Performance Evaluation of Flapping-Wing Energy Harvester in Confined Duct Environments

Abstract

This study investigates the impact of different duct designs on the energy-harvesting performance of oscillating-wing systems in both partially and fully confined environments. Numerical simulations were conducted to examine the effects of straight, convergent–straight, and convergent–divergent duct configurations on the aerodynamic forces and overall energy extraction efficiency. Under partial confinement, the convergent–divergent duct demonstrated a significant improvement of 67.5% in power output over the ductless baseline configuration. This enhancement is attributed to the increased incoming flow velocity and amplified pressure difference around the wing, which improve the effectiveness of energy generation. However, the straight and convergent–straight ducts reduced the harvester’s performance due to the diminished flow velocity within each duct. Under full confinement, all duct configurations substantially enhanced energy-harvesting performance, with the convergent–straight duct providing the highest efficiency gain (84.9%). This improvement is primarily due to the increased velocity and pressure differential across the wing surfaces, which maximise the heaving force and overall energy generation performance. These findings highlight the critical role of duct geometry in optimising energy-harvesting performance, both in partially confined and fully confined flow environments.

1. Introduction

The quest for renewable energy sources has led to the development of multiple innovative methods to harness energy from natural phenomena. One such emerging technology is the flapping-foil energy harvester, which imitates the movement of biological swimmers and fliers to convert the kinetic energy of fluid flow into electrical energy [1,2,3,4,5]. Unlike traditional wind turbines or hydroelectric generators, flapping-foil devices operate based on unsteady flow dynamics, offering potential advantages in low-speed, turbulent environments, such as rivers, ocean currents, and atmospheric winds. These devices consist of foils or wings that oscillate in response to fluid flow and pressure differentials, driving energy conversion. The potential for high efficiency, a compact design, and adaptability to various environmental conditions makes flapping-foil systems an attractive area for research and development [6,7,8]. This introduction lays the groundwork for understanding the fundamental concepts, design considerations, and performance metrics related to flapping-foil energy harvesters.

The exploration of flapping-foil technology began with pioneering investigations demonstrating its practical feasibility. McKinney and DeLaurier [9] conducted the first experimental investigations, showing that energy could be extracted from an aerofoil oscillating in a wind tunnel. Building on this foundation, Triantafyllou et al. [10] explored the hydrodynamic principles underlying fish propulsion, highlighting the potential for harnessing energy through the oscillatory motions of aerofoils. This foundational research led to the development of prototypes and experimental setups for applying these principles to engineering systems. Subsequent numerical simulations and experimental studies by Jones and Platzer [2] and Jones et al. [11] confirmed the effectiveness of energy harvesting at low wind speeds and small angles of attack, with the results closely matching those from earlier wind-tunnel experiments. Zhu and Peng [12] further explored the energy-harvesting potential of flapping foils by analysing the coupling of pitching and heaving motions, focusing on vortex control and optimal axis placement. Conversely, Peng and Zhu [13] examined flow-induced oscillations in a foil system, identifying periodic pitching as the key for efficient energy capture. Dumas and Kinsey [14] and Kinsey and Dumas [15] expanded on these insights by conducting numerical simulations, which showed that, while foil shape has a lesser effect on energy generation, motion-related parameters have a significant impact, with up to 35% efficiency being achievable under optimal operating conditions.

