Numerical Study on the Energy Harvesting Performance of a Flapping Foil with Attached Flaps

Abstract

A flapping foil, which mimics the flapping wings of birds and the locomotion of aquatic organisms, is an alternative to a conventional turbine for the harvesting of renewable energy from ubiquitous flows in the atmosphere, oceans, and rivers. In this work, the energy harvesting performance of flapping foils with attached flaps at the trailing edge is numerically studied by using an immersed boundary–lattice Boltzmann method (IB-LBM) at a Reynolds number of 1100. Three different configurations are considered, namely, a clean NACA0015 foil, a NACA0015 foil with a single flap, and a NACA0015 foil with two symmetric flaps. The results show that the flap attached to the trailing edge is able to enhance the energy harvesting efficiency, and the two symmetric flaps can achieve more enhancements than its single-flap counterpart. The mechanism of such enhancements is attributed the separation of the interactions of vortexes generated at the upper and bottom surfaces of the foil. To further obtain the optimal configurations of the two symmetric flaps, the angle between the two flaps ($\alpha$) and the length ($l_f$) of the flap are systematically studied. The results show that the optimal energy harvesting performance is achieved at $\alpha ={60}^{\circ }$ and $l_f=0.1c$ (c denotes the chord length of the foil). Compared with the baseline case, namely, the clean NACA foil, the optimal configuration can achieve an improvement of efficiency up to 19.94%. This study presents a strategy by adding two symmetric flaps at the trailing edge of the foil to enhance the energy harvesting performance of a flapping foil, which contributes to advancing the development of simple and efficient clean energy harvesting by using a flapping foil.

1. Introduction

The rapid growth of the global population and advances in civilization have led to an exponential growth in demand for energy [1]. Although unsustainable fossil fuels have severe environmental and health issues, they are still the main energy sources [2]. Renewable energy is urgently needed to replace coal, oil, and gas in the generation of electricity and the reduction of the emissions of greenhouse gases such as methane, carbon dioxide, and nitrous oxide. Many different strategies have been explored such as wind turbines and solar photovoltaic cells. In the last decades, energy harvesters that mimic the flapping wings of birds and the locomotion of aquatic organisms have attracted significant attention for their high efficiency in power harvesting, even at the cut-in speed of conventional wind turbines, where the efficiency of wind turbines drops rapidly and very limited energy can be extracted [3]. This is because the conventional rotary turbines rely on the attachment of flow on the turbine surface to achieve high efficiency, while the flapping foils ubiquitously seen in nature generate significantly high instantaneous forces by exploiting the low-pressure core of the leading-edge vortex (LEV) [4].

