# The Effect of Damping Coefficient, Spring Coefficient, and Mass Ratio on the Power Extraction Performance of a Semiactive Flapping Wing

Authors: Jianyang Zhu
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

The effect of varying damping coefficientC∗, spring coefficient K∗, and mass ratio M∗ on the semiactive flapping wing power extraction performance was numerically studied in this paper. A numerical code based on Finite Volume method to solve the two-dimensional Navier-Stokes equations and coupled with Finite Center Difference method to solve the passive plunging motion equation is developed. At a Reynolds number of 3400 and the pitching axis at quarter chord from the leading edge of the wing, the power extraction performance of the semiactive flapping wing with different damping coefficient, spring coefficient, and mass ratio is systematically investigated. The optimal set of spring coefficient is found at a value of 1.00. However, the variation of mass ratio M∗ cannot increase the maximum mean power coefficient and power efficiency, but it can influence the value of damping coefficient C∗ at which the wing achieves the maximum mean power coefficient and power efficiency. Moreover, insensitivity of the mean power coefficient and power efficiency to the variation of damping coefficient C∗ is observed for the wing with smaller mass ratio, which indicates the wing with smaller M∗ has better working stability.


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## Body

## 1. Introduction

The application of flapping wing for energy extraction is inspired by insects and birds, who exhibit excellent aerodynamic performance by extracting wind energy through flapping their wings. Comparing to the conventional rotation energy extraction turbines, flapping wing power generator possesses many advantages, such as simpler design, less construction costs, more efficiency in low stream speeds, and more friendly to the flying creatures in nature [1]. Therefore, more and more recently researches have been focused on this type of power generator [2–5].Up to now, there are three types of flapping wing power generator, named fully active [6], semiactive, and purely passive system, respectively [7]. Among them, the semiactive flapping wing power generator which has one of the flapping motion (pitching) is actuated and another motion (plunging) is induced by free stream fluid and is considered as a more feasible approach in industry, because it is easy to implement and control [8, 9]. This type of flapping wing power generator is characterized by the interaction coupling of fluid, driven wing pitching, and passive wing plunging, which results the flow around the generator very complex, and the energy extraction performance is still the main research focus of this type of power generator.To investigate the energy extraction performance of a semiactive flapping wing near solid walls, the mechanical parameters of the wing to achieve power extraction efficiency were optimized by Wu et al. [10]. They concluded that the spring constant at a value of 1.0 and damping coefficient at a value of π are an optimal choice to achieve net power extraction efficiency of the wing, when the wing has fixed wing mass at a value of 1.0 and Reynolds number at a value of 1100. Under the same Reynolds number, Zhan et al. [11] performed a similar numerical study to optimize the energy extraction performance of a semiactive flapping wing with fixed wing mass at a value of 1.0 and reduced frequency f∗=0.2. It was found that the parameters for the wing that achieved best power extraction are the difference with different pitching amplitude. For the wing with pitching amplitude θm≤67.5, the optimal power extraction efficiency is achieved when the wing has the spring constant at a value of 5.0 and damping coefficient at a value of π, while for the wing with pitching amplitude θm>67.5, the optimal power extraction efficiency is achieved when the wing has the spring constant at a value of 5.0 and damping coefficient at a value of 0.5π. Moreover, based on the numerical study by Zhu et al. [12], the optimal power extraction performance is the system where there is no spring on the plunge motion and the damping coefficient is at a value of π.On the other hand, to explore an inertial effect on the power extraction performance of a semiactive flapping wing, Deng et al. [13] conducted a strong fluid-structure coupling method to study the power extraction performance of a semiactive flapping wing with different mass ratios (ranging from 0.125 to 100), and it was found that the energy harvesting efficiency decreases monotonically with increasing mass ratio. However, the amount of power extraction changes very little when the wing has mass ratio less than 10. While, according to the experiment study on a flapping wing hydroelectric power generation system by Abiru and Yoshitake [14], the wing with lager mass ratio is needed to excite the hydroelastic response; therefore, the wing with lager mass ratio has better power extraction performance.As discussed above, the effect of damping coefficient, spring constant, and mass ratio on the power extraction performance of the semiactive flapping wing is still not clearly understood. Therefore, in this paper, a numerical code based on Finite Volume method to solve the two-dimensional Navier-Stokes equations and coupled with Finite Center Difference method to solve passive plunging motion equation is employed. The flow around a semiactive flapping wing is simulated with different damping coefficient, spring constant, and mass ratio. The NACA0015 airfoil is employed to represent the two-dimensional section wing, and the power extraction performance as well as the fluid around the wing are studied details in the following.