The recent research has focused on optimising flapping-foil systems for better efficiency and durability. Shimizu et al. [16] performed multi-objective optimisation using an adaptive neighbouring search algorithm to maximise power and efficiency. They used self-organising maps to visualise trade-offs between power and efficiency, identifying crucial relationships among design variables and objective functions. They found that the heaving amplitude, reduced frequency, and phase-delay angle significantly influence performance. Specifically, when prioritising power, the system exhibited large heaving motions at low frequencies, while emphasising efficiency led to smaller-amplitude motions at higher frequencies. Liu et al. [17] utilised a multi-fidelity evolutionary algorithm to optimise the kinematics of flapping foils, observing that leading-edge vortex behaviour significantly impacts energy extraction. Experimental studies by Kim et al. [18] revealed that, while the cross-sectional shape of the hydrofoil minimally affects power generation, variations in the heaving motion and aspect ratio significantly influence efficiency. The heaving motion contributes more to efficiency at lower frequencies and optimal pitching amplitudes, while the efficiency from pitching decreases with reduced frequency due to the delayed development of the leading-edge vortex. Increasing the aspect ratio of the hydrofoil enhances the heaving efficiency but has a negligible effect on the pitching efficiency. Furthermore, the advancements in flexible-foil designs [19,20,21,22,23,24,25,26,27], the integration of flaps [28,29,30,31,32,33,34,35,36,37,38], and the exploration of different motion profiles [39,40,41,42] and tandem-foil configurations [43,44,45,46,47,48,49,50,51] continue to refine and improve the power extraction capabilities of flapping-foil systems, establishing flapping foils as a promising alternative for wind energy conversion.

Although the extensive exploration of various optimisation techniques has highlighted the potential for enhancing the energy conversion efficiency of flapping-foil systems, these systems are theoretically constrained by the “Lanchester–Betz” limit, which caps the efficiency at 59% for an unconfined turbine [52]. However, theoretical studies [53] have shown that duct effects can significantly increase power extraction, allowing efficiencies to surpass this limit in unconfined rotary turbines. Ducts, whether fully or partially confined, play a crucial role in this enhancement. For instance, Gauthier et al. [54] found that a fully confined straight duct could increase efficiency by nearly 50% at a 25% blockage ratio and almost double it when the blockage ratio is 50%, due to the blockage effect of the flapping foil. Conversely, experiments by Karakas and Fenercioglu [55] indicated that performance declines when the flapping foil is too close to the side walls, which underscores the importance of optimal spacing. Building on this idea, Iverson et al. [56] suggested applying a blockage correction to water tunnel data at higher reduced frequencies to align such data with the numerical results obtained from unconfined environments. Furthermore, Yang et al. [57] explored the effects of confining straight ducts on oscillating hydrofoils, finding that a proper foil–wall spacing can enhance efficiency by 19.34%. These findings suggest that, with a strategic duct design and optimal spacing, flapping-foil systems can achieve efficiencies beyond conventional limits, offering potential for further development. However, studies related to fully confined flapping-foil harvesters have primarily focused on straight ducts, with other duct types remaining underexplored.

Partial duct confinement introduces additional complexities. Liu et al. [58] reported that, in partially straight ducts, efficiency decreases as the gap between the confining plates and the foil narrows; this reduces the low-pressure area on the suction side, thereby lowering the pushing forces and power output. Additionally, they observed potential power losses due to the reduced mass flow in the duct, caused by the flapping foil. However, studies on partially divergent ducts [59] show that the power output can be significantly increased—by up to 35.8%—compared with an unconfined flapping-foil energy harvester. Divergent ducts enhance the mass flow through their entrance by creating a low-pressure area near the outlet, which is crucial for increasing the power output of wind or water turbines. Since the power output of a turbine is proportional to the cube of the flow speed, even a slight increase in the flow speed can significantly boost power. This intricate interplay highlights the complex relationship between spatial confinement and energy-harvesting performance, particularly under partial-confinement conditions.

While studies have explored straight or divergent ducts in either fully or partially confined setups, they have generally focused on isolated configurations. To the best of our knowledge, no study has systematically evaluated and compared the aerodynamic performance of multiple duct geometries—specifically, straight, convergent–straight, and convergent–divergent—under both partially and fully confined conditions. We address this gap by conducting a comprehensive computational-fluid-dynamics-based analysis across these configurations. Our dual-level confinement framework, combined with the different duct designs, provides new insights into the role of spatial confinement in modulating vortex dynamics and enhancing the energy capture. Thus, this study offers a novel and integrated perspective that expands the design space for efficient flapping-foil energy harvesters and contributes to the advancement of ducted renewable energy systems.