One of the earliest studies on power harvesting using flapping foils can be dated back to the 1980s, conducted by McKinney and DeLaurier [5]. In this work [5], the innovative concept is theoretically analyzed based on unsteady wing aerodynamics and aeroelasticity and compared with a wind tunnel test. The consistent results demonstrated that the flapping-foil power harvester is able to achieve comparable efficiency to rotary designs. This concept was also then explored by Platzer’s group. For example, David [6] systematically examined a range of kinematics, including reduced frequencies, plunge amplitudes, pivot locations, phase angles, and angles of attack, by using an unsteady panel method (UPM). The UPM is based on the potential flow assumption, it enforces the Kutta condition at the trailing edge, and no separations over the airfoil surface are allowed. However, both an over-prediction and an under-prediction of the efficiency were observed in comparison with the experimental measurements in [5], which was attributed to the onset of dynamic stall for the higher-pitch amplitudes. Lindsey [7] further studied the effect of phase between the plunging and pitching motion, angle of attack magnitude, and foil thickness using the UPM. The limited accuracy of the UPM inspired researchers to develop more accurate methods to study the aerodynamics/hydrodynamics of flapping foils. To take advantage of low computational cost and remove the constraints associated with the attached flow resulting from UPM, a dynamic stall model and an improved discrete vortex method (DVM) were combined to study the power harvesting performance of a passive flapping foil in [8]. It was found that such an improved reduced-order method is able to predict the force and power generation reasonably compared with the experimental results in [9]. Liu [10] combined the improved DVM and a multi-fidelity evolutionary algorithm to search for high-energy-extraction performance solutions for a flapping-foil power generator. It is also noted that the reduced-order method of DVM is still not able to accurately predict the force generation compared with the high-fidelity results from computational fluid dynamics (CFD) [8]. On the other hand, Peng and Zhu [11,12] adopted a Navier–Stokes equation-based method to directly solve the fluid–structure interaction in a passive flapping foil and identified four different modes that depend on the configuration of the system and the mechanical parameters (including the stiffness of the rotating spring and the location of the pitching axis). As an alternative method to the conventional Navier–Stokes equation-based methods in CFD, the lattice Boltzmann method (LBM) is robust and efficient, especially for the incompressible and weakly compressible fluids involved in the flapping-foil power generator [13]. The incorporation of the immersed boundary method (IBM) into the LBM presents a highly efficient way to solve complex FSIs such as in the flapping foil [14]. For example, the IB-LBM has been adopted to study the energy harvesting performance of flapping foils with confinement effects [15].

In addition, bio-inspired features have also been explored to further improve the energy harvesting performance. For example, Le et al. [16] examined the effect of corrugation combined with the camber of the foil. Inspired by the flexible wings of insects and the fins of fish, Liu et al. [17] studied the flapping foil with active deformation or morphing in energy harvesting and identified the improved performance in power extraction. This was also confirmed by taking the power required to morph the foils into account [18]. Liu et al. [19] studied the flapping-foil power generator performance with a spring-connected tail numerically and obtained the optimal kinematics. Inspired by the pop-up feather of a bird during landing, a simplified configuration, i.e., a NACA foil with a vertically attached flap, has been numerically studied, and it was found that the bio-inspired flap is able to increase the lift generation [20] and reduce the noise generation [21]. It is natural to expand this simple modification to the flapping foil for energy harvesting. Although the spring-connected flap is considered in [19], this work is more focused on the kinematics and the stiffness of the torsional spring, leaving the length and configuration of the flap unexplored. Moreover, the utilization of rigidly attached flaps is simpler than the spring-connected ones that do not induce extra instability to the system, indicating that the rigid flap is more favorable if comparable improvements can be achieved. It should be noted that the improved energy harvesting performance of the flapping foil is very important, especially for the practical application in real-world scenarios for more sustainable development. It should also be noted that the combination of deep learning and CFD may be used to achieve fine tuning in some practical applications of the flap in the real world [22,23].

In this paper, the energy harvesting performance of a flapping foil with attached flaps are numerically studied by using an IB-LBM, and three different configurations are considered, namely, the clean NACA0015 foil, the NACA0015 foil with a flap attached at the trailing edge, and the NACA0015 foil with double symmetric flaps attached at the trailing edge, as shown in Figure 1. The rest of this paper is organized as follows: the definition of the physical problem is described in Section 2; the numerical method used for the simulation is introduced in Section 3; the effects of the flaps are firstly studied and analyzed in Section 4.1; the parametric study on the effects of flap length and the angle between the double flaps are then studied in Section 4.2; and a conclusion is given in Section 5.

