At specified Reynolds numbers , this study investigates the power-extraction regime of a flapping-foil turbine executing a figure-eight trajectory. This study thoroughly explored the impacts of trajectory shape, heave and pitch amplitudes, phase difference, and pitch axis location on the power-extraction regime of a flapping turbine. A figure-eight trajectory substantially bolsters the energy harvesting capacity, achieving a peak efficiency of up to 50%. This trajectory capitalizes on the energy generated from the drag encountered by the flapping foil, thereby marking a significant efficiency breakthrough. This could denote a considerable progression for flapping foils tailored for heave and pitch motions since the free stream can be optimally harnessed by the trajectory we have established. In scenarios where the flapping foil undergoes the same maximum effective angle of attack, a wake diffusion spectrum aligns with the Betz limit threshold. Crucially, the closeness to this efficiency frontier suggests a universal maximum effective angle of attack—a consistent law that remains valid for the conventional flapping turbine design. It implies that selections for stroke and pitch amplitudes should be methodical rather than arbitrary. In addition, the positioning of the pitch axis ought to be modified in tandem with phase difference adjustments to bolster the synchronization between heaving motions and resultant lift. The ideal phase difference is variable, contingent on pitch amplitude and pitch axis position. Even with low pitch amplitudes, fine-tuning the phase difference guarantees that the energy harvesting efficiency does not fall below 30%. Such an enhancement would significantly broaden the operational envelope and the parameter space for flapping turbines.
I. INTRODUCTION
Flapping-foil systems have emerged as a highly active research area in recent years.1,2 The flight of insects, the swimming motion of fish, and the harmonic beating of foils provide particularly interesting examples of such systems. These common modes of motion in nature are not only useful for propelling objects but also for energy harvesting.3–5 Compared to traditional windmills or water wheel designs, flapping-foil energy harvesting devices have numerous advantages, such as being environmentally friendly, easy to deploy, and producing low noise.6 The development of renewable energy utilization technology is a boon for the decreasing reliance on fossil fuels,7 and enhancing the energy harvesting performance of flapping foils adds a vibrant facet to renewable energy technologies.
The pursuit of energy harvesting efficiency in flapping foils has never ceased, and breakthroughs in this area will facilitate their industrial applications.8,9 Since McKinney and DeLaurier first researched the feasibility of a windmill consisting of a rigid horizontal wing oscillating in plunge (vertical translation) and pitch,10 an efficiency of up to 28.3% has been achieved in a wind tunnel model. At low Reynolds numbers, Kinsey and Dumas established an energy harvesting regime and realized an efficiency of 34%.11 It seems that the efficiency limit for flapping foils operating in heave and pitch modes has been reached. Following this, researchers have shown significant interest in the study of low and moderate Reynolds number flapping-foil turbines,9,12,13 with an emphasis on enhancing their fluidic properties and energy harvesting capabilities. Researchers have begun to improve efficiency through various approaches, including modifying the geometric shape of the airfoils,14,15 adopting non-sinusoidal trajectories,16–18 and employing flow control.19,20 Despite these efforts, the highest efficiency achievable by flapping foils is still significantly short of the Betz limit,21 indicating that more breakthrough methods need to be adopted. In fact, in the natural world, the motion trajectory of birds' wings is often figure-eight shaped, rather than merely performing heave and pitch motions. The limitation of motion patterns will result in the drag on the flapping foil not being utilized, and the mode of harvesting energy solely through lift needs to be broken. Under an elliptical motion trajectory, Swain and others22 greatly enhanced the energy extraction capability of tandem flapping foils. Their research showed that the energy enhancement of tandem beating wings largely depends on the formation of the wake or deflected wake in the presence of shed vortices. Under favorable conditions, the formation of dipoles was observed, indicating that the trajectory can have a profound impact on the wake. Additionally, this mode of motion is similar to that of a fish's tail,23 with the application inspired by the natural world.
In this study, the choice of the figure-eight trajectory was not arbitrary. In the limit-cycle oscillations, the unsteady drag frequency experienced by the flapping foil is twice the frequency of the unsteady lift.24 The selection of the trajectory needs to take into account the laws of force change, which is conducive to energy harvesting. Moreover, for a cylinder undergoing two-degree-of-freedom oscillations, the trajectory it follows often forms a figure-eight shape.25 This suggests that the figure-eight trajectory in fluid dynamics is consistent with the motion pattern that facilitates energy exchange between the fluid and the solid.
In the general assumption, the inflow is uniform and flows over the airfoil surface from a horizontal plane. The airfoil has a certain angle with the inflow, leading to the production of lift, but the use of drag is often overlooked. Drag can still be considered as part of the inflow with energy, which should be harnessed rather than discarded; hence, more energy can be utilized. Furthermore, a substantial amount of research has been concentrated on low to moderate Reynolds numbers,26–28 with relatively fewer studies focused on the middle-to-high Reynolds number range. The energy harvesting efficiency of flapping turbines increases with higher Reynolds numbers, and practical engineering applications tend to favor operations in the middle-to-high Reynolds number range due to the potential for greater power generation. Despite the myriad of strategies aimed at improving energy harvesting, a unified improvement mechanism seems elusive, providing limited guidance for engineering applications. Additionally, studies on fluidic parameters often explore single variables, leaving the interactions among multiple parameters largely unknown. Conclusions drawn from controlling a single variable may be abstract and lack universality, overlooking the intricate interplay among parameters. Considering the comprehensive impact of multiple parameters is essential for gaining more valuable insights, and discovering more universally applicable principles is imperative.