## 2. Problem Definition and Methodology

### 2.1. Problem Definition

In this paper, NACA0015 airfoil with chord lengthd and mass M is employed to model the flapping wing. A damper with damping coefficient C and spring constant K which is attached to the wing is employed to mimic the power extractor. The schematic view of the semiactive flapping wing is shown in Figure 1, where the profile θt is defined to drive the pitching motion of the wing; the plunging motion htis induced by the lift force of the wing. The pitching point O is located at the center line of the wing with a distance 0.25d from the leading edge. To simplify the problem, a cosine pitching motion mode is employed, and the center of mass of the wing is also designed to coincide with the pitching point O. Then, the governing equation of the pitching motion θt and passive plunging motion ht can be defined as
(1)θt=−θmcos2πft,(2)Mh¨+Cḣ+Kh=FL,where θm is the pitching amplitude, f is the pitching frequency, FL is the lift force in Y direction as shown in Figure 1. The left and right sides of equation (2) is divided by the mass, then, it can be described as
(3)h¨+CMḣ+KMh=FLM.Figure 1
The schematic view of the semiactive flapping wing.It is seen from equation (1) to equation (3) that if we fix the pitching amplitude θm and pitching frequency f, the kinematic of the semiactive flapping wing is determined by the following three characteristic parameters:
(4)C∗=CM,K∗=K/M2πf,M∗=ρsdρd,where ρ is the fluid density, ρs is the wing density, M=ρsS, and S is the area of the wing. Moreover, the Reynolds number Re and reduced frequency f∗ can be defined as
(5)Re=ρU∞dμ,f∗=2πfdU∞,where U∞ is the stream velocity and ν is the fluid kinematic viscosity.When the semiactive flapping wing is working, the power extracted from the fluid can be described as(6)PE=Cḣ2,and the power which is needed to input the system to drive the pitching motion can be described as
(7)PI=−MTω,where MT is the fluid torque about the pitching axis O and ω is the pitching velocity. Then, the net power is given by
(8)PT=PE−PI,and the power coefficient is described as
(9)CP=PT0.5ρU∞3d.The mean energy coefficient can be defined as(10)CP¯=1T∫tt+TCPdt,where T is the pitching period.Then, the power extraction efficiency of the generator can be defined as(11)η=PT¯0.5ρU∞3hmax=CP¯dhmax,where hmax is the overall vertical extent of the wing motion (the distance in Y direction between the highest position and the lowest position reached by the wing, either the leading edge or the trailing edge). It is defined as the maximum value of the following equation:
(12)hmax=maxamplitudeh+0.75dsinθ,amplitudeh−0.25dsinθ.Note that the drag force has not been mentioned in this work, because it does not contribute to the power production.


### 2.2. Numerical Method

Based on the free stream velocity and the chord length of the wing, the Reynolds number and Mach number are fixed at a value of 3400 and 0.0015, respectively. According to our previous work [5], at Reynolds number 3400, the flow assuming laminar, the calculating results can match experimental results very well for flapping wing. Therefore, the fluid around the wing is assumed as laminar and incompressible. The governing equation can be described as
(13)∇⋅V=0,(14)∂V∂t+V⋅∇V=−1ρ∇p+μρ∇2V,where V is the velocity vector and P is the pressure. Fluent ver. 6.3 proprietary solver is employed to solve equation (13) and equation (14). It has been well validated for simulating flapping wing problems [15, 16]. The second-order upwind algorithm is adopted for the space discretization, and first-order implicit algorithm is selected for time discretization. Meanwhile, SIMPLE algorithm is selected for solving the coupling between the pressure and the velocity. The convergence criterion for the solver is satisfied with residual less than 10-3.Finite Center Difference method is employed to solve passive plunging motion governing equation (3), which is embedded in Fluent ver. 6.3 by using a user-defined function (UDF). Dynamic mesh technique is also employed to update the wing’s position (active pitching and passive plunging) at each time step. For more details about the coupling method to solve interaction of fluid and the semiactive flapping wing, one can refer to our previous work [5, 17].The computational domain of this work has C type shape. According to the works of Zhu and Peng [18], the distance between the outer boundary and the airfoil is at about 8d lengths; flow field and the aerodynamic performance of the semiactive flapping airfoil are not significant by the far field boundary. Therefore, the radius of the semicircle on the left side of the computational domain is fixed 20d, and the size of the rectangle on the right side of the computational domain is set as L=20d, H=40d. A triangular mesh system is employed where a C type computational domain (shown in Figure 2) containing an inner flapping domain and an outer fixed domain, and the inner domain which moves according to the wing kinematics, and the first grid is located at 0.001c from the wing surface, the growth rate of the grid size is 1.08.Figure 2
Schematic of the computational grid and boundary conditions.Figure 3
The power coefficient of a semiactive flapping wing with different meshes and iteration time steps.The no-slip wall boundary condition is applied at the semiactive flapping wing. The inlet velocity defined asu=U∞ is imposed on the left, up, and low side of the domain. Pressure-outlet boundary condition which has zero static pressure is imposed on the right side of the domain.