2. Physical Problem

In this paper, the power generator of a NACA0015 foil with a combined heaving and pitching motion is considered, defined as

$$\theta ={\theta }_{0}sin\left(2\pi ft\right),h=h_{0}sin(2\pi ft+\varphi ),$$

(1)

where $h_0$ and ${\theta }_0$ are, respectively, the amplitudes of the heaving and pitching motion, f is the frequency, and $\varphi$ is the phase difference. As shown in Figure 1, the pivot point has a distance of $c/3$ to the leading edge of the foil. One or two flaps are attached to the trailing edge of the foil, where $\alpha =0$ means that there is a single flap and $\alpha >0$ means that there are two symmetric flaps. The flap has a length of $l_f$, and the foil has a length of c. A uniform flow with a velocity of $u_0$ is applied at the inlet; the far-field boundary condition is applied for all other boundaries. In this work, $h_0=c$, ${\theta }_0=76.3^{\circ }$, $\varphi ={90}^{\circ }$, the Reynolds number $Re=u_{0}c/\nu =1100$, and a reduced frequency $f^{*}=fc/u_0=0.14$ is used, according to the optimal parameters from Refs. [9,19]; the focus here is on the effects of the attached flaps. The hydrodynamic performance of the flapping-foil power generator can be quantified by the mean power coefficient and efficiency, defined as

$$\overline{C_P}={\displaystyle \frac{1}{nT}}{\int }_0^{nT}{C}_P\left(t\right)dt,\eta =C_P{\displaystyle \frac{c}{d}},$$

(2)

where T is the period of the flapping motion, n is an integer representing how many periods are used for the averaging ($n=4$ is used, which is large enough to remove the noise over multiple periods; the averaging is applied after the initial developing stage), d is the peak-to-peak excursion of the trailing edge, and the instantaneous power coefficient is defined as

$$C_P\left(t\right)=-{\displaystyle \frac{\int \mathit{f}·\mathit{u}}{0.5u_0^{3}c}},$$

(3)

with $\mathit{f}$ representing the instantaneous aerodynamic/hydrodynamic force on the foil and flap, $\mathit{u}$ representing the instantaneous velocity, and the integral being taken over the space, which can be simplified as a sum in the current solver, as both the foil and the flap are democratized as nodes in the current IB-LBM solver. In addition to the total power coefficient $C_P$, the power coefficient contributed by the foil and flap are denoted by $C_{P,foil}$ and $C_{P,flap}$, respectively.

3. Numerical Methods

In this paper, FSIs in weakly compressible fluid flow are considered. Specifically, the LBM is employed for the fluid dynamics and the complex no-slip boundary conditions are handled by using an IB method. The numerical methods are briefly introduced in this section, and the details can be found in Refs. [14,24,25].

3.1. LBM for the Fluid Flow

In the multi-relaxation-time (MRT)-based LBM with BGK approximation, the evolution of the velocity distribution function $g_i$ along the i-th direction at a position $\mathit{x}$ is given as [26]

$$g_i(\mathit{x}+{\mathit{e}}_i\Delta t,t+\Delta t)=g_i(\mathit{x},t)-{\mathsf{\Omega }}_i(\mathit{x},t)+\Delta t{G}_i,i=0,1,\dots ,8,$$

(4)

where $\Delta t$ denotes the time step. The collision operator and body force effects on the distribution function are, respectively, represented by ${\mathsf{\Omega }}_i$ and $G_i$ in the collision operator, and they can be defined as

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_i& =& -{\left(M^{-1}SM\right)}_{ij}[g_j(\mathit{x},t)-g_j^{eq}(\mathit{x},t)]\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill G_i& =& {\left[M^{-1}(I-S/2)M\right]}_{ij}{F}_j\hfill \end{array}$$

(6)

where M is a transform matrix with a dimension of $19\times 19$ for the D2Q9 model used here, and S is a diagonal matrix that is related to the fluid dynamic viscosity. The macro variables, including the density and momentum, can be directly calculated from the distribution functions as follows:

$$\rho =\sum _{i=0}^8{g}_i,\rho \mathit{u}=\sum _{i=0}^8{g}_i{\mathit{e}}_i+{\displaystyle \frac{1}{2}}\mathit{f}\Delta t.$$

(7)