This study explores the shape, stroke, pitch axis position, and phase difference of the figure-eight trajectory flapping turbine, which is accompanied by changes in pitch amplitude. The changes in lift generated are studied with the help of the effective angle of attack, and the efficiency improvement mechanism is combined with the flow field analysis. Pay more attention to the universal law of the power-extraction regime of a flapping turbine, and put it on traditional flapping turbine for verification. The trajectory set in this study may be more universal. After all, the movement pattern of traditional flapping turbine is only a special case in this study. In addition, considering that the total power coefficient is occasionally negative, the flapping foil is phase-adjusted to reduce energy loss. Taking the influence of phase difference and pitch axis position into consideration, a large number of cases are comparatively analyzed, and there are a few studies in this part. This study considered the comprehensive impact of multiple parameters and obtained some conclusions that are not detectable by changes in a single parameter.
The main purpose of this study is to explore the power-extraction regime of a figure-eight trajectory flapping foil and analyze the law of efficiency changes. Section II introduces the physical model and numerical methods adopted, including trajectory setting, efficiency calculation, numerical methods, and verification. Section III is the main research content. First, different figure-eight trajectories are studied, and then, the impact of the stroke is investigated. Finally, the influence of the pitch axis position was studied. This study involved thousands of cases, examined the impact of multiple parameters, and provided solutions to some of the problems that emerged. Section IV reviews and summarizes all work.
II. FORMULATION AND NUMERICAL METHOD
A. Motion description
The motion pattern similar to the figure-eight trajectory is shown in Fig. 1. The trajectory of the pitching center of the flapping foil forms a figure-eight shape, and the flapping foil rotates around the pitch axis. In this study, the setup of the motion equation deviates from traditional motion patterns, primarily due to the additional streamwise motion involved. Considering the characteristics of the figure-eight trajectory, the motion equation is set as follows:
The feathering parameter significantly influences the foil's operational regime. At = 0, the foil is within the energy harvesting regime. When , the foil is in the feathering regime (no power extraction and no propulsion). Conversely, when , the foil is in the propulsive regime.
The feathering parameter significantly influences the foil's operational regime. At = 0, the foil is within the energy harvesting regime. When , the foil is in the feathering regime (no power extraction and no propulsion). Conversely, when , the foil is in the propulsive regime.
B. Numerical method and validation
1. Governing equation
2. Meshing
High-resolution 2D unsteady computations were carried out at a Reynolds number using the commercial finite volume code Fluent, adhering to essentially the same strategy as described in our previous study.20,36 To facilitate the study of energy harvesting performance of the flapping foil with freed streamwise degree of freedom, structured grids were employed for the numerical simulations, as shown in Fig. 3. In the current study, a symmetrical two-dimensional airfoil NACA0015 was used, executing periodic surging, pitching, and heaving motions. The grid was divided into the background grid and the airfoil grid. The background grid remained stationary, while the airfoil grid underwent both heaving and pitching motions, controlled by user-defined functions (UDF). Grid nodes where the background grid and the airfoil grid overlapped were deleted. The overset boundary condition was used for the periphery of the airfoil grid. An overview of the computational domain and boundary conditions is illustrated. The entire computational domain spanned 75 chord lengths from inlet to outlet, with a total width of 60 chord lengths for the symmetric boundaries above and below, and the rotation center of the airfoil was located 30 chord lengths from the inlet boundary. The entire domain was structured with a grid that was refined around the airfoil to accurately capture the flow field characteristics, including the addition of a boundary layer at the airfoil edges. To improve computational efficiency, a coarser structured grid was utilized for the background grid. To ensure that the grid size in the overlapping region between the background and the airfoil grids was similar, the background grid was refined in a rectangular area centered around the rotation center of the airfoil, measuring 9c × 8c. The airfoil grid was circular with a diameter of three times the chord length. The inflow velocity at the inlet was dependent on the desired Reynolds number, and the outlet was set as a pressure outlet at standard atmospheric pressure, with symmetric boundary conditions set for the top and bottom boundaries.
3. Validation
This study was conducted at a Reynolds number of , and in order to validate the accuracy of the numerical methods used, comparisons were made with the numerical simulation results of Kinsey and Dumas.37 The validation parameters were set at , , and . The percentage error in the efficiency, average lift power coefficient, and average torque power coefficient for different turbulence models and different meshes—Mesh1 (14w), Mesh2 (34w), and Mesh3 (56w)—compared to Kinsey's data is shown in Fig. 4. When using the k-epsilon turbulence model, the variation of different coefficients with the mesh is minimal, indicating good mesh independence in this study. However, there is a notable error in the torque power coefficient, which is influenced by the combined effects of lift and drag. Additionally, under the shear stress transport (SST) turbulence models, coefficients exhibit some fluctuations around the results of Kinsey and Dumas,37 but the errors are relatively small. These differences may arise from variations in the grid and turbulence model used compared to Kinsey and Dumas.37 In their study, Spalart–Allmaras (S–A) turbulence models and unstructured grids were employed. Therefore, in comparison, the numerical results using the S–A turbulence model in this research align closely with Kinsey et al.'s findings; thus, the S–A turbulence model was chosen. Figure 5 presents the comparison of lift, torque power coefficients, and total power coefficient over one cycle for different meshes. It was found that with Mesh1, instantaneous values could oscillate locally, and since Mesh2 has fewer grid points than Mesh3, resulting in higher computational efficiency, Mesh2 was chosen as shown in Fig. 5(b). Additionally, the numerical scheme adopted in this study matches well with that of Kinsey et al., as illustrated in Fig. 5(a). Furthermore, the percentage error comparison of the instantaneous lift, instantaneous torque energy coefficient, and instantaneous energy with the results of Kinsey et al. for different time steps with Mesh2 is shown in Table I. The numerical results show little error at different time steps, and when the time step is set to t/T = 2000, the results are in very close agreement. Moreover, the selection of the cycle during the limit-cycle oscillations of the flapping is also crucial, as shown in Fig. 6. To ensure computational efficiency without losing accuracy, the data from the seventh cycle were selected with a time step of 2000 after comparing computational data with different cycle values and time steps.