### 2.3. Validation

To validate the independence of the numerical results on the special-discretization and time-discretization scheme, computations with different meshes and iteration time steps are simulated. These checks are carried out on the representative case ofθm=45°, f∗=0.32, Re=3400, C∗=2.00, K∗=1.0, and M∗=60. The power coefficient calculated by the coarse mesh (the first grid is located at 0.005c from the wing surface, the growth rate of the grid size is 1.08, total 4.22×104 cells), medium mesh (the first grid is located at 0.001c from the wing surface, the growth rate of the grid size is 1.08, total 8.39×104 cells), and fine mesh (the first grid is located at 0.0005c from the wing surface, the growth rate of the grid size is 1.08, total 1.51×105 cells) with iteration time steps ∆t=T/2000 and ∆t=T/4000 is shown in Figure 3. It is seen from this figure that the meshes and time steps have slightly influenced the computation results. Therefore, the medium mesh is sufficiently dense and iteration time steps ∆t=T/2000 is sufficiently small to grasp the main flow features over the wing, and they are employed for the next simulations.To validate the coupling method for the study on the semiactive flapping wing, a typical case of a semiactive flapping wing (wing shape is also NACA0015) which was numerical studied by Teng et al. [19] is simulated. The simulation parameters are set the same as the parameters used by Teng et al. [19]: Re=1000, θm=45° and f∗=0.16, C∗=π, K∗=0.0, and M=1.0 kg. The resulting lift coefficient CL=FL/0.5ρU∞2d of the wing is plotted against time as shown in Figure 4. Comparing to the result by Teng et al. [19], good agreement is achieved.Figure 4
The lift coefficient of a semiactive flapping wing.


## 2.1. Problem Definition

In this paper, NACA0015 airfoil with chord lengthd and mass M is employed to model the flapping wing. A damper with damping coefficient C and spring constant K which is attached to the wing is employed to mimic the power extractor. The schematic view of the semiactive flapping wing is shown in Figure 1, where the profile θt is defined to drive the pitching motion of the wing; the plunging motion htis induced by the lift force of the wing. The pitching point O is located at the center line of the wing with a distance 0.25d from the leading edge. To simplify the problem, a cosine pitching motion mode is employed, and the center of mass of the wing is also designed to coincide with the pitching point O. Then, the governing equation of the pitching motion θt and passive plunging motion ht can be defined as
(1)θt=−θmcos2πft,(2)Mh¨+Cḣ+Kh=FL,where θm is the pitching amplitude, f is the pitching frequency, FL is the lift force in Y direction as shown in Figure 1. The left and right sides of equation (2) is divided by the mass, then, it can be described as
(3)h¨+CMḣ+KMh=FLM.Figure 1
The schematic view of the semiactive flapping wing.It is seen from equation (1) to equation (3) that if we fix the pitching amplitude θm and pitching frequency f, the kinematic of the semiactive flapping wing is determined by the following three characteristic parameters:
(4)C∗=CM,K∗=K/M2πf,M∗=ρsdρd,where ρ is the fluid density, ρs is the wing density, M=ρsS, and S is the area of the wing. Moreover, the Reynolds number Re and reduced frequency f∗ can be defined as
(5)Re=ρU∞dμ,f∗=2πfdU∞,where U∞ is the stream velocity and ν is the fluid kinematic viscosity.When the semiactive flapping wing is working, the power extracted from the fluid can be described as(6)PE=Cḣ2,and the power which is needed to input the system to drive the pitching motion can be described as
(7)PI=−MTω,where MT is the fluid torque about the pitching axis O and ω is the pitching velocity. Then, the net power is given by
(8)PT=PE−PI,and the power coefficient is described as
(9)CP=PT0.5ρU∞3d.The mean energy coefficient can be defined as(10)CP¯=1T∫tt+TCPdt,where T is the pitching period.Then, the power extraction efficiency of the generator can be defined as(11)η=PT¯0.5ρU∞3hmax=CP¯dhmax,where hmax is the overall vertical extent of the wing motion (the distance in Y direction between the highest position and the lowest position reached by the wing, either the leading edge or the trailing edge). It is defined as the maximum value of the following equation:
(12)hmax=maxamplitudeh+0.75dsinθ,amplitudeh−0.25dsinθ.Note that the drag force has not been mentioned in this work, because it does not contribute to the power production.


## 2.2. Numerical Method

Based on the free stream velocity and the chord length of the wing, the Reynolds number and Mach number are fixed at a value of 3400 and 0.0015, respectively. According to our previous work [5], at Reynolds number 3400, the flow assuming laminar, the calculating results can match experimental results very well for flapping wing. Therefore, the fluid around the wing is assumed as laminar and incompressible. The governing equation can be described as
(13)∇⋅V=0,(14)∂V∂t+V⋅∇V=−1ρ∇p+μρ∇2V,where V is the velocity vector and P is the pressure. Fluent ver. 6.3 proprietary solver is employed to solve equation (13) and equation (14). It has been well validated for simulating flapping wing problems [15, 16]. The second-order upwind algorithm is adopted for the space discretization, and first-order implicit algorithm is selected for time discretization. Meanwhile, SIMPLE algorithm is selected for solving the coupling between the pressure and the velocity. The convergence criterion for the solver is satisfied with residual less than 10-3.Finite Center Difference method is employed to solve passive plunging motion governing equation (3), which is embedded in Fluent ver. 6.3 by using a user-defined function (UDF). Dynamic mesh technique is also employed to update the wing’s position (active pitching and passive plunging) at each time step. For more details about the coupling method to solve interaction of fluid and the semiactive flapping wing, one can refer to our previous work [5, 17].The computational domain of this work has C type shape. According to the works of Zhu and Peng [18], the distance between the outer boundary and the airfoil is at about 8d lengths; flow field and the aerodynamic performance of the semiactive flapping airfoil are not significant by the far field boundary. Therefore, the radius of the semicircle on the left side of the computational domain is fixed 20d, and the size of the rectangle on the right side of the computational domain is set as L=20d, H=40d. A triangular mesh system is employed where a C type computational domain (shown in Figure 2) containing an inner flapping domain and an outer fixed domain, and the inner domain which moves according to the wing kinematics, and the first grid is located at 0.001c from the wing surface, the growth rate of the grid size is 1.08.Figure 2
Schematic of the computational grid and boundary conditions.Figure 3
The power coefficient of a semiactive flapping wing with different meshes and iteration time steps.The no-slip wall boundary condition is applied at the semiactive flapping wing. The inlet velocity defined asu=U∞ is imposed on the left, up, and low side of the domain. Pressure-outlet boundary condition which has zero static pressure is imposed on the right side of the domain.