The local equilibrium distribution function $g_i^{eq}$ and the force term $G_i$ can be calculated by

$$g_i^{eq}={\omega }_i\rho [1+{\displaystyle \frac{{\mathit{e}}_i·\mathit{u}}{c_s^2}}+{\displaystyle \frac{\mathit{u}\mathit{u}:({\mathit{e}}_i{\mathit{e}}_i-c_s^2\mathit{I})}{c_s^4}}],$$

(8)

$$G_i={\omega }_i[{\displaystyle \frac{{\mathit{e}}_i-\mathit{u}}{c_s^2}}+{\displaystyle \frac{{\mathit{e}}_i·\mathit{u}}{c_s^4}}{\mathit{e}}_i]·\mathit{f},$$

(9)

where ${\omega }_i$ are the weights [27]. The sound speed $c_s=\Delta x/(\sqrt{3}\Delta t)$, and $\mathit{f}$ is the force exerted on the fluid node by the structure nodes close to it. For the two-dimensional simulation, the D2Q9 model is adopted in this work. The relaxation time is related to the kinematic viscosity $\nu$ in the Navier–Stokes equations according to $\nu =(\tau -0.5)c_s^2\Delta t$. In addition, the multi-block technique developed by Yu et al. [28] is combined with a geometry-adaptive method to improve the computational efficiency.

3.2. The IB Method for FSI

Here, the penalty immersed boundary (pIB) method proposed by Kim and Peskin [29] is employed to handle the no-slip boundary conditions at the fluid–structure interface. In this method, a body force is added to both the fluid and structure solver, where the interaction force between the fluid and the structure can be calculated according to the feedback law [29]:

$$\mathit{F}=\alpha {\int }_0^t({\mathit{U}}_{ib}-\mathit{U})dt+\beta ({\mathit{U}}_{ib}-\mathit{U}),$$

(10)

where ${\mathit{U}}_{ib}$ is the velocity of immersed boundary nodes (which is obtained from the interpolation of the fluid domain), $\mathit{U}$ is the actual velocity of the structure, and $\alpha$ and $\beta$ are positive constants for the coupling ($\alpha =0$ and $\beta =2.0$ are used here). The no-slip boundary conditions at the fluid–structure interface are achieved by directly adding a body-force term into Equation (9), and this body force is obtained by distributing the force calculated by Equation (10) to the ambient fluid nodes of the Lagrange nodes. Compared with the sharp-interface IB method [30,31], the pIB method treats all the grid nodes in the computational domain by using a unified equation, which makes it particularly suitable here. In addition, the direct-force pIB is simpler when handling complex geometry compared with the sharp-interface IB method. The force transformation between the Lagrange and Euler variables is accomplished by using the Dirac delta function. The interpolation of velocity on the immersed boundary nodes from the fluid domain and the spreading of the Lagrange force to the ambient fluid nodes can be expressed as

$$\begin{array}{c}\hfill {\mathit{U}}_{ib}(s,t)={\int }_V\mathit{u}(x,t){\delta }_h(\mathit{X}(s,t)-\mathit{x})d\mathit{x},\end{array}$$

(11)

$$\begin{array}{c}\hfill \mathit{f}(\mathit{x},t)=-{\int }_{\mathsf{\Gamma }}\mathit{F}(s,t){\delta }_h(\mathit{X}(s,t)-\mathit{x})ds,\end{array}$$

(12)

where $\mathit{u}$ is the velocity of fluid nodes, $\mathit{X}$ and $\mathit{x}$ are, respectively, the coordinates of structural nodes and fluid nodes, s is the arc coordinate, V denotes the fluid domain, and $\mathsf{\Gamma }$ denotes the structure domain. The smooth function ${\delta }_h$ is used to approximate the Dirac delta function. In this work, the delta function proposed by Peskin [32] is used. The fluid domain has a dimension of $32c\times 32c$, and 6 blocks are used to discretize the whole domain, as shown in Figure 2. The coarsest mesh spacing is $0.16c$, and it is refined with a factor of 2; therefore, the fluid domain close to the foil is resolved with a mesh spacing of $0.005c$. The foil and the flap are discretized with a mesh spacing of $0.004c$ to match the resolution of the fluid domain. This numerical solver and its previous versions have been extensively validated in our previous work, and more details can be found in [19,24,25,33].