The disparity in numerical results under different meshes and turbulence models.
A comparison of the lift, torque power coefficient, and total power coefficient within a cycle. (a) Comparison of the numerical results of this study with the results of Kinsey and Dumas37 and (b) comparison of the numerical simulation results using different meshes.
A comparison of the lift, torque power coefficient, and total power coefficient within a cycle. (a) Comparison of the numerical results of this study with the results of Kinsey and Dumas37 and (b) comparison of the numerical simulation results using different meshes.
The efficiency η, the average lift power coefficient , the average torque power coefficient , and their respective errors , , and under different cycles.
Time steps . | η . | . | . | . | . | . | |
---|---|---|---|---|---|---|---|
500 | 0.456 | 2.11 | −0.51 | 2.3% | 2.1% | 0.6% | |
2000 | 0.453 | 2.10 | −0.51 | 1.7% | 1.4% | 0.0% | |
4000 | 0.457 | 2.11 | −0.51 | 2.4% | 2.1% | 0.7% |
Time steps . | η . | . | . | . | . | . | |
---|---|---|---|---|---|---|---|
500 | 0.456 | 2.11 | −0.51 | 2.3% | 2.1% | 0.6% | |
2000 | 0.453 | 2.10 | −0.51 | 1.7% | 1.4% | 0.0% | |
4000 | 0.457 | 2.11 | −0.51 | 2.4% | 2.1% | 0.7% |
The error in efficiency η, average lift power coefficient , and average torque power coefficient compared to the results of Kinsey and Dumas37 at different cycle numbers.
The error in efficiency η, average lift power coefficient , and average torque power coefficient compared to the results of Kinsey and Dumas37 at different cycle numbers.
III. RESULTS AND DISCUSSION
This study extensively explores various parameters, encompassing pitch amplitude , surge motion amplitude , heave amplitude , pitch-heave phase difference , surge-heave phase difference , and pitch axis position , as shown in Table II. The variation in pitch amplitude is a pervasive theme throughout this research, and its interaction with changes in other parameters collectively investigates the energy harvesting characteristics. The surge motion amplitude is held constant, and the shape and size of the trajectory rely on the control of heave amplitude . Changes in surge-heave phase difference will alter the wing's motion trajectory, discussed first, with the impact of heave amplitude considered subsequently. Finally, the effects of pitch axis position , pitch amplitude , and pitch-heave phase difference are also taken into account, as these parameters collectively influence the energy harvesting characteristics of the flapping wing.
The parameter variables considered in this study include pitch amplitude , surge motion amplitude , heave amplitude , pitch-heave phase difference , surge-heave phase difference , and pitch axis position .
/deg . | . | . | /deg . | /deg . | . |
---|---|---|---|---|---|
60–100 | 0.1 | 0.3–1.7 | 75–120 | 0–180 | 1/5–1/2 |
/deg . | . | . | /deg . | /deg . | . |
---|---|---|---|---|---|
60–100 | 0.1 | 0.3–1.7 | 75–120 | 0–180 | 1/5–1/2 |
A. Trajectory effect
Different streamwise phase differences will affect the flapping trajectory of the foil, as shown in Fig. 7. Here, we may as well first choose a streamwise amplitude of and a heaving amplitude of . Due to the multitude of varying parameters, we choose a heaving phase difference of , a choice that allows for a relatively high energy harvesting efficiency of the foil.31,32 Moreover, the pitching axis is positioned at a distance of one-third chord length ( = 1/3) from the leading edge of the foil. It is not hard to see that as the phase difference increases, the center of the flapping motion trajectory gradually moves toward the positive direction of the x axis. When the phase difference is and , the flapping reciprocating motion trajectories overlap, with their paths being a C-shape and an inverse C-shape, respectively. When the phase difference is between , the flapping trajectory during the reciprocating process is different, forming a figure-eight trajectory. As the phase difference increases, the intersection of the figure-eight trajectory gradually moves backward. When the phase difference is , the flapping trajectory forms a regular figure-eight. For motion trajectories under different phase differences, they are symmetrical about the (for example, the shapes of the trajectories at and are mirror images). There has not yet been a study on the energy harvesting performance of the foil for different trajectories, and these contents will be discussed below.
Flapping-foil pitching center's trajectory variation with phase difference.
To find the optimal mode for energy harvesting, a decay frequency of can be selected. In the study by Kinsey et al.,30 the highest energy harvesting efficiency was found at a decay frequency of . For different maximum effective angles of attack , this frequency could appear within the range of optimal efficiency. The variation of energy harvesting efficiency for the figure-eight-shaped trajectory flapping foil within a cycle is shown in Fig. 8. At a phase difference of 0°–90°, the optimal pitching amplitude of the flapping foil gradually increases with the increase in phase difference. In addition, as the phase difference increases, the maximum efficiency that the flapping foil can reach gradually increases, and the speed of efficiency decay gradually decreases. At a phase difference of 90°–180°, the optimal pitching amplitude of the flapping foil gradually decreases, which is the opposite of the case when 0°–90°. As the phase difference increases, the maximum efficiency that the flapping foil can reach gradually decreases, and the speed of efficiency decay gradually increases. The optimal phase difference for the flapping foil is , and its efficiency change characteristics are mirrored relative to , which may be related to the characteristic that the trajectory changes are symmetrical relative to . The subtle differences between them are mainly due to the differences and similarities in their motion direction relative to the inflow, which also makes the energy harvesting characteristics of the flapping foil different under different phase differences. This suggests that the energy harvesting efficiency under a regular figure-eight trajectory can reach a maximum value ( ), which is 47.6%. This efficiency is already higher than the maximum efficiency that Kinsey et al.30 could achieve at a Reynolds number of . Benefiting from increased degrees of freedom, the drag coefficient of the flapping wing can be utilized. The efficiency improvement at this point originates from the contribution of the drag power coefficient, reaching 0.18. This achievement is unattainable in traditional flapping wing motions, indicating that the figure-eight trajectory can enhance the energy harvesting capability of the flapping foil. Additionally, although different trajectories will cause changes in the optimal pitching amplitude , the effective angle-of-attack amplitude when the flapping foil reaches maximum efficiency is always around 37.5°, indicating the existence of an optimal effective angle-of-attack amplitude that maximizes the efficiency. Theoretically, there is a peak range of energy harvesting efficiency for the flapping foil, which also means that there is a potential optimal effective angle-of-attack amplitude. After all, a certain increase in the effective angle of attack is conducive to the increase in lift, but too large an effective angle of attack will cause severe dynamic stall. This will be explained in the subsequent study [Fig. 10(a)].