## 2.3. Validation

To validate the independence of the numerical results on the special-discretization and time-discretization scheme, computations with different meshes and iteration time steps are simulated. These checks are carried out on the representative case ofθm=45°, f∗=0.32, Re=3400, C∗=2.00, K∗=1.0, and M∗=60. The power coefficient calculated by the coarse mesh (the first grid is located at 0.005c from the wing surface, the growth rate of the grid size is 1.08, total 4.22×104 cells), medium mesh (the first grid is located at 0.001c from the wing surface, the growth rate of the grid size is 1.08, total 8.39×104 cells), and fine mesh (the first grid is located at 0.0005c from the wing surface, the growth rate of the grid size is 1.08, total 1.51×105 cells) with iteration time steps ∆t=T/2000 and ∆t=T/4000 is shown in Figure 3. It is seen from this figure that the meshes and time steps have slightly influenced the computation results. Therefore, the medium mesh is sufficiently dense and iteration time steps ∆t=T/2000 is sufficiently small to grasp the main flow features over the wing, and they are employed for the next simulations.To validate the coupling method for the study on the semiactive flapping wing, a typical case of a semiactive flapping wing (wing shape is also NACA0015) which was numerical studied by Teng et al. [19] is simulated. The simulation parameters are set the same as the parameters used by Teng et al. [19]: Re=1000, θm=45° and f∗=0.16, C∗=π, K∗=0.0, and M=1.0 kg. The resulting lift coefficient CL=FL/0.5ρU∞2d of the wing is plotted against time as shown in Figure 4. Comparing to the result by Teng et al. [19], good agreement is achieved.Figure 4
The lift coefficient of a semiactive flapping wing.


## 3. Results and Discussions

As conducted above, the main parameters were related to the response of the semiactive flapping wing including the pitching amplitudeθm, reduced frequency f∗, Reynolds number Re, damping coefficient C∗, spring coefficient K∗, and mass ratio M∗. To simplify the problem, the factor θm is fixed at 45°, f∗ is set at 0.32, and Re keeps a value of 3.4×103. Consequently, the key parameters that affect the power extraction performance of the semiactive flapping wing are the damping coefficient C∗, spring coefficient K∗, and mass ratio M∗. The details of these three parameters studied in this paper are summarized in Table 1. Note that the parameter spaces in this work are almost covering all the parameters studied in the literature as discussed in Introduction, but some controversial conclusions are conducted; therefore, it needs to be studied further.Table 1
Parameters considered in this paper.
Damping coefficientC∗
Spring coefficientK∗
Mass ratioM∗
0.0625~18.0
0.00~3.00
15~240