4. Results and Discussions

In this section, the effects of the flap on the power harvesting performance are firstly studied in Section 4.1. Here, a clean NACA foil, a NACA foil with a single attached flap, and a NACA foil with two symmetric flaps are considered. To keep the study simple, this work starts from a fixed length of flap $l_f=0.1c$ and a fixed angle between the two symmetric flaps $\alpha ={60}^{\circ }$. The effects of the flap length $l_f$ and the angle $\alpha$ are studied in Section 4.2.

4.1. Enhancement Effects of the Flap

Figure 3 shows the time histories of the total lift coefficient $C_L$ and power coefficient $C_P$ generated by the clean NACA foil, the NACA foil with a single flap, and the NACA foil with two symmetric flaps ($\alpha ={60}^{\circ }$). It shows that the flap does not change the overall trends of these two major coefficients, i.e., one cycle for $C_L$ and two cycles for $C_P$ in one period. Meanwhile, the flaps show an evident effect on both the lift and harvested energy. Specifically, the peaks of $C_L$ increased significantly from around 1.87 to 2.27 when a flap is attached, and they further increased to 2.94 when two symmetric flaps are adopted. It is also noted that both peaks are slightly delayed by the flap, i.e., the first peak is delayed from $t=0.09T$ to $0.12T$ and the second one is delayed from $t=0.41T$ to $0.44T$. Such a delay effect is also consistent with that observed in the power coefficient $C_P$, and the significant increase in the peaks in $C_P$ indicates that the flap attached to the trailing edge benefits the energy harvesting.

To quantify the effects of the flap, the peak-to-peak excursion d, time-averaged power coefficient $\overline{C_P}$, efficiency $\eta$, and enhancement ratio $\Delta \eta$ (defined as the increment compared with the clean NACA baseline case) are shown in

Table 1. Peak-to-peak excursion d, time-averaged power coefficient $\overline{C_P}$, efficiency $\eta$, and enhancement ratio $\Delta \eta$.

Sources	$\mathit{d}/\mathit{c}$	$\overline{{\mathit{C}}_{\mathit{P}}}$	$\mathsf{\eta }$	$\mathbf{\Delta }\mathsf{\eta }$
Clean NACA	2.565	0.850	0.331	-
NACA with a flap	2.698	0.992	0.368	11.17%
NACA with two flaps	2.753	1.092	0.397	19.94%

. It is clearly seen that the time-averaged power coefficient increases significantly from 0.850 to 0.992 when a flap is attached to the trailing edge, and such an increase is further enhanced by the two-symmetric-flap configuration. By looking at the efficiency, enhancements of 11.17% and 19.94% are achieved, respectively, by the single-flap and double-flap configurations. The 19.94% enhancement is even higher than that obtained by a flexible flap in [19], which indicates that such a configuration is very attractive for its simplicity. It is noted that the peak-to-peak excursion of the trailing edge is slightly increased when the flap is attached. However, such an increase does not reduce the enhancement of efficiency, as the time-averaged power coefficient $C_P$ has a stronger increase. It is also noted that the $C_P$ and $\eta$ obtained by the present solver (0.85 and 0.331) are very close to those (0.86 and 0.337) from [9], which confirms the reliability of our solver.

