick and blunt airfoil compared to conventional aircraft designs. However, since the wing used for energy harvesting does not move forward, drag force is not a primary concern. Rather than other factors, emphasis is placed on amplifying the thrust produced in the heaving direction, as it plays a critical role in improving the efficiency of energy harvesting.

3.2. Impact of the Shape of the Flapping Wing on the Energy Extraction Performance

Table 3. Calculated results of power coefficient and efficiency for various cases.

Case Number	Pitch Angle	Thickness	${\mathbf{C}}_{\mathbf{p}\mathbf{l}}$	${\mathbf{C}}_{\mathbf{p}\mathbf{m}}$	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$	η (%)	${\mathbf{C}}_{\mathbf{p}\mathbf{t}}$ (%) Increment	η (%) Increment
Baseline case	75°	4%c	0.96	0.08	1.04	40.31
Case 1	85°	32%c	1.163	0.043	1.206	45.61	15.96	13.14
Case 2	85°	32%c	0.656	−0.086	0.570	19.86	−45.19	−50.73
Case 3	85°	32%c	1.194	0.020	1.214	45.46	16.73	12.77
Case 4	85°	32%c	0.789	0.075	0.865	31.59	−16.82	−21.63
Case 5	85°	32%c	0.521	0.435	0.956	35.80	−8.07	−11.18
Case 6	85°	32%c	0.866	−0.227	0.639	23.93	−38.55	−40.63

shows the power output and efficiency for the various cases examined, highlighting that a thickness of T1 = 32%c combined with a pitch angle of 85° achieved the highest power output among all the configurations, thereby enhancing energy harvesting. These results are compared with the baseline case, which features a rectangular flat plate with a thickness of ${\mathrm{t}}_{\mathrm{w}}$ = 4%c set at a pitch angle of 75°. Among the six cases analyzed, case 3 showed the highest power output of 1.214 with the highest increment in output power of 16.73% and efficiency of 45.46%, followed closely by case 1, which achieved a power output of 1.206 and efficiency of 45.61% and the highest increment in efficiency of 13.14%.

In contrast, case 2 exhibited the lowest power output (0.570) and efficiency (19.86%), indicating a significant performance drop compared to the baseline case. Cases 4 and 5 showed a moderate performance, with power outputs of 0.865 and 0.956, and efficiencies of 31.59% and 35.80%, respectively. Meanwhile, case 6 produced a power output of 0.639 with an efficiency of 23.93%.

3.3. The Effect of Leading- and Trailing-Edge Shapes

To further investigate the impact of edge modifications, this study compares the best-performing case (case 3) with case 4, which features the opposite edge shapes. This comparison aims to highlight the significance of the leading-edge shape in enhancing energy harvesting performance over modifications to the trailing edge. Figure 8 shows the calculated results for ${\mathrm{C}}_{\mathrm{L}}$, ${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$, ${\mathrm{C}}_{\mathrm{M}}$, ${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$, and ${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$ over a single cycle. The comparison between case 3 in the black line and case 4 in the red line reveals distinct differences in their aerodynamic performance. In terms of the pushing force coefficient (${\mathrm{C}}_{\mathrm{L}}$), case 3 shows higher pushing force values from t/T = 0.0 to t/T = 0.35, while case 4 shows larger values from t/T = 0.35 to t/T = 0.5. The pushing power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{l}}$) follows a similar trend to the pushing force coefficient for both cases; however, case 3 exhibits higher peaks, while case 4 maintains low values throughout. While the moment coefficient (${\mathrm{C}}_{\mathrm{M}}$) and the moment power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{m}}$) for cases 3 and 4 show similar values, the results indicate that the moment coefficient is not sensitive to the shape of the leading or trailing edge. Therefore, the total power coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$) is highly dependent on the pushing force generated by the shape of the leading or trailing edge, with case 3 producing higher total power coefficients.