### 3.1. Effect of Damping Coefficients and Spring Coefficients

To examine the effect of damping and spring coefficient, the mass ratioM∗ is fixed at 60 first. Figure 5 plots the mean power coefficient and power efficiency against spring coefficient K∗ with different damping coefficient C∗. Obviously, in this figure that at a given C∗ both mean power coefficient and power efficiency increase first then decrease and achieve a maximum at K∗=1.0. However, when K∗ is larger than 1.0, both mean power coefficient and power efficiency of the wing decrease sharply, which implies the power extraction performance of the wing is deteriorated. In addition, for K∗=1.0, the maximum mean power coefficient of the wing monotonically decrease with C∗, while the maximum power efficiency of the wing increase first then decrease and achieve a maximum at C∗=4.0. This conclusion is different with the results obtained by Zhu et al. [12], at which they concluded that the best power extraction performance of the wing is achieved, when the spring coefficient is at 0.00 and the damping coefficient is at a value of π. These difference can be due to the potential flow theory used by Zhu et al. [12], for which the viscosity effect and the interaction between wing and vortex are neglected.Figure 5
The mean power coefficient and power efficiency against spring coefficientK∗ with different damping coefficient C∗; (a) mean power coefficient; (b) power efficiency.
(a)
(b)To explore the mechanism of how the spring coefficients and damping coefficients influence the power extraction performance of the semiactive flapping wing in detail, we first investigate the performance of the wing with fixedC∗=1.0, and three typical cases with K∗=0.0, 1.0, and 3.0 are studied. The time variation of power coefficient of the above considered cases is shown in Figure 6. It is clear in this figure that the wing with K∗=1.0 has the largest power coefficient amplitude, then is the wing with K∗=0.00, and the wing with K∗=3.0 has the smallest power coefficient amplitude, which is consistent with the mean power generation of the wing as shown in Figure 5. Meanwhile, there exists phase difference between the three wings. The wing with K∗=1.0 has the earliest time to achieve maximum power coefficient, then is the wing with K∗=0.00, and the wing with K∗=3.0 has the latest to achieve maximum power coefficient.Figure 6
The time variation of power coefficient of the considered cases withC∗=1.0 and K∗=0.0, 1.0, and 3.0.Besides the power coefficient, the power efficiency of the wing is also influenced by the plunging motionht. Figure 7 shows the time variation of ht of the above three considered cases, where the defined pitching motion θt is also illustrated for comparison. It is found from this figure that the amplitude of the ht first increase then decrease with K∗, and the phase difference between pitching and ht is -0.305T, -0.205T, and 0.475T for the wing with K∗=0.0, 1.0, and 3.0, respectively. Moreover, nonharmonic time variation of ht is observed for the wing with K∗=3.0, which indicates the complex lift force induced by the wing with K∗=3.0.Figure 7
The time variation ofh/d of the considered cases with K∗=0.0, 1.0, and 3.0, C∗=1.0.Figure8 shows the time variation of lift coefficient CL=FL/0.5ρU∞2d of the above three considered cases. It is seen from this figure that different with the time variation of ht, the amplitude of the CL first decrease then increase with K∗. The wing with larger amplitude of CL has smaller amplitude of ht.Figure 8
The time variation of lift coefficient of the considered cases withK∗=0.0, 1.0, and 3.0, C∗=1.0.To explore the physical mechanism of the effect of variation of spring coefficient, Figure9 shows the vorticity contours during a semipitching cycle (t=0.10T-0.40T). Due to the symmetrical flapping of the wing, semipitching cycle is considered. It is seen from this figure that the spring coefficient can influence the vortex structure of the wing significantly. Comparing with the wing with K∗=0.0, there are less separated vortex in the wake for the wing with K∗=1.0, and the separated vortex can reattach to the wing surface during flapping; moreover, there are weaker leading edge vortex generating for the wing with K∗=1.0, which results the wing to have smaller lift coefficient as shown in Figure 8. Moreover, the separated vortex in the wake indicates more power disappeared, which is the reason why the wing with K∗=1.0 has larger mean power coefficient and efficiency. On the other hand, for the wing with K∗=3.0, obviously, there are more separated vortex in the wake comparing with the wing with K∗=0.0; however, the stronger leading edge vortex are generated by the wing with K∗=3.0, which is the reason why the wing with K∗=3.0 has larger lift coefficient but smaller mean power coefficient and efficiency.Figure 9
The vorticity contours of the considered cases withK∗=0.0, 1.0, and 3.0, C∗=1.0.After investigating the effect of spring coefficient on the power extraction performance of the semiactive flapping wing, the effect of damping coefficient on the power extraction performance of the semiactive flapping wing is studied. To this end, we study the performance of the wing with fixedK∗=1.0, and two typical cases with C∗=1.0 and 4.0 are investigated in details. The time variation of power coefficient of the above considered cases is shown in Figure 10. It is clear in this figure that the wing with C∗=1.0 has the larger power coefficient than the wing with C∗=4.0 almost during the whole flapping cycle, which is consistent with the mean power generation of the wing as shown in Figure 5.Figure 10
The time variation of power coefficient of the considered cases withK∗=1.0, C∗=1.0 and 4.0.Figure11 shows the time variation of ht of the above two considered cases. The same as the power coefficient, the amplitude of the ht of the wing with C∗=1.0 has larger amplitude than the wing with C∗=4.0; however, comparing with the pitching motion, the two considered wings have almost identical phase difference.Figure 11
The time variation ofh/d of the considered cases with C∗=1.0 and 4.0, K∗=1.0.Figure12 shows the time variation of lift coefficient of the two considered wings with C∗=1.0 and 4.0, K∗=1.0. It is seen from this figure that contrary with the time variation of ht, the wing with C∗=4.0 has larger amplitude of lift coefficient than the wing with C∗=1.0.Figure 12
The time variation of lift coefficient of the considered cases withC∗=1.0 and 4.0, K∗=1.0.Figure13 shows the vorticity contours of the two considered wings with C∗=1.0 and 4.0, K∗=1.0 during a semipitching cycle. Obviously, in this figure, there exists a vortex separated for the wing with C∗=4.0 (see t=0.20T), while for the wing with C∗=1.0, no obviously separated vortex is observed, which results that the wing with C∗=4.0 has smaller mean power coefficient. It is also found that the passive plunging amplitude of the wing with C∗=1.0 has larger value than the wing with C∗=4.0. According to equation (11), when the increased passive plunging amplitude of the wing is larger than the increased mean power coefficient, the efficiency of the wing will fall. This is the reason why the wing with C∗=4.0 has smaller mean power coefficient but has larger power efficiency.Figure 13
The vorticity contours of the considered cases withC∗=1.0 and 4.0, K∗=1.0.