2. Mathematical Modelling

2.1. Problem Description

In this study, we investigate and compare the energy-harvesting performance of an oscillating wing across various duct configurations, as illustrated in Figure 1. In the unconfined configuration, the oscillating wing operates without a duct (Figure 1a). In the straight duct case, the wing is positioned within a straight duct that guides the incoming flow (Figure 1b). The convergent–straight duct case involves a duct with a convergent section leading into a straight duct (Figure 1c). Finally, in the convergent–divergent duct case, the wing is enclosed within a duct featuring a convergent section followed by a divergent section (Figure 1d).

Each configuration is analysed under two flow confinement conditions, as shown in Figure 2. The first scenario represents a partially confined setup within a 50c-wide channel, where c denotes the chord length. In this configuration, the duct throat remains 3.5c wide, whereby fluid can enter both through the duct and around it, which simulates an open environment (Figure 2a). For both the convergent–straight and convergent–divergent ducts, the convergent section is 1.5c in length, with a 15° convergence angle. In the convergent–straight case, the convergent section is followed by a 3c-long straight section, while, in the convergent–divergent case, it transitions into a 3c-long divergent section with a 15° divergence angle. These geometric parameters were selected to ensure consistency with previous studies and to capture representative flow conditions with minimal wall-induced effects [60].

The second scenario involves a fully confined setup within a narrower channel with a 3.5c width, which forces all incoming fluid to pass through the duct (Figure 2b), thereby eliminating any external bypass flow. The duct geometry remains identical to that in the wide-channel case. This configuration enables a focused investigation into the effects of geometric confinement on flow structures and energy-harvesting behaviour, elucidating the differences between confined and unconfined environments.

2.2. Kinematics

To analyse the performance of the oscillating wing inside the ducts, we selected an NACA0015 aerofoil with a chord length of c, which has been shown to deliver superior performance compared with other aerofoil thicknesses [15]. The wing undergoes simultaneous heaving and pitching motions, as illustrated in Figure 3. The pitching motion follows the sinusoidal pattern described by Equation (1) [15]:

$$\theta \left(t\right)={\theta }_{\mathrm{o}} sin\left(\omega t\right)$$

(1)

where $\theta \left(t\right)$ is the pitch angle at time t, ${\theta }_{\mathrm{o}}$ is the maximum pitch angle, and $\omega$ is the angular frequency, defined as 2πf (where f is the flapping frequency). The reduced frequency, f*, which characterises the unsteadiness of the oscillating motion, is defined as f* = fc/U∞, where c is the chord length of the foil, and U∞ is the free-stream velocity.

The heaving motion, which is the vertical movement of the foil, also follows a sinusoidal pattern, with a phase shift $\phi$ that accounts for any delay in the motion:

$$h\left(t\right)=H_{\mathrm{o}} sin(\omega t+\phi )$$

(2)

where $h\left(t\right)$ represents the vertical position at any instant, and $H_{\mathrm{o}}$ is the maximum vertical displacement. The total average power output, $\overline{P}$, is the sum of the average power contributions from the heaving (${\overline{P}}_y$) and pitching (${\overline{P}}_m$) motions:

$$\overline{P}=\overline{P_y}+{\overline{P}}_m$$

(3)

This power output can be expressed as the following integral over a period $T$:

$$\overline{P}={\int }_0^T\left[F_Y\left(t\right)V_y\left(t\right)+M\left(t\right)\Omega \left(t\right)\right]dt$$

(4)

where $F_Y$ is the heaving force, $V_y$ is the vertical velocity, $M$ is the pitching moment, and $\Omega$ is the angular velocity.

The overall average power coefficient (${\overline{C}}_{pt}$) is calculated by summing the respective power coefficients for heaving (${\overline{C}}_{py}$) and pitching (${\overline{C}}_{pm}$):

$${\overline{C}}_{pt}={\overline{C}}_{py}+{\overline{C}}_{pm}$$

(5)

Finally, ${\overline{C}}_{pt}$ is expressed as the ratio of the total average power output to the kinetic energy of the incoming flow, which depends on the fluid density, flow velocity, and foil dimensions:

$${\overline{C}}_{pt}={\displaystyle \frac{\overline{P}}{0.5\rho U_{\infty }^{3}bc}}$$

(6)

where b is the wingspan (equal to 1 in 2D simulations), and d indicates the overall vertical displacement of the wing. For fully constrained configurations, the inflow energy is evaluated using the cross-sectional area b × d.