To understand the source of the enhancement in energy harvesting when the flap is attached, the $C_L$ and $C_P$ are further decomposed into two sections, i.e., $C_{L,foil}$/$C_{P,foil}$ and $C_{L,flap}$/$C_{P,flap}$, where the subscripts $foil$ and $flap$, respectively, denote the contributions from the NACA foil and flap. The decomposed results are shown in Figure 4. For the lift coefficient $C_L$, it is found that the lift generated by the NACA foil is enhanced when the flap is attached, and the two symmetric flaps are able to enhance the lift generation more than the single-flap counterpart. It is also found that the single flap shows more significant gain in lift generation at the pitching-reversal stage, i.e., $t=0.5T$ and T, while the two symmetric flaps show almost negligible lift gain. This indicates that the lift enhancement for the single-flap case has two major sources, i.e., the augmentation effects on the foil due to the flap and the flap itself, while the two-symmetric-flap case has only one major source, which is the augmentation effect of the flap on the foil. Such a mechanism is also similar for the power coefficient, although the energy harvested by the flap is negligible compared with the foil. For example, the maximum $C_P$ achieved by the flap is only around 0.04, while the $C_P$ achieved by the foil is up to 2.0. Another interesting phenomenon is that the two-flap case shows an asymmetric history in one period, i.e., an obvious energy harvest at $t=11T$, while it is not observed at $t=10.5T$. This is attributed to the chaotic flow fields, and it is probably related to the starting position of the flapping motion. As it does not affect the enhancement effects in lift generation and energy harvesting, it is not further analyzed.

To further analyze the mechanism of the flap in enhancing the performance of energy harvesting, the instantaneous pressure coefficient (defined as $c_p=(p-p_{\infty })/\left(0.5\rho u_0^2\right)$, with p being the instantaneous pressure and $p_{\infty }$ being the pressure in the far field) at four typical times is shown in Figure 5. First, these contours show that the major flow features are very similar for all cases, i.e., the LEV is first generated and then propagates downstream with the flapping of the foil, as shown in Figure 6, while the major differences are observed for the amplitudes of the local pressure and the area that these low-/high-pressure bubbles cover. The major effect of the flap is that it separates the interactions of the vortexes generated on the upper and bottom surface, which enhances the amplitudes of the generated LEV and maintains the vortexes for a longer time, preventing them from detaching to the downstream flow. This effect can be clearly seen in Figure 5 at $t=2T/16$ by the enhanced high pressure on the upper surface and low pressure on the bottom surface; it also shows that such an effect is much more effective when two symmetric flaps are attached to the trailing edge. Similarly, this effect can be seen in Figure 5 at $t=7T/16$ for the enlarged low-pressure bubble on the bottom surface of the foil, which augments the aerodynamic/hydrodynamic force acting on the foil and, thus, enhances the energy harvesting, as shown in Figure 4. The second major effect of the flap is that it interacts with the vortex, as shown in Figure 5, and generates extra hydrodynamic force, which can be seen by the high lift peaks generated by the flap at around $t=10T$ and $11T$ in Figure 4. It should be also noted that the overall flow fields are very similar for the considered cases, and the flap mainly serves to mitigate the amplitude of the local pressure fields, which are the major sources of the aerodynamic force.

Apart from the qualitative demonstration of the flow features, shown using the instantaneous vorticity contours in Figure 6 and pressure coefficient contours in Figure 5, a direct comparison in terms of the distributions of the power coefficient along the surface of the foil at $t=4T/16$ and $7T/16$ are shown in Figure 7. Only $t=4T/16$ and $7T/16$ are shown here, as these are when the peaks occur with the most significant difference. As the major source of the enhancement of the power coefficient is from the foil (as identified before), only the $C_P$ on the foil is shown here. It is found that the foil with attached flaps obtains a slightly higher power coefficient on the upper surface, and the enhancement becomes less evident near the leading edge, as the influence from the flap decreases quickly from the trailing edge to the leading edge. In terms of the bottom surface, more complex effects are observed for $t=4T/16$ and $7T/16$. When $t=4T/16$, a consistent enhancement is observed for $x/c<0$ and a negligible effect is observed for $x/c>0$; this can be explained by the location of the LEV, which is close to the leading edge. The stronger low-pressure bubble induced by the flap therefore enhances the power coefficient for $x/c<0$, while the low-pressure region at $x/c>0$ is very weak and contributes negligibly to the power coefficient. When $t=7T/16$, the flap first deteriorates the power coefficient at $x/c<0.1$ and then switches to enhance the power coefficient. It should be noted that the pitching velocity (in the negative y direction) is dominating at this stage; therefore, the stronger low-pressure bubble induced by the flap contributes to the positive $C_P$, which explains the enhancement of $C_P$ at $x/c>0.1$. Meanwhile, the pitching motion of the leading edge generates a positive velocity in the y direction, which makes the low pressure contribute to the negative power coefficient. However, as the pitching is dominating, the flap enhances the power generation on the bottom surface at $t=7T/16$.