(a) Flapping-foil energy harvesting efficiency and (b) maximum effective angle of attack with trajectory variation.
(a) Flapping-foil energy harvesting efficiency and (b) maximum effective angle of attack with trajectory variation.
At different phase differences , the motion trajectories of the flapping foil vary significantly, especially at and . Additionally, the motion trajectory forms a regular figure-eight shape at . The energy harvesting characteristics of three typical motion trajectories are shown in Fig. 9. Although the motion trajectories of the flapping foil differ, the flapping foil primarily relies on lift for energy harvesting, consistent with traditional flapping foils. For different phase differences , when the average lift power coefficient reaches its peak, the efficiency η also reaches a peak. The contribution of drag to efficiency is small, and it makes a certain contribution at 90°. The main reason for this is that the regular figure-eight shape trajectory aligns with the interaction between the fluid and solid, similar to the spontaneous formation of a figure-eight shape by an oscillating cylindrical body.38 Therefore, as the pitch amplitude increases, the drag coefficient power gradually increases, leading to an increased contribution of drag to power, as shown in Fig. 9(b). Moreover, the torque coefficient mostly contributes negatively to efficiency, and its impact on efficiency is small. Additionally, for different motion trajectories, the pitching amplitudes at which the flapping foil achieves maximum efficiency are similar. To achieve a breakthrough in efficiency, focusing more on increasing the lift power coefficient will be reliable.
At phase differences of (a) , (b) , and (c) , variation of efficiency , average drag power coefficient , average lift power coefficient , and average torque power coefficient with pitch amplitude .
At phase differences of (a) , (b) , and (c) , variation of efficiency , average drag power coefficient , average lift power coefficient , and average torque power coefficient with pitch amplitude .
From the above analysis, we fully recognize that lift plays a dominant role in contributing to efficiency. Of course, for different motion trajectories, we also recognize that the peak efficiency occurs within a definite effective angle-of-attack amplitude range of 35°–40°. The variations of the lift coefficient, effective angle of attack, and lift power coefficient within one cycle of the flapping foil are shown in Fig. 10. As the pitching amplitude increases below a certain threshold, the peak lift also gradually increases. When the pitching amplitude reaches a certain value, the lift coefficient reaches its peak, and a significant dynamic stall occurs after the moment when the lift reaches its peak, as shown in Fig. 10(a). As the pitch amplitude increases, the peak lift coefficient of the flapping foil gradually increases, leading to efficiency gains. However, beyond a pitch amplitude of 85°, the change in the peak lift coefficient becomes less significant and may even be compromised by the intense dynamic conditions (t = 0.7T–0.9T). During this time range, with increasing pitch amplitude, the mean lift coefficient gradually decreases, indicating a potential loss in efficiency. Therefore, the selection of pitch amplitude should be appropriate, necessitating the choice of an optimal pitch amplitude to maximize efficiency gains. With the increase in the pitching amplitude, the effective angle-of-attack amplitude is 18.2°, 27.2°, 36.7°, and 46.6°, respectively. It is worth noting that the variation of the lift coefficient and the effective angle of attack are synchronized. As the pitching amplitude increases, the moment when the lift coefficient exceeds zero is gradually delayed, which is synchronized with the change of the effective angle of attack, as shown in Fig. 10(b). This synchronous behavior can be utilized because full synchronization of the lift coefficient Cy and the heaving speed Vy would benefit energy harvesting. The variation in the lift power coefficient caused by the desynchronization of the lift coefficient Cy and the effective angle of attack is shown in Fig. 10(c). The peak of the lift power coefficient does not occur at the peak of the lift, which is due to the asynchrony of the heaving speed Vy and the lift coefficient Cy at the time they reach their peaks. Furthermore, the asynchrony in the sign of the lift coefficient Cy and the heaving speed Vy can cause the lift power coefficient to perform negative work. Although increasing the pitching amplitude improves the synchronization rate of the lift coefficient Cy and the heaving speed Vy, the decay of the peak lift power coefficient will reduce the energy harvesting capability of the flapping foil. Overall, relying on the pitching amplitude to change the effective angle of attack is one-dimensional. It is unrealistic to solely depend on pitching amplitude to achieve an increase in the peak lift and maximize the synchronization rate. Of course, this also offers potential schemes for efficiency improvement.
At a phase difference of , variation of the flapping foil's (a) lift coefficient , (b) effective angle of attack , and (c) lift power coefficient over one cycle.
At a phase difference of , variation of the flapping foil's (a) lift coefficient , (b) effective angle of attack , and (c) lift power coefficient over one cycle.