The efficiency (η) is calculated as

$$\eta ={\overline{C}}_{pt}{\displaystyle \frac{c}{d}}$$

(7)

2.3. Numerical Methodology

We conducted 2D unsteady numerical simulations in ANSYS Fluent 21 to assess the energy harvester’s performance at a Reynolds number (Re) of 5 × 10⁵, a value typically relevant for small-scale applications [45,61]. To accommodate the dynamic motion of the aerofoil, an overset-mesh approach was employed within the computational domain, similar to previous studies [40,59]. We analysed the fluid flow around the oscillating wing in each of the four configurations shown in Figure 1, under the two conditions illustrated in Figure 2a,b. Figure 4 and Figure 5 depict the computational domains and mesh configurations used for the partially and fully confined scenarios, respectively.

For the partially confined channel, the computational domain extends from $-20c$ to $50c$ in length and from $-25c$ to $25c$ in width (Figure 4a). A velocity inlet is applied 20c upstream from the wing’s hinge point, with a pressure outlet positioned 50c downstream [59]. The top and bottom boundaries are designated as slip walls, which allow inviscid flow without frictional interaction. The subgrid region (Figure 4b) comprises a stationary grid for the duct and a moving grid for the aerofoil. These grids are finely meshed to accurately capture the fluid dynamics around the oscillating wing and duct. The space between the duct walls is fixed at 3.5c across all configurations, which creates a partially confined environment influencing the aerodynamic forces acting on the wing.

For the fully confined channel (Figure 5), the domain spans from $-20c$ to $50c$ in length and from $-1.75c$ to $1.75c$ in width (at the throat), similar to the fully confined channels investigated in previous studies [54,57]. The inlet is positioned −20c upstream of the foil’s hinge point, while the outlet extends 50c downstream from the hinge point. As with the partially confined channel, velocity inlet and pressure outlet boundary conditions are applied, and the upper and lower boundaries are designated as walls; this setup simulates a fully confined channel. The moving subgrid for the oscillating wing (pink) accurately captures the wing’s fluid dynamics, with its motion controlled by a user-defined function in both scenarios.

The flow field is solved using Reynolds-averaged Navier–Stokes equations, coupled with the $k-\omega$ shear stress transport turbulence model, which is well-suited for capturing the complex turbulence phenomena around oscillating aerofoils [40,62]. We utilised a coupled algorithm for the numerical simulations to ensure a stable and accurate pressure–velocity coupling. Regarding spatial discretisation, the least-squares cell-based method was applied for the gradient, while second-order upwind schemes were used for the pressure, momentum, turbulent kinetic energy, and specific dissipation rate. These high-order discretisation techniques were chosen to enhance the accuracy of the flow solution, particularly in resolving intricate details around the moving aerofoil. This combination of methods ensured reliable and precise simulations, even in the challenging dynamic environment imposed by the foil’s oscillating motion. As shown in

Table 1. Physical parameters of oscillating-wing energy harvester.

Parameter	Symbol	Value
Profile	–	NACA0015
Chord length	c	1.0
Pitch centre	xp	c/3
Heave amplitude	Ho/c	1
Pitch amplitude	θo	75°
Frequency	f*	0.14
Reynolds number	Re	5 × 105
Phase angle	$\varphi$	90°
Duct width at throat	w	3.5c
Convergence angle	–	15°
Divergence angle	–	15°

, the same key parameters were adopted across all configurations to ensure meaningful comparisons.

To confirm the mesh independence of the numerical model, we tested three distinct mesh densities—designated as D1, D2, and D3—for the flapping foil operating without any duct confinement (

Table 2. Comparison of average power coefficients between different mesh densities in unconfined turbine configuration.