To conclude, this section presents three typical configurations, i.e., a clean NACA foil, a NACA foil with a single attached flap, and a NACA foil with two symmetric attached flaps. The results show that the flap is able to increase the energy harvesting efficiency significantly, by up to 19.94%, when double symmetric flaps are used, which is even more than that achieved by using single a flexible flap, as reported by [19]. The major mechanism is attributed to the better performance of the double flap in separating the interaction of the LEV generated on the upper and bottom surfaces of the foil, and the extra power harvested by the flap is negligible compared with that harvested by the flap.

4.2. Effects of the Flap Length and the Angle between the Two Symmetric Flaps

In this section, the effects of the flap length and the angle between the two symmetric flaps on the performance of the energy harvesting are further studied. Both the single-flap and double-flap configurations are studied, and the angle between the double flap is varied from ${60}^{\circ }$ to ${120}^{\circ }$, with an interval of ${30}^{\circ }$. In addition to $l_f=0.1c$, as examined in Section 4.1, three more lengths are considered: $0.04c$, $0.15c$ and $0.2c$. These combinations give a total of 16 simulations.

Figure 8 shows the time-averaged power coefficient $\overline{C_P}$ and the energy harvesting efficiency $\eta$. First, for the two-flap scenario, it is found that both $\overline{C_P}$ and $\eta$ increase first when increasing $l_f$ from $0.04c$ to $0.1c$, and then decrease when further increasing $l_f$ up to $0.2c$. To have a clear understanding of the effects induced by different flap lengths, the time histories of $C_L$ and $C_P$ for $\alpha ={60}^{\circ }$ with four different flap lengths are shown in Figure 9. It is found that the lengths considered here do not alter the overall shape of both $C_L$ and $C_P$; only some differences are observed in some local areas, e.g., the peaks of $C_L$ at $t=10T$ and $10.5T$ increase with the increase in $l_f$, which can be explained by the increasing chord length effects. Such large forces actually induce a negative power coefficient, which explains the worse energy harvesting performance when longer flaps are used. However, when a very short flap is used, e.g., $l_f=0.04c$, it is less effective in separating the interactions of the vortexes generated on the upper and bottom surfaces of the NACA foil, and a lower force is generated during almost the entire flapping period.