Figure 9 shows that the vorticity contours for cases 3 and 4 at the best pitch angle of 85° reveal distinct differences in the formation and shedding of vortices at various time steps. At t/T = 0.05, cases 3 and 4 generate clockwise vorticity near the leading edge due to the decreasing pitching motion. However, the rounded leading-edge shape in case 3 delays flow separation, resulting in the vorticity forming closer to the trailing edge compared to case 4. In contrast, the sharp leading edge in case 4 induces strong vorticity almost immediately, which remains concentrated near the leading edge. At the midpoint of the downstroke (t/T = 0.25), the wing begins to pitch upward. At this stage, case 4 generates strong counterclockwise vorticity, whereas case 3 does not exhibit a significant formation of vortices. As the motion progresses to t/T=0.35, case 4 retains a larger and stronger vortex on the lower surface of the wing near the hinge point compared to case 3.

As shown in Figure 9, the shape of the leading edge has a more significant impact on the overall energy harvesting performance compared to the trailing edge. This is primarily because the flow encounters the leading edge first in a relatively uniform and undisturbed state, allowing its geometry to directly influence the initial flow behavior. In contrast, the trailing edge interacts with flow that has already been modified by the leading edge, reducing its relative influence on performance.

The streamline in Figure 10 shows the flow patterns around the airfoil for cases 3 and 4 at a pitch angle of 85° and different time steps. At early time steps (t/T = 0.05), the streamlines appear relatively smooth, indicating minimal flow separation, which is consistent with the vorticity observed in Figure 9. As time progresses (0.25 $\le$ t/T $\le$ 0.45), the streamlines become more distorted, reflecting the formation and shedding of vortices, as evidenced by the stronger vorticity contours. Case 3 shows better energy harvesting performance through stable leading-edge vortex (LEV) attachment, maintaining proximity to the airfoil surface. This promotes consistent lift generation via sustained low-pressure regions while minimizing energy losses. By t/T = 0.50, the streamlines exhibit pronounced recirculation zones, coinciding with the peak vorticity values, highlighting the impact of unsteady flow separation and vortex interaction on the overall flow structure. Case 4 shows premature LEV detachment and larger-scale vortex shedding (Figure 10b), resulting in chaotic flow structures that reduce efficiency. These findings directly confirm that controlled LEV development is critical for maximizing power extraction efficiency.

The generation of pushing power is primarily driven by the pressure difference across the flapping wing, which is influenced by variations in wing configuration and pitch angle. Figure 11 shows the pressure distribution for cases 3 and 4. At t/T = 0.05 and t/T = 0.15, case 3 exhibits a relatively smoother pressure distribution on the upper surface than case 4. At the point of the leading edge, the pressure gradient shows a generally rounded profile, but a very sharp variation is observed in case 4. The rapid pressure increases on the upper surface of case 4 induce strong separation, with the separation position located more at the leading edge at t/T = 0.05 as shown in Figure 11b. As the time progresses to t/T = 0.25, the pressure coefficient of case 4 experiences abrupt fluctuations, with rapid decreases and increases, while case 3 maintains a steady and considerable pressure difference. At t/T = 0.35, the pressure distribution continues to change, with case 4 displaying a greater pressure difference compared to case 3. At the time steps (t/T = 0.45 to t/T = 0.50), both cases exhibit the same horizontal projection length.

4. Conclusions

This study investigated the effects of modifying the leading- and trailing-edge shapes of an oscillating wing to evaluate its aerodynamic performance. Unsteady simulations were performed utilizing the overset mesh technique, in combination with a k−ω SST turbulence model. The wing’s pitch angles were varied at 80°, 85°, and 90°. The results indicated that the highest power was achieved with the following modifications: a semicircular leading edge combined with a rectangular trailing edge, as seen in case 3, at a pitch angle of 85°. The maximum power output was obtained at a thickness corresponding to 32% of the chord length. The peak power output coefficient (${\mathrm{C}}_{\mathrm{p}\mathrm{t}}$) of 1.214 indicated a 16.73% improvement over the rectangular flat plate, while the maximum efficiency of 45.46% reflected a 12.77% increase. This research highlights the significant impact of modifying the leading-edge shape over the trailing-edge shape for enhanced power extraction. The rounded shape of the leading edge provided a superior power output compared to other shapes, while the rectangular trailing edge outperformed the sharp trailing-edge shape, offering higher performance across multiple configurations. The current results indicate that this configuration can significantly enhance energy harvesting efficiency, while also offering the added benefit of enhanced structural integrity.