Mesh Density	Grid Cells in Stationary Domain	Grid Cells in Moving Domain	${\overline{\mathit{C}}}_{\mathit{p}\mathit{y}}$	${\overline{\mathit{C}}}_{\mathit{p}\mathit{m}}$	${\overline{\mathit{C}}}_{\mathit{p}\mathit{t}}$	% Difference in ${\overline{\mathit{C}}}_{\mathit{p}\mathit{t}}$
D1	55.6 × 103	32.8 × 103	1.074	−0.088	0.986	−
D2	91.3 × 103	54.1 × 103	1.101	−0.108	0.993	0.7
D3	125.8 × 103	82.5 × 103	1.108	−0.112	0.996	0.3

). Notable variations in the computed values were observed when the mesh was refined from D1 to D2. In contrast, transitioning from D2 to D3 resulted in only marginal differences, which indicates that additional mesh refinement beyond D2 would not alter the simulation outcomes significantly. Consequently, D2 was selected for the simulations as it provided the optimal balance between computational efficiency and accuracy.

For time discretisation, we used a constant time-step size, corresponding to 2000 steps per flapping cycle. This temporal resolution was chosen to accurately capture the unsteady flow structures and transient force dynamics. The selected time-step size aligns with values commonly reported in the literature [15,63], optimising the trade-off between computational cost and temporal accuracy. Each simulation was run for 10 complete flapping cycles to ensure the convergence of the periodic behaviour. The cycle-to-cycle variation between the 9th and 10th cycles was found to be less than 0.5%, which confirms the temporal stability of the solution.

To verify the accuracy of the numerical model and mesh, we compared our results with those reported by Kinsey et al. [63]. The validation process involved an examination of an NACA0015 aerofoil. Figure 6 displays the simulation results for this aerofoil at an Re of 5 × 10⁵, a heave amplitude-to-chord ratio (H_o/c) of 1.0, a pitch amplitude (θ_o) of 75°, and a reduced frequency (f*) of 0.14. The drag coefficient (C_x), pushing force coefficient (C_y), and moment coefficient (C_m) computed by our model align closely with the values obtained by Kinsey et al. [63]. This agreement confirms the precision of the numerical model and mesh used in this study.

Furthermore, to validate the findings regarding the wall confinement effect observed in this study, the NACA0015-based flapping-wing energy harvester was simulated in a partially confined space between two thin straight walls. The wall length and distance were 20c and 6c, respectively, while the foil was hinged at 5c from the wall inlet, similar to the configuration employed by Liu et al. [58]. The simulations were performed using optimal kinematic parameters to achieve high efficiency (H_o/c = 1.0, θ_o = 75°, f* = 0.14, x_p = 0.33c, and φ = 90°) at a Reynolds number of 1100. Figure 7 compares the pushing force and moment coefficients simulated in this study with the corresponding values reported by Liu et al. [58]. The congruence in the C_y and C_m values between the two sets of results validates the proposed computational model and methodologies, confirming that our simulations accurately captured the relevant fluid–structure interactions.

3. Results and Discussion

3.1. Duct Configurations Without Oscillating Wing

Figure 8 presents a comparative analysis of the pressure coefficient and velocity magnitude contours across the straight, convergent–straight, and convergent–divergent duct configurations under partially and fully confined conditions.

The straight duct configuration showed the same pressure contours for the partially and fully confined channels, as shown in Figure 8a,b. This uniformity was mirrored in the velocity magnitude contours, which showed negligible differences between the two channels (Figure 8c,d). In the convergent–straight duct configuration under the partially confined condition, the convergent section reduced the pressure and increased the velocity of the flow (Figure 8a,c). This reflects the fluid’s natural tendency to flow along the path of least resistance, with the amount of flow entering the duct being reduced in the partially confined channel. In the fully confined channel, the convergent–divergent section significantly increased the incoming fluid velocity since the mass flow rate was constant through the duct; this resulted in a pressure reduction in the straight section (Figure 8b,d).

In the partially confined convergent–divergent duct, the convergent–divergent section reduced the pressure and increased the velocity. While this pattern was consistent across both channel types, the increase in velocity was notably higher in the partially confined channel than in the fully confined one. These findings indicate that the geometry of the duct directly influences the flow behaviour inside it in both confinement scenarios.