On the other hand, it is noted that the drop induced by increasing $l_f$ is more profound when large angles are used, e.g., the $C_P$ drops from 1.04 to 0.3 when increasing $l_f$ from 0.1 to 0.2 with $\alpha ={120}^{\circ }$, while the drop for $\alpha ={60}^{\circ }$ is only from 1.09 to 1.01. To further understand how the angle $\alpha$ affects the performance of energy harvesting, the time histories of $C_L$ and $C_P$ at $l_f=0.1c$ and $0.2c$ are, respectively, shown in Figure 10 and Figure 11, where three different angles, i.e., $\alpha ={60}^{\circ }$, ${90}^{\circ }$, and ${120}^{\circ }$, are shown for comparison for each flap length. For the short flap with $l_f=0.1c$, very small differences are observed for all three angles considered, and the major difference is observed between $t=10.4T$ and $10.5T$. The larger $\alpha$ first enhances the $C_P$ and then deteriorates the $C_P$ slightly, and the combined effects show that $\alpha ={60}^{\circ }$ achieves the best performance. Conversely, for the longer flaps, e.g., $l_f=0.2c$, the differences are evident, and the energy harvesting performance decreases continuously with the increase in $\alpha$ during almost the entire flapping period. To have a clear picture on how the angle $\alpha$ affects the aerodynamics/hydrodynamics, the instantaneous vorticity and pressure fields at two typical times ($2T/16$ and $4T/16$) are shown in Figure 12 and Figure 13, respectively. First, the flow fields for all three cases are very similar, i.e., LEV is generated and propagated along the foil, while small differences are observed near the flap. For example, smaller vortex cores are generated close to the flap at $\alpha ={60}^{\circ }$ compared with those at $\alpha ={90}^{\circ }$ and ${120}^{\circ }$, due to the large area in which flaps interact with the flow. The large face area at high values of $\alpha$ also delays the vortex shedding, which keeps a low-pressure core between the two flaps, and it plays a negative role in force generation. For the pressure fields, it can be seen that the low-pressure region on the bottom of the foil decreases continuously when increasing $\alpha$ from ${60}^{\circ }$ to ${120}^{\circ }$ at $t=2T/16$ and $4T/16$, although the difference in positive pressure is not very evident. This is probably due to the larger effective length of the foil at small $\alpha$ values, and such effects contribute to the higher values of $C_L$ and $C_P$ at $\alpha ={60}^{\circ }$, as shown in Figure 11.

To summarize all results, the contours of the efficiency $\eta$ for all 16 cases considered here are shown in Figure 14. The figure shows that there is an optimal configuration that achieves the highest efficiency, i.e., $\alpha ={60}^{\circ }$ and $l_f=0.1$. The corresponding optimal efficiency is increased by up to 19.94% compared with the baseline clean NACA foil, which is comparable with the increase achieved by the flexible flap considered in [19]. This makes the double-symmetric-flap configuration favorable for its simplicity and robustness.

5. Conclusions

In this paper, the energy harvesting performance of a flapping foil with attached flaps is numerically studied by using an IB-LBM, and three different configurations are considered, namely, the clean NACA0015 foil, the NACA0015 foil with a flap attached at the trailing edge, and the NACA0015 foil with double symmetric flaps attached at the trailing edge. It is found that the flap is able to enhance the energy harvesting performance by up to 19.94% when two symmetric flaps are attached to the NACA0015 foil compared with the clean NACA0015 foil. The two flaps outperform the single-flap configuration, mainly induced by the stronger separation effects on the interactions of LEVs formed on the bottom and upper surfaces of the foil.

The parametric study on the length of the flap (from $0.04c$ to $0.2c$) and the angle between the two symmetric flaps (from ${60}^{\circ }$ to ${120}^{\circ }$) shows that increasing the length of the flap first enhances and then deteriorates the energy harvesting performance. The best performances are achieved at $l_f=0.1c$ and $0.15c$ for the double-flap and single-flap configurations, respectively. For the short flaps with $l_f=0.04c$, the larger angle of $\alpha$ performs slightly better, but the increasing length of the flap also deteriorates the performance more quickly for larger $\alpha$ values. Overall, the best energy harvesting performance is achieved at $l_f=0.1c$ and $\alpha ={60}^{\circ }$, with an improvement by up to 19.94% compared with the baseline clean NACA foil.

The current results provide a simple strategy to improve the energy harvesting performance of a flapping foil by adding double symmetric flaps at the trailing edge of the foil, which is simple and effective without bringing in extra complexity and instability like that introduced by a flexible flap. This study should be able to offer some insights to advance the development of simple and effective flapping-foil power generators. An experimental study on the proposed strategy is crucial for practical applications and will be the focus of future work.