At different pitching amplitudes, the energy harvesting characteristics of the flapping foil vary greatly, which is related to changes in the flow field environment. For different pitching amplitudes, the pressure distribution of the flapping foil with a regular figure-eight trajectory is shown in Fig. 11. Here, we focus on the moments when the flapping foil reaches the peak lift [at time t = 0.08T, as shown in Fig. 10(a)] and during dynamic stall (at time t = 0.24T). At time t = 0.08T, the angle of the foil changes with the pitching amplitude. Naturally, the greater the pitching amplitude, the greater the turning angle of the foil at this time. The change in the turning angle alters the angle between the airfoil and the oncoming flow, which affects the frontal area. The foil facing the oncoming flow from infinity converts the kinetic energy of the flow into pressure potential energy. An increase in the turning angle will undoubtedly increase the frontal area, thereby increasing the pressure on the pressure side. With the increase in the turning angle, this is also accompanied by a decrease in the pressure on the vortex side. This explains why the peak lift increases with the increase in pitching amplitude. At time t = 0.24T, the frontal area of the foil first increases and then decreases with the increase in pitching amplitude. When the pitching amplitude exceeds a certain threshold, the frontal area will decrease. With the increase in pitching amplitude, the pressure on the pressure side of the foil first increases and then decreases. In addition, for foils that exceed the critical angle of attack, the leading-edge vortex (LEV) generated on the vortex side will spread to the pressure side, leading to a reduction in the pressure difference across the foil surface. At excessively large angles of attack, severe dynamic stall occurs in the foil, as shown in Fig. 10(a). This also means that there is an appropriate maximum angle of attack that balances lift enhancement and stall process mitigation.
Pressure coefficient distribution of the standard figure-eight trajectory flapping foil at different time instants.
Pressure coefficient distribution of the standard figure-eight trajectory flapping foil at different time instants.
Facing different pitching amplitudes, the wake structure of the flapping foil also undergoes a transformation, as shown in Fig. 12. As analyzed above, an increase in pitching amplitude means an increase in the energy absorption of the airfoil from the flow. Obviously, the kinetic energy of the oncoming flow does not increase or decrease out of nowhere, meaning that the kinetic energy in the wake of the foil will decay. In fact, the research by Betz21 provides valuable reference. To achieve efficiency close to the Betz limit, the average flow speed in the wake of the foil must be one-third of the oncoming flow at infinity. This is mainly because the flowability within the flow tube is also very important. For the flapping foil, to meet such conditions, it is undoubtedly a matter of selecting the appropriate maximum effective angle of attack and allowing it to occur at the right moment. A pair of counter-rotating vortices shed from the foil in one cycle. With the increase in pitching amplitude, the diffusion range gradually increases, which is also accompanied by a decrease in the streamwise spacing between vortices. A decrease in the streamwise spacing means a reduction in the average flow speed in the wake. Too low wake flow speed will lead to a decrease in efficiency. In addition, when the effective angle of attack exceeds the critical value ( ), the stability of the wake deteriorates ( ), which may be related to flow separation at high angles of attack.
At time instant , dimensionless wake structure of the standard figure-eight trajectory flapping foil at different pitch amplitudes (scale represents multiples of chord length).
At time instant , dimensionless wake structure of the standard figure-eight trajectory flapping foil at different pitch amplitudes (scale represents multiples of chord length).
The figure-eight trajectory takes advantage of the streamwise motion of the foil, increasing the relative speed of the foil to the oncoming flow. Figure 13 provides a schematic diagram of the vortex formation and shedding for the figure-eight trajectory. Over time, the LEV forms at time t = 0.20T and develops, moving from the leading edge of the foil to the trailing edge and eventually shedding at t = 0.50T. In fact, the shedding of vortices is continuous, spanning the entire cycle of the airfoil. The pair of vortices considered in Fig. 13 is part of a Kármán vortex street. The figure-eight trajectory takes advantage of the vortex shedding time to move in the direction of the oncoming flow, increasing the relative speed to the flow. In this way, the flowability of the wake of the foil is improved. When the LEV is formed, the motion of the airfoil is in the same direction as the oncoming flow, reinforcing the formation of the low-pressure zone. This movement pattern “stores” and “releases” the flow speed, and thus, the figure-eight trajectory greatly improves the utilization rate of the oncoming flow.
Formation and shedding of the LEV on the standard figure-eight trajectory flapping foil.
Formation and shedding of the LEV on the standard figure-eight trajectory flapping foil.
B. Stroke amplitude effect
In the analysis above, under the standard figure–eight trajectory ( ), the flapping foil demonstrates better energy harvesting characteristics, which can be maintained and used for subsequent research. As shown in the study by Kinsey and others,11 the adjustment of the flapping foil's stroke has an impact on energy harvesting efficiency. This means that there is an optimal heave amplitude value that allows the flapping foil to achieve maximum energy harvesting efficiency. Here, we first fix the amplitude of the surge direction and study the effect of lateral amplitude on energy harvesting efficiency. Efficiency , effective angle-of-attack amplitude , and feathering parameter change with pitch amplitude as shown in Fig. 14. For different pitch amplitudes, the flapping foil reaches maximum efficiency with different heave strokes, and the efficiency varies with the heave stroke as shown by the solid dots in Fig. 14(a). Although the efficiency increases with pitch amplitude and then decreases (after a threshold), the moment when the threshold appears is difficult to judge. Here, the effective angle-of-attack amplitude is introduced to help analyze its change characteristics, as shown in Fig. 14(b). For a given pitch amplitude, there is an optimal effective angle-of-attack amplitude 37.5° that maximizes efficiency. In addition, the feathering parameter is accurate in judging the power-extraction regime, as shown in Fig. 14(c). When the feathering parameter reaches 0, the flapping foil no longer harvests energy. For different heave-pitch combinations, the optimal feathering parameter exponent approaches 1.6. Here, another interesting phenomenon is worth noting. For a given pitch amplitude, there is an optimal heave-pitch combination that makes the efficiency reach a threshold. There is an optimal pitch amplitude , which makes the threshold of efficiency (solid dot) reach its peak. Under the optimal heave-pitch motion combination and , the energy harvesting efficiency of the flapping foil is as high as 48.1%.