### 3.2. Effect of Mass Ratio

To examine the effect of mass ratio, the spring coefficientK∗ is fixed at 1.00, which has the best energy extraction performance under different damping coefficient as section above discussed.Figure14 plots the mean power coefficient and power efficiency against damping coefficient C∗ with different mass ratio M∗. There are three interesting conclusions that can be concluded from this figure. Firstly, at a given M∗, both mean power coefficient and power efficiency increase first then decrease; moreover, the value of C∗ at which the wing achieves maximum mean power coefficient is smaller than the wing at which achieves maximum power efficiency. Secondly, the maximum mean power coefficient and power efficiency, respectively, are almost identical for the wing with different M∗; however, the value of C∗ at which the wing achieves the maximum mean power coefficient and power efficiency decreases with M∗ increasing. Thirdly, for the wing with smaller M∗, both the mean power coefficient and power efficiency are insensitive to the variation of damping coefficient C∗C∗=5~18, which indicates that the wing with smaller M∗ has better working stability that, therefore, leads the wing to have better power extraction performance.Figure 14
The mean power coefficient and power efficiency against damping coefficientC∗ with different mass ratio M∗; (a) mean power coefficient; (b) power efficiency.
(a)
(b)To explore the mechanism of how the mass ratio influence the power extraction performance of the semiactive flapping wing in detail, two typical cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0 are considered. Figure 15 shows the time variation of power coefficient of the above two considered cases. It is clear in this figure that for the wing with M∗=15, C∗=3.0, it has larger power coefficient than the wing with M∗=240, C∗=3.0 almost the whole pitching cycle, and it also has earlier time to achieve maximum power coefficient.Figure 15
The time variation of power coefficient of the considered cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure16 shows the time variation of h/d of the above two considered cases. The wing with M∗=15, C∗=3.0, has larger amplitude of h/d than the wing with M∗=240, C∗=3.0; meanwhile, the phase difference between pitching and h of the two considered wings is identical; the value is 0.30T.Figure 16
The time variation ofhmax/d of the considered cases with M∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure17 shows the time variation of lift coefficient CL=FL/0.5ρU∞2d of the above two relation cases. Different from the time variation of power coefficient and the passive plunging motion h, the wing with M∗=15, C∗=3.0, has smaller amplitude of lift coefficient than the wing with M∗=240, C∗=3.0; meanwhile, comparing to the wing with M∗=240, C∗=3.0, there exists an obvious phase delay for the wing with M∗=15, C∗=3.0, to achieve maximum lift coefficient.Figure 17
The time variation of lift coefficientCL=FL/0.5ρU∞2d of the considered cases with M∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure18 shows four instantaneous vorticity contours of the above considered wings during a semipitching cycle. It is seen from this figure that the mass ratio can influence the flow field of the wing significantly. Similar as the wing with different damping coefficient (as shown in Figure 13), there exists a vortex separated for the wing with M∗=240(see t = 0.10T to 0.30T), while for the wing with M∗=15, no obviously separated vortex is observed, which results the wing with M∗=240 has smaller mean power coefficient. It is also found that the passive plunging amplitude of the wing with M∗=15 has larger value than the wing with M∗=240. According to equation (11), when the increased passive plunging amplitude of the wing is larger than the increased mean power coefficient, the efficiency of the wing will fall. This is the reason why the wing with M∗=240 has smaller mean power coefficient but has larger power efficiency.Figure 18
The vorticity contours of the considered cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0.