Figure 9 compares the pressure coefficients and velocity magnitudes between all the duct configurations under partial and full confinement. Figure 9a,b illustrate the pressure coefficient variations along the duct centreline (X-axis) in the partially and fully confined channels, respectively. Under partial confinement (Figure 9a), the straight duct maintained a nearly constant pressure coefficient throughout its length. The pressure coefficient in the convergent–straight configuration decreased as the flow approached the convergent section, whereafter it stabilised. Similarly, in the convergent–divergent duct configuration, the pressure coefficient dropped near the convergent section before recovering in the divergent section.

Under full confinement (Figure 9b), the inlet pressure in the straight duct was slightly higher than the outlet pressure because of the existence of the duct wall. However, the convergent–straight configuration induced a more pronounced drop in the pressure coefficient due to the geometric constriction, downstream of which the pressure stabilised. The convergent–divergent configuration exhibited a comparable drop near the convergent section, followed by a partial recovery, but with less variation than in the partially confined channel.

Figure 9c,d show the changes in velocity along the centreline (X-axis) of the ducts under partial and full confinement, respectively. Under partial confinement (Figure 9c), the straight duct exhibited a uniform flow velocity throughout its length. In the convergent–straight configuration, the velocity was noticeably reduced within the convergent section. The convergent–divergent duct configuration displayed a marked increase in velocity, which peaked at the throat section. Under full confinement (Figure 9d), the inlet velocity was the same across all three duct configurations; however, the velocity at the geometric discontinuities varied significantly, especially in the convergent–straight configuration.

These observations emphasise the significant impact of duct geometry on fluid flow dynamics, with the channel width influencing the velocity profiles considerably.

3.2. Energy-Harvesting Performance with Different Duct Designs Under Partial Confinement

The energy-harvesting performance of the flapping foil in different channel and confinement configurations was evaluated with the following parameters kept constant (Table 1): pivot point position (x_p = 0.33c), heave amplitude-to-chord ratio (H_o/c = 1.0), phase angle (φ = 90°), reduced frequency (f* = 0.14), and maximum wing pitch angle (θ_o = 75°).

Table 3. Overall power extraction coefficient for oscillating wing with different wide-channel duct configurations and differences in ${\overline{C}}_{pt}$ (%) compared with unconfined turbine.

	Unconfined (Baseline)	Straight	Convergent–Straight	Convergent–Divergent
${\overline{C}}_{py}$	1.101	0.982	1.002	1.514
${\overline{C}}_{pm}$	−0.108	−0.186	−0.193	0.092
${\overline{C}}_{pt}$	0.993	0.795	0.808	1.606
$\Delta {\overline{C}}_{pt}$ (%)	–	−19.8	−18.4	67.5
$\eta$ (%)	38.9	31.2	31.7	62.9

compares the average heaving power coefficient (${\overline{C}}_{py}$), average moment power coefficient (${\overline{C}}_{pm}$), and average total power coefficient (${\overline{C}}_{pt}$) of an oscillating-wing energy harvester across all the duct configurations considered. The percentage change in the total power coefficient ($\Delta {\overline{C}}_{pt}$) relative to the baseline configuration (without a duct) is also included, alongside the energy efficiency (η) for each configuration.

The straight duct configuration showed a marginal reduction in ${\overline{C}}_{py}$ but a more pronounced increase in the magnitude of ${\overline{C}}_{pm}$. It also exhibited a notable decline in ${\overline{C}}_{pt}$, which signifies a reduction in overall performance, with the efficiency being 31.2%. This suggests that straight ducts may inhibit the fluid dynamics that contribute to efficient energy harvesting. Conversely, the convergent–straight duct yielded a slightly better ${\overline{C}}_{py}$ but a marginally higher ${\overline{C}}_{pm}$ compared with the straight duct. ${\overline{C}}_{pt}$ displayed a minor improvement over the straight duct configuration, and the efficiency increased slightly to 31.7%, although the total power coefficient ${\overline{C}}_{pt}$ decreased.