Energy harvesting characteristics of the standard figure-eight trajectory flapping foil. (a) Efficiency , (b) maximum effective angle of attack , and (c) feathering parameter as a function of pitch amplitude .
Energy harvesting characteristics of the standard figure-eight trajectory flapping foil. (a) Efficiency , (b) maximum effective angle of attack , and (c) feathering parameter as a function of pitch amplitude .
In fact, this universal law is still applicable to traditional flapping foils and, comparatively speaking, the trajectory studied in this research is more complex. To illustrate the universality of this rule, this study examined 405 cases and selected two typical Reynolds numbers, denoted as and . At the same Reynolds numbers as this study, the flapping foil achieved maximum efficiency at a maximum effective angle of attack of , despite choosing different pitch amplitudes, as shown in Fig. 15. It is evident that the figure-eight-shaped trajectory is helpful in improving the peak of efficiency. In addition, the influence of airfoil thickness will also be taken into account, as shown in Fig. 16. At the same Reynolds number, the peak energy harvesting efficiency of the NACA0009 airfoil occurs at an effective angle of attack of about . Airfoils with different thicknesses exhibit similar patterns of variation, although the corresponding maximum effective angle of attack is slightly smaller. The influence of the geometric model may result in a slightly different maximum effective angle of attack for adaptation, but this does not deviate from the general rules. The peak efficiency is always closely associated with the maximum effective angle of attack. Moreover, for cases with Reynolds number , the maximum effective angle of attack corresponding to the peak of efficiency is still , as shown in Fig. 17. This universal law indicates that there is a certain correlation between pitch amplitude and heave amplitude, and arbitrary combinations will lead to efficiency loss. In such cases, this provides a guiding role for further related research. The heave amplitude should not be a fixed choice;39–43 instead, combinations of different pitch amplitudes and heave amplitudes should be considered.
Energy harvesting characteristics of the conventional flapping foil ( ) at the same Reynolds number. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Energy harvesting characteristics of the conventional flapping foil ( ) at the same Reynolds number. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Energy harvesting characteristics of the NACA0009 airfoil at the same Reynolds number. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Energy harvesting characteristics of the NACA0009 airfoil at the same Reynolds number. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Energy harvesting characteristics of the conventional flapping foil at a Reynolds number of 1100. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Energy harvesting characteristics of the conventional flapping foil at a Reynolds number of 1100. (a) Efficiency and (b) maximum effective angle of attack as a function of pitch amplitude .
Due to the existence of an optimal effective angle-of-attack amplitude , the same selection was applied to the cases below to isolate the influence of angle-of-attack amplitude on the energy harvesting characteristics. Here, cases with different pitch amplitudes from the above scenarios were selected for comparison. The variations in lift, effective angle of attack, lift power coefficient, and the dimensionless total power coefficient of the foil within one cycle are shown in Fig. 18. For the same angle-of-attack amplitude , the lift coefficient curves under different pitch amplitudes showed similar trends, as depicted in Fig. 18(a). Minor differences between the lift coefficients are mainly influenced by the angle-of-attack curve, as shown in Fig. 18(b). When the angle-of-attack variation curve approximates a sinusoidal profile, it exhibits better energy harvesting characteristics. Since a larger pitch amplitude requires a larger heave amplitude to match, the lift power coefficient of the foil increases with the pitch amplitude, as shown in Fig. 18(c). Here, the lift power coefficient has a portion below zero, indicating that the lift and heave displacement are not completely in sync. The peak of the foil's efficiency appears at a certain , as shown in Fig. 18(d). As the pitch amplitude increases, the heave amplitude also increases, leading to an increase in the vertical distance d. Therefore, the efficiency will not continuously increase with the increase in pitch amplitude, which explains the rationality of the existence of a threshold. Macroscopically, the curvature of the angle of attack could become one of the factors affecting the characteristics of energy harvesting. The theory of aerodynamics proposed by Theodorsen30 takes into account the influence of this parameter on unsteady forces. This was also demonstrated in our cases. Adjustment strategies should consider both the maximum effective angle of attack and its curvature on the unsteady forces of the flapping foil.
Variation of (a) lift coefficient, (b) effective angle of attack, (c) lift power coefficient, and (d) the dimensionless total power coefficient of the flapping foil over one cycle at different pitch amplitudes.
Variation of (a) lift coefficient, (b) effective angle of attack, (c) lift power coefficient, and (d) the dimensionless total power coefficient of the flapping foil over one cycle at different pitch amplitudes.
For the flapping foil experiencing the same angle-of-attack amplitude, the wake structure is as shown in Fig. 19. The foil following a regular figure-eight trajectory sheds two vortices within one cycle during the limit-cycle oscillation, forming a von Kármán vortex street in the wake. The pitch amplitude mainly affects the spread range of the wake but does not change the wake pattern. Interestingly, the effective angle-of-attack amplitude cannot determine the stability of the wake. When the pitch amplitude is too large, the instability caused by flow separation has a negative impact on the propulsion of the wake. As the pitch amplitude increases, the spread range of the wake also increases. According to the Betz limit,21,44 the flow velocity in the wake of the foil should be one-third of the incoming flow at infinity. However, this conclusion is derived based on streamlines and flow tubes. According to the continuity equation, the spread range of the wake should be certain given the flow velocity in the wake. Since the spread range of the wake increases with the pitch amplitude, the corresponding heave amplitude should change accordingly. When the flapping foil experiences the same effective angle-of-attack amplitude, the value is around one-third, which gives a reasonable explanation for this universal rule. This also indicates that the effective angle-of-attack amplitude is an effective parameter for controlling the spread range of the wake.