## 3.1. Effect of Damping Coefficients and Spring Coefficients

To examine the effect of damping and spring coefficient, the mass ratioM∗ is fixed at 60 first. Figure 5 plots the mean power coefficient and power efficiency against spring coefficient K∗ with different damping coefficient C∗. Obviously, in this figure that at a given C∗ both mean power coefficient and power efficiency increase first then decrease and achieve a maximum at K∗=1.0. However, when K∗ is larger than 1.0, both mean power coefficient and power efficiency of the wing decrease sharply, which implies the power extraction performance of the wing is deteriorated. In addition, for K∗=1.0, the maximum mean power coefficient of the wing monotonically decrease with C∗, while the maximum power efficiency of the wing increase first then decrease and achieve a maximum at C∗=4.0. This conclusion is different with the results obtained by Zhu et al. [12], at which they concluded that the best power extraction performance of the wing is achieved, when the spring coefficient is at 0.00 and the damping coefficient is at a value of π. These difference can be due to the potential flow theory used by Zhu et al. [12], for which the viscosity effect and the interaction between wing and vortex are neglected.Figure 5
The mean power coefficient and power efficiency against spring coefficientK∗ with different damping coefficient C∗; (a) mean power coefficient; (b) power efficiency.
(a)
(b)To explore the mechanism of how the spring coefficients and damping coefficients influence the power extraction performance of the semiactive flapping wing in detail, we first investigate the performance of the wing with fixedC∗=1.0, and three typical cases with K∗=0.0, 1.0, and 3.0 are studied. The time variation of power coefficient of the above considered cases is shown in Figure 6. It is clear in this figure that the wing with K∗=1.0 has the largest power coefficient amplitude, then is the wing with K∗=0.00, and the wing with K∗=3.0 has the smallest power coefficient amplitude, which is consistent with the mean power generation of the wing as shown in Figure 5. Meanwhile, there exists phase difference between the three wings. The wing with K∗=1.0 has the earliest time to achieve maximum power coefficient, then is the wing with K∗=0.00, and the wing with K∗=3.0 has the latest to achieve maximum power coefficient.Figure 6
The time variation of power coefficient of the considered cases withC∗=1.0 and K∗=0.0, 1.0, and 3.0.Besides the power coefficient, the power efficiency of the wing is also influenced by the plunging motionht. Figure 7 shows the time variation of ht of the above three considered cases, where the defined pitching motion θt is also illustrated for comparison. It is found from this figure that the amplitude of the ht first increase then decrease with K∗, and the phase difference between pitching and ht is -0.305T, -0.205T, and 0.475T for the wing with K∗=0.0, 1.0, and 3.0, respectively. Moreover, nonharmonic time variation of ht is observed for the wing with K∗=3.0, which indicates the complex lift force induced by the wing with K∗=3.0.Figure 7
The time variation ofh/d of the considered cases with K∗=0.0, 1.0, and 3.0, C∗=1.0.Figure8 shows the time variation of lift coefficient CL=FL/0.5ρU∞2d of the above three considered cases. It is seen from this figure that different with the time variation of ht, the amplitude of the CL first decrease then increase with K∗. The wing with larger amplitude of CL has smaller amplitude of ht.Figure 8
The time variation of lift coefficient of the considered cases withK∗=0.0, 1.0, and 3.0, C∗=1.0.To explore the physical mechanism of the effect of variation of spring coefficient, Figure9 shows the vorticity contours during a semipitching cycle (t=0.10T-0.40T). Due to the symmetrical flapping of the wing, semipitching cycle is considered. It is seen from this figure that the spring coefficient can influence the vortex structure of the wing significantly. Comparing with the wing with K∗=0.0, there are less separated vortex in the wake for the wing with K∗=1.0, and the separated vortex can reattach to the wing surface during flapping; moreover, there are weaker leading edge vortex generating for the wing with K∗=1.0, which results the wing to have smaller lift coefficient as shown in Figure 8. Moreover, the separated vortex in the wake indicates more power disappeared, which is the reason why the wing with K∗=1.0 has larger mean power coefficient and efficiency. On the other hand, for the wing with K∗=3.0, obviously, there are more separated vortex in the wake comparing with the wing with K∗=0.0; however, the stronger leading edge vortex are generated by the wing with K∗=3.0, which is the reason why the wing with K∗=3.0 has larger lift coefficient but smaller mean power coefficient and efficiency.Figure 9
The vorticity contours of the considered cases withK∗=0.0, 1.0, and 3.0, C∗=1.0.After investigating the effect of spring coefficient on the power extraction performance of the semiactive flapping wing, the effect of damping coefficient on the power extraction performance of the semiactive flapping wing is studied. To this end, we study the performance of the wing with fixedK∗=1.0, and two typical cases with C∗=1.0 and 4.0 are investigated in details. The time variation of power coefficient of the above considered cases is shown in Figure 10. It is clear in this figure that the wing with C∗=1.0 has the larger power coefficient than the wing with C∗=4.0 almost during the whole flapping cycle, which is consistent with the mean power generation of the wing as shown in Figure 5.Figure 10
The time variation of power coefficient of the considered cases withK∗=1.0, C∗=1.0 and 4.0.Figure11 shows the time variation of ht of the above two considered cases. The same as the power coefficient, the amplitude of the ht of the wing with C∗=1.0 has larger amplitude than the wing with C∗=4.0; however, comparing with the pitching motion, the two considered wings have almost identical phase difference.Figure 11
The time variation ofh/d of the considered cases with C∗=1.0 and 4.0, K∗=1.0.Figure12 shows the time variation of lift coefficient of the two considered wings with C∗=1.0 and 4.0, K∗=1.0. It is seen from this figure that contrary with the time variation of ht, the wing with C∗=4.0 has larger amplitude of lift coefficient than the wing with C∗=1.0.Figure 12
The time variation of lift coefficient of the considered cases withC∗=1.0 and 4.0, K∗=1.0.Figure13 shows the vorticity contours of the two considered wings with C∗=1.0 and 4.0, K∗=1.0 during a semipitching cycle. Obviously, in this figure, there exists a vortex separated for the wing with C∗=4.0 (see t=0.20T), while for the wing with C∗=1.0, no obviously separated vortex is observed, which results that the wing with C∗=4.0 has smaller mean power coefficient. It is also found that the passive plunging amplitude of the wing with C∗=1.0 has larger value than the wing with C∗=4.0. According to equation (11), when the increased passive plunging amplitude of the wing is larger than the increased mean power coefficient, the efficiency of the wing will fall. This is the reason why the wing with C∗=4.0 has smaller mean power coefficient but has larger power efficiency.Figure 13
The vorticity contours of the considered cases withC∗=1.0 and 4.0, K∗=1.0.