The convergent–divergent duct configuration excelled, displaying a marked increase in ${\overline{C}}_{py}$ and achieving a positive ${\overline{C}}_{pm}$ value. This led to a considerable surge in ${\overline{C}}_{pt}$, which signifies a substantial improvement over the baseline configuration and highlights the significant influence of duct geometry on the harvesting efficiency; this is also evident from the efficiency of 62.9%. The significantly higher efficiency of this configuration suggests an optimal interplay between the wing and the fluid flow, which maximises the energy extraction. This is ascribed to the convergent–divergent configuration acting as a Venturi: the convergent section raises the local dynamic pressure, intensifying the leading-edge vortex on the suction side, while the divergent diffuser delays the flow detachment and maintains a large spanwise pressure differential until mid-stroke.

Figure 10 illustrates the effects of oscillating wings with different duct configurations on the pushing force coefficient (C_y), pushing power coefficient (C_py), moment power coefficient (C_pm), and total power generated (C_pt) within an oscillation cycle. Figure 10a and 10(b) display the variations in C_y and C_py, revealing that, when the flow entered the straight and convergent–straight ducts, C_y decreased relative to the baseline case during the upward and downward movement of the oscillating wing. However, the variation in C_y and C_py significantly increased in the convergent–divergent duct configuration. Figure 10c,d present the variations in C_pm and C_pt throughout a cycle across all duct configurations. The variation trend for C_pm was similar across the configurations until 0.3t/T; from 0.3t/T to 0.5t/T, C_pm decreased in the straight and convergent–straight configurations, whereas it increased in the convergent–divergent configuration, which resulted in a positive ${\overline{C}}_{pm}$ value for the latter. C_pm fluctuated between negative and positive values over the oscillation cycle. Consequently, the time-averaged ${\overline{C}}_{pm}$ value was relatively small and had less influence than ${\overline{C}}_{pt}$, as shown in Table 3. The C_pm peak in the unconfined configuration differed from that in the other configurations, possibly due to the effect of the channel wall. The reflected pressure from the channel wall varies depending on the wall angle and the distance of the wall from the wing. As shown in Figure 11 and Figure 4, the pressure contours also differed significantly in the presence of the channel wall.

During the time interval T, C_pt also decreased in the straight and convergent–straight ducts compared with the baseline, whereas it increased in the convergent–divergent duct. Therefore, the maximum ${\overline{C}}_{pt}$ was achieved by the convergent–divergent configuration, as seen in Table 3.

Figure 11 showcases the streamlines and velocity profiles, along with the pressure contours around the foil’s surface, for various duct configurations at 0.25t/T, an instant at which significant differences in the force, moment, and power coefficient were observed across the configurations. At this instant, the straight and convergent–straight ducts showed no discernible differences, while the convergent–divergent configuration exhibited a notable expansion of the high-velocity region on the wing’s lower surface (Figure 11a). The pressure contours in Figure 11b display similar patterns for the bare, straight, and convergent–straight configurations, which suggests that a more in-depth analysis is required in order to understand the pressure distribution on the wing surface at this phase of the cycle. Conversely, a significant pressure reduction in the divergent section was evident in the convergent–divergent duct.

To examine the pressure disparities across the duct configurations in greater depth, we plotted the corresponding pressure coefficients under partial confinement. Figure 12 presents the pressure distribution along the X-axis of the wing for all wide-channel configurations at 0.25t/T. The data vividly illustrate a considerable drop in the pressure on the lower wing surface for the convergent–divergent duct; this corresponds to an increased pressure differential across the wing surfaces, which enhanced the heaving force and power output. By contrast, a marginal increase in pressure on the lower wing surface was observed for the straight and convergent–straight ducts, which resulted in a lower pressure difference compared with the baseline configuration and, consequently, a reduction in the heaving force and power output.

3.3. Energy-Harvesting Performance with Different Duct Under Full Confinement

The energy-harvesting performance of the oscillating wing with various duct configurations under full confinement was comparatively analysed against the baseline scenario.

Table 4. Overall power extraction coefficient for oscillating wing with different duct configurations under full confinement and differences in ${\overline{C}}_{pt}$ (%) compared with the unconfined turbine.