Wake structure of flapping foils at different pitch amplitudes experiencing the same time step t = 0.26T.
Wake structure of flapping foils at different pitch amplitudes experiencing the same time step t = 0.26T.
C. Pitch axis effect
The influence of the pitching axis position on the energy harvesting performance of flapping foils has received considerable attention, yet breakthroughs in this area are still pending.6 Kinsey and others have achieved a maximum efficiency of 44.6% for flapping foils at a Reynolds number of , but their research did not consider the impact of pitching axis and phase difference on energy harvesting behavior, which suggests potential efficiency improvement schemes. Zhu6 studied the optimal frequency of operation for flapping-foil turbines at low Reynolds numbers, emphasizing in their conclusions the importance of the effects of pitching axis position and phase difference. However, most current studies14,45,46 have fixed the choice of phase difference and pitching axis position at , and whether this choice is optimal remains unknown. In general cases, the position of the pitching axis is given without explaining the reason for its selection. Indeed, in the above analysis, we observed that the lift and heaving velocity are not completely synchronized, which could lead to energy loss. This chapter will study a large number of cases (576 cases), considering the influence of pitching axis position (1/5–1/2) and phase difference ( ).
To facilitate emphasis and comparison, efficiency analysis is performed at pitching axis positions 1/5, 1/4, 1/3, and 1/2 away from the leading edge, as shown in Fig. 20. The solid lines in the figure still correspond to the peak efficiencies at different pitching amplitudes. In this study, the phase difference is used to adjust synchronous behavior. Of course, there are multiple choices for adjusting the synchronization rate, such as using phase difference or other parameters. At different values, the maximum efficiencies achieved by the flapping foil differ slightly, but the highest efficiency occurs at the aerodynamic center of NACA0015 at , which is 50%. This suggests that the maximum efficiency achievable by the flapping foil may not vary significantly with changes in the position of the pitching axis. However, it is important to note that without phase difference adjustment, fixing the phase difference at will cause huge efficiency disparities. For example, at and a pitching amplitude of , the optimal phase difference is , which deviates significantly from the general choice of . Although lower pitching amplitudes are not conducive to energy harvesting, a reasonable match can ensure that the energy harvesting efficiency of the flapping foil is not less than 30%, which is a significant efficiency difference compared to cases without phase difference adjustment. The optimal phase difference is clearly not fixed and is affected by both pitching amplitude and phase difference. The smaller the , the larger the phase difference needed to adjust synchronous behavior and thus enhance energy harvesting capability. As decreases, its sensitivity to phase difference increases. For all cases, the larger the pitching amplitude of the flapping foil, the smaller the phase difference required to achieve maximum efficiency. At a pitching amplitude of , all cases reached peak efficiency, consistent with the conclusions of the above cases. After adjusting the phase difference in combination with the pitching axis position, the energy harvesting efficiency of the flapping foil can reach 50%, which is very close to the Betz limit (59%). At the same time, this shows that the pitching axis position, phase difference, and pitching amplitude are interrelated, and a single analysis may not be reliable.
Influence of phase difference on the energy harvesting efficiency of flapping foils at different pitch axis locations: (a) at 1/5 chord length from the leading edge, (b) at 1/4 chord length from the leading edge, (c) at 1/3 chord length from the leading edge, (d) at 1/2 chord length from the leading edge.
Influence of phase difference on the energy harvesting efficiency of flapping foils at different pitch axis locations: (a) at 1/5 chord length from the leading edge, (b) at 1/4 chord length from the leading edge, (c) at 1/3 chord length from the leading edge, (d) at 1/2 chord length from the leading edge.
From the extensive cases above, it is evident that the adjustment of the phase difference is effective in improving efficiency. It is worthwhile to compare the cases before and after the phase adjustment to analyze the enhancement mechanism. Figure 21 presents the changes in the total power coefficient per unit of the figure-eight trajectory flapping foil within a cycle and selects two typical pitching amplitudes for analysis. First, for the lower pitching amplitude , the adjustment of the phase significantly improved the curve of the total power coefficient variation of the flapping foil, although the improvement effect is different under different pitching amplitudes, as shown in Fig. 21(a). After the phase adjustment, there is a certain enhancement in the maximum value of the total power coefficient, and there is an improvement in the minimum value of the total power coefficient, which has reduced the part of negative work done. Moreover, with the increase in Lc, the greater the phase difference needed to be adjusted. However, the position of the pitching axis may not have a significant impact on the energy harvesting efficiency of the flapping foil. For the flapping foil that has undergone phase adjustment, the efficiency difference of different pitching axis positions is less than 1%. The adjustment of the phase allows the efficiency of the flapping foil under lower pitching amplitudes to still reach 34%, which significantly surpasses the general choice .11,37,46,47
Variation of the dimensionless total power coefficient of an eight-shaped trajectory flapping foil over a cycle at different pitch amplitudes: (a) at lower pitch amplitudes , (b) at optimal pitch amplitudes . Solid lines represent results after phase difference adjustment, and dashed lines represent general selections .
Variation of the dimensionless total power coefficient of an eight-shaped trajectory flapping foil over a cycle at different pitch amplitudes: (a) at lower pitch amplitudes , (b) at optimal pitch amplitudes . Solid lines represent results after phase difference adjustment, and dashed lines represent general selections .
Second, for the optimal pitching amplitude , the influence of phase difference and pitching axis position on the unit total power coefficient is shown in Fig. 21(b). With the increase in , the peak value of the total power coefficient of the flapping foil gradually decreases, which also reduces the efficiency loss brought about by negative work. The main function of phase adjustment has become to reduce the part of the total power coefficient that does negative work, which is the benefit brought by the improvement of the synchronization rate of the lift coefficient and the sinking speed.