## 3.2. Effect of Mass Ratio

To examine the effect of mass ratio, the spring coefficientK∗ is fixed at 1.00, which has the best energy extraction performance under different damping coefficient as section above discussed.Figure14 plots the mean power coefficient and power efficiency against damping coefficient C∗ with different mass ratio M∗. There are three interesting conclusions that can be concluded from this figure. Firstly, at a given M∗, both mean power coefficient and power efficiency increase first then decrease; moreover, the value of C∗ at which the wing achieves maximum mean power coefficient is smaller than the wing at which achieves maximum power efficiency. Secondly, the maximum mean power coefficient and power efficiency, respectively, are almost identical for the wing with different M∗; however, the value of C∗ at which the wing achieves the maximum mean power coefficient and power efficiency decreases with M∗ increasing. Thirdly, for the wing with smaller M∗, both the mean power coefficient and power efficiency are insensitive to the variation of damping coefficient C∗C∗=5~18, which indicates that the wing with smaller M∗ has better working stability that, therefore, leads the wing to have better power extraction performance.Figure 14
The mean power coefficient and power efficiency against damping coefficientC∗ with different mass ratio M∗; (a) mean power coefficient; (b) power efficiency.
(a)
(b)To explore the mechanism of how the mass ratio influence the power extraction performance of the semiactive flapping wing in detail, two typical cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0 are considered. Figure 15 shows the time variation of power coefficient of the above two considered cases. It is clear in this figure that for the wing with M∗=15, C∗=3.0, it has larger power coefficient than the wing with M∗=240, C∗=3.0 almost the whole pitching cycle, and it also has earlier time to achieve maximum power coefficient.Figure 15
The time variation of power coefficient of the considered cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure16 shows the time variation of h/d of the above two considered cases. The wing with M∗=15, C∗=3.0, has larger amplitude of h/d than the wing with M∗=240, C∗=3.0; meanwhile, the phase difference between pitching and h of the two considered wings is identical; the value is 0.30T.Figure 16
The time variation ofhmax/d of the considered cases with M∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure17 shows the time variation of lift coefficient CL=FL/0.5ρU∞2d of the above two relation cases. Different from the time variation of power coefficient and the passive plunging motion h, the wing with M∗=15, C∗=3.0, has smaller amplitude of lift coefficient than the wing with M∗=240, C∗=3.0; meanwhile, comparing to the wing with M∗=240, C∗=3.0, there exists an obvious phase delay for the wing with M∗=15, C∗=3.0, to achieve maximum lift coefficient.Figure 17
The time variation of lift coefficientCL=FL/0.5ρU∞2d of the considered cases with M∗=15, C∗=3.0 and M∗=240, C∗=3.0.Figure18 shows four instantaneous vorticity contours of the above considered wings during a semipitching cycle. It is seen from this figure that the mass ratio can influence the flow field of the wing significantly. Similar as the wing with different damping coefficient (as shown in Figure 13), there exists a vortex separated for the wing with M∗=240(see t = 0.10T to 0.30T), while for the wing with M∗=15, no obviously separated vortex is observed, which results the wing with M∗=240 has smaller mean power coefficient. It is also found that the passive plunging amplitude of the wing with M∗=15 has larger value than the wing with M∗=240. According to equation (11), when the increased passive plunging amplitude of the wing is larger than the increased mean power coefficient, the efficiency of the wing will fall. This is the reason why the wing with M∗=240 has smaller mean power coefficient but has larger power efficiency.Figure 18
The vorticity contours of the considered cases withM∗=15, C∗=3.0 and M∗=240, C∗=3.0.


## 4. Conclusion

In this paper, the power extraction performance of the semiactivated flapping wing with different damping coefficient, spring coefficient, and mass ratio is numerically examined. A numerical code based on Finite Volume method to solve the two-dimensional Navier-Stokes equations and coupled with Finite Center Difference method to solve passive plunging motion equation is developed.We first examine the effect of damping coefficient and spring coefficient and fix the mass ratio at 60. It is found that an optimal set of spring coefficient (K∗=1.00) is obtained, for which both the high mean power coefficient and power efficiency are achieved no matter what the damping coefficient value of the wing is. Second, the effect of mass ratio is examined, and the spring coefficient is fixed at 1.00. It is concluded that the variation of mass ratio cannot increase the maximum mean power coefficient and power efficiency of the semiactivated flapping wing, but it can influence the value of damping coefficient C∗ at which the wing achieves the maximum mean power coefficient and power efficiency; moreover, insensitivity of the mean power coefficient and power efficiency to the variation of damping coefficient C∗ is observed for the wing with smaller mass ratio, which indicates that the wing with smaller M∗ has better working stability. More specifically, the flow field around the wing is also investigated, and for the wing with appropriate damping coefficient, spring coefficient, and mass ratio (K∗=1.00, C∗=4.0, and M∗=60), the separated vortex can reattach to the wing surface during flapping, which results in no obvious vortex disappeared in the wake. This is the reason for the wing to have better power extraction performance.



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*Source: 5151808-2019-04-16.xml*