	Unconfined (Baseline)	Straight	Convergent–Straight	Convergent–Divergent
${\overline{C}}_{py}$	1.101	1.695	2.007	1.925
${\overline{C}}_{pm}$	−0.108	0.121	0.158	0.197
${\overline{C}}_{pt}$	0.993	1.817	2.165	2.122
$\Delta {\overline{C}}_{pt}$ (%)	–	82.9	118.1	113.7
$\eta$ (%)	38.9	71.2	84.9	83.2

summarises the average coefficients of power (${\overline{C}}_{py}$, ${\overline{C}}_{pm}$, and ${\overline{C}}_{pt}$), the percentage change in ${\overline{C}}_{pt}$ (denoted as $\Delta {\overline{C}}_{pt}$), and $\eta$ for each duct configuration.

With a narrow channel, all three duct configurations significantly enhanced ${\overline{C}}_{py}$ and ${\overline{C}}_{pm}$ compared with the baseline configuration. While the straight duct configuration substantially improved the power output and overall system efficiency, the convergent–straight duct performed even better, delivering the most significant gains in both efficiency and power output. These improvements underscore the effectiveness of duct modifications, especially convergent geometries, in harnessing energy more efficiently by optimising the fluid dynamics in narrow channels.

Figure 13 presents the variations in the force, moment, and power coefficients throughout an oscillation cycle for various narrow-channel duct configurations. Figure 13a,b depict the C_y and C_py variations over a cycle. Notably, C_y and C_py exhibited higher values in the convergent–straight duct.

Figure 13c,d present the variations in C_pm and C_pt throughout a cycle for all duct configurations. Evidently, the variation pattern of C_pm was similar across all cases until 0.2t/T; however, from 0.2t/T to 0.5t/T, C_pm increased for all three duct configurations. The variation in C_pt was the maximum in the convergent–straight configuration during most of the cycle.

Figure 14 presents the streamlines, velocity profiles, and pressure contour plots around the wing surfaces for all duct configurations at 0.25t/T. The flow around the foil was altered significantly by the ducts relative to the unconfined configuration. The downward pitching of the wing increased the pressure on the upper surface across all duct configurations, as shown in Figure 14b. Correspondingly, the velocity increase along the lower surface reduced the pressure, thereby amplifying the pressure differential across the wing surfaces, which is beneficial for increasing the pushing force.

This change in the pressure distribution is quantified in the plots of the pressure coefficient along the wing’s X-direction (Figure 15) for all duct configurations at 0.15t/T. The straight and convergent–straight ducts produced the highest pressure on the upper wing surface. The convergent–straight duct provided the lowest pressure on the lower surface, with the convergent–divergent duct producing a slightly higher pressure. Thus, the largest pressure differential was observed in the convergent–straight duct configuration, which produced the maximum pushing force at the aforementioned instant.

4. Conclusions

This study investigated the crucial role of duct design in enhancing the energy extraction efficiency of oscillating-wing systems, particularly under different levels of confinement. The findings explain how various duct geometries affect the aerodynamic forces and energy conversion performance in both partially confined and fully confined environments.

In the partially confined wide channel, the convergent–divergent configuration emerged as the most effective design, delivering a significant 67.5% increase in power output compared with the baseline. This improvement is primarily attributed to the duct’s ability to accelerate the incoming flow, which enhances the energy conversion efficiency. However, the results also revealed that the straight and convergent–straight configurations compromised performance, with the reductions in flow velocity within the duct negatively impacting the system’s energy extraction capabilities. These findings underscore the complexities associated with partially confined systems, where the optimal choice of duct geometry is critical for avoiding unfavourable flow interactions that can diminish the pushing forces and overall efficiency.

In the fully confined narrow channel, all duct configurations—straight, convergent–straight, and convergent–divergent—yielded substantial improvements in power output and efficiency. Specifically, the convergent–straight duct provided the most significant efficiency gain, with an increase of 84.9% over the baseline. This enhancement is largely due to the increased flow velocity and the resulting pressure differential across the wing surfaces, which maximise the heaving force and overall energy extraction performance. The findings thus confirm that fully confined duct designs, especially those that accelerate flow and enhance pressure distribution effectively, are highly effective in boosting the performance of oscillating-wing energy harvesters.

Importantly, this study relied on 2D simulations, which do not capture 3D effects, such as tip vortices and spanwise flow. These effects may alter the results slightly, particularly in confined environments. Future studies will investigate these aspects.