As we have described in Fig. 10, changing the synchronization rate can synchronize the effective angle of attack and the heaving velocity. The reason why such a strategy was not used in this study for analysis is that the torque and drag do not always do positive work, resulting in the adjusted total power coefficient still showing some negative values, and the results of a large number of cases are more convincing. In addition, there are currently no good strategies for controlling torque and drag, so the optimal phase difference may not occur when the effective angle of attack and heaving speed are completely synchronized. The degree of negative values in the total power coefficient is affected by the pitching amplitude. When the pitching amplitude is larger, the negative work done by the torque will be greater, which leads to an increase in the range of negative values of the total power coefficient. Increasing Lc can reduce the impact of negative work done by torque and also reduce the peak value of the total power coefficient. Overall, the benefits brought by the pitching axis position are not significant, with the maximum difference in efficiency being only 3.7%. If structural design is a priority, the pitching axis position can be installed as an adjustable parameter, but it is necessary to pay attention to the matching phase.
The effect of phase adjustment strategy on the energy harvesting efficiency of traditional flapping foils is shown in Fig. 22, where a lower pitching amplitude is selected (the optimal case corresponding to Fig. 15). Under different pitching axis positions, the optimal phase difference of the flapping foils varies. As Lc increases, the optimal phase difference gradually decreases, with the corresponding optimal phase differences being 120°, 117°, 108°, and 87°, respectively. However, the maximum efficiency of the flapping foils is almost consistent for different pitching axis positions, which also indicates that the position of the pitching axis may not help improve efficiency. Yet after phase adjustment, the efficiency of traditional flapping foils can all exceed 30%, which allows the flapping foils to work effectively at lower pitching amplitudes.
Energy harvesting efficiency of traditional flapping foils at different pitch axis positions as a function of phase difference when the pitch amplitude is .
Energy harvesting efficiency of traditional flapping foils at different pitch axis positions as a function of phase difference when the pitch amplitude is .
IV. CONCLUSION
This study has investigated the power-extraction regime of a flapping turbine following a figure-eight trajectory. It involved over a thousand cases, representing a comprehensive examination of the operational parameters for flapping turbines. For flapping foils experiencing different trajectory shapes, pitching amplitudes, plunging amplitudes, and pitching axis positions, there is an universal effective angle-of-attack amplitude suitable for energy harvesting modes. This universal law is still applicable to traditional flapping turbines, as the traditional cases correspond to the situation in this study where the surge motion coefficient is zero. The unified effective angle-of-attack amplitude poses requirements for the flapping motion's maximum swept distance d and the wake's diffusion range dv, which align with the objective requirements of reaching the Betz limit. This rule provides schemes for the design of flapping turbines. Moreover, lift contributes mainly to efficiency, with its oscillating process synchronized with the effective angle of attack, but drag and torque are difficult to predict. Improvements in this area may rely on the development of unsteady linear potential flow theory. Baik et al.,32 based on the unsteady linear potential flow theory proposed by Theodorsen30 and Carrick,33 made good predictions of the lift on pitching and plunging flat-plate airfoils, so further development of linear potential flow models can be reasonably expected.
This study focused on cases with the same effective angle of attack, primarily expressing dissatisfaction with the occasional negative work of the flapping turbine. Therefore, control over the pitching axis position and phase difference was conducted, finding that the pitching axis position may have a small impact on efficiency peaks, but the phase difference had a significant effect. Different pitching axis positions and pitching amplitudes are suitable for different phase differences, mainly due to the phase difference's adjustment of the synchronization behavior of lift and heaving velocity. Analysis of the cases found that the pitching amplitude has an effect on the energy harvesting regime of the flaps. An increase in the pitching amplitude will increase the pressure on the pressure side, thereby improving the lift and power peak of the flaps. However, an excessively large pitching amplitude may cause severe dynamic stall, thereby increasing the chances of negative work by the flapping turbine. There is an optimal pitching amplitude for energy harvesting, but the matching of phases must be considered. A reasonable choice of phase greatly reduces the possibility of negative work by the flapping turbine, thereby enhancing the energy harvesting efficiency. The adjustment of phase has significant benefits for the flapping turbine, ensuring that its energy harvesting efficiency is not less than 30%, which improves the optimal operational parameter space of the turbine. Moreover, since the trajectory we designed includes the motion trajectory of traditional flapping turbines, the rules obtained have more universality. Reasonable trajectory design can make use of drag, enabling the maximum efficiency of the flapping turbine to reach 50%, which is very close to the Betz limit (59%).
Finally, this study mapped efficiency to multiple parameter spaces, and the results showed that efficiency is affected comprehensively, and it is not reasonable to analyze a single parameter alone. Since the heaving amplitudes and phase differences that need to be matched with different pitching amplitudes are different, fixing the phase difference and heaving amplitude behavior will lead to a decrease in efficiency. In addition, future work should consider the decay frequency and examine the power-extraction regime under three-dimensional conditions. In a three-dimensional environment with more degrees of freedom, it will exhibit a richer range of fluid dynamics behaviors, accompanied by an increase in computational complexity. Consideration can be given to the use of reduced-order modeling48 or deep learning49,50 for accelerating computations, effectively addressing the computational power requirements.
ACKNOWLEDGMENTS
Y.B. acknowledges the support of the National Natural Science Foundation of China (No. 11402115) and the Jiangsu Province Natural Science Foundation of China (No. BK20130782).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Min Zheng: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Huimin Yao: Formal analysis (equal); Validation (equal). Yalei Bai: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal). Qin Bo: Conceptualization (equal); Funding acquisition (equal). Xu Chi: Investigation (equal). Jinyan Chen: Investigation (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.