# Aerodynamic Performance of the Three-Dimensional Lumped Flexibility Bionic Hovering Wing

Authors: Zhu Jianyang; Jiang Lin; Hou Yu
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

Understanding the effect of flexibility on the aerodynamic characteristics of the wing is one of the most important considerations for successfully designing a flapping wing micro aero vehicle (FMAV). This paper aims at providing a systematic synthesis on the flexibility effects on the hovering performance of the bionic wing based on the numerical analysis approach. We construct a novel three-dimensional bionic wing, which has lumped flexibility at the root, and develop an iterative coupling program to simulate the interaction between the flexible wing and fluid. The effects of flexibility on the hovering performance of the three-dimensional flapping wing are investigated, and the results indicate that the best performance of the wing is achieved when the wing flaps at resonance and has the density close to the natural insect wing. The feasibility of using lumped flexibility wings driven by a simple harmonic flapping for designing efficient FMAV is also concluded in this study.


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## Body

## 1. Introduction

Flapping wing aero vehicle has become a popular research topic in recent years because it has better aerodynamic performance than the fixed-wing and rotary-wing aero vehicle under low Reynolds number flight regime [1–3]. Understanding the effect of flexibility on the aerodynamic characteristics of the wing is one of the most important considerations for successfully designing this type of vehicle. Therefore, many studies have been carried out to investigate this problem, and considerable understandings have been achieved [4–7].The lumped flexibility wing for insect flight was first proposed by Ishihara et al. in [8]. In their study, the two-dimensional wing was modeled by a rectangular plate, and the wing’s flexibility was lumped into the spring at the leading edge. It was found that the studied wing can generate sufficient lift force to support the insect’s weight. Therefore, they concluded that the pitching motion can be passive in the dipteran’s flapping flight. Later, they again experimentally and numerically [9, 10] investigated the performance of the three-dimensional lumped flexible wing, and it was also found that if the wing has appropriate flexibility, the lumped flexible wing can generate sufficient lift force to support the corresponding insect weight.Michelin and Llewellyn Smith [11] studied the influence of the flexibility on the performance of a heaving wing. They found that the maximum values of the mean thrust and efficiency are obtained when the resonance occurs between the forcing frequency and a natural frequency of the system. Masoud and Alexeev [12] conducted a numerical study to focus on the aerodynamic characteristics of a flexible planar hovering wing which was flapped at resonance. They again concluded that large-amplitude resonance flapping wing can drastically enhance aerodynamic lift and efficiency of the wing under low Reynolds number. Moore [13] also found that dramatic performance improves when the wing is driven near resonance and deteriorates when it is driven at higher frequencies. Zhu et al. [14, 15] studied the aerodynamic performance of a 2D flexible wing in forward flight by solving the N-S equations coupled with structural dynamic equation and the flapping motion of the wing. They concluded that if the wing flapped at a frequency close but below the wing’s natural vibration frequency, the flexibility can increase the aerodynamic force and the energy efficiency.On the other hand, Chen et al. [16] performed a study to investigate the aerodynamic performance of a flapping and passively rotating insect wing by using experimental and computational methods. A novel wing which had a single flexible hinge allowing passive pitching under load was devised in their study. The results indicated that the wing’s kinematic and flapping efficiency depends on the stiffness of a passive compliant hinge. It was not the resonance wing but the stiffness wing which had larger lift force production. Cho et al. [17] developed a computational framework to simulate a three-dimensional flapping flexible wing under simultaneous pitching and plunging motion, and the results indicated that the flexibility can be beneficial for the wing with moderate flexibility. Yeh and Alexeev [18] carried out a numerical study to investigate the performance of the plunging elastic plates with aspect ratios ranging from 0.5 to 5. They showed that the maximum velocity occurs near the first natural frequency regardless of aspect ratio, whereas the maximum economy occurs away from the first natural frequency.In summary, the flexibility is beneficial for the flapping wing as shown in the above literatures; however, a full understanding on the mechanism of how the flexibility affects the aerodynamic characteristics of the wing is still not achieved, especially the best performance of the wing flapping at resonance or not. To fill in these gaps, a novel three-dimensional bionic wing which has lumped flexibility is constructed, and an iterative coupling program to simulate the interaction between the flexible wing and fluid is developed. Then the details of the lift force, hovering efficiency, and vortex of the flexible bionic wing are studied.


## 2. The Novel Three-Dimensional Bionic Wing Model

Bumblebee is one of the typical insects which have amazing flying capabilities. The shape, structure, and aerodynamic characteristics of this insect’s wing have been studied extensively [19–21]. In this paper, the planform of the bionic bumblebee wing is selected similar to the experimental model studied by Nagai et al. [22] for comparison. The schematic of the wing planform is shown in Figure 1, where two coordinate systems are also defined: one is the inertial coordinate system (OXYZ) and the other is the wing-fixed coordinate system (OGxGyGz). The symbol b = 0.100 m is the span, br = 0.024 m is the distance from wing base to the flapping y-axis, and c is the reference chord which is defined as a length at two-thirds of the span location (c = 0.01575 m). The area of the wing S is fixed at a value of 2.984 × 10−3 m2, and the thickness of the wing is fixed at S/(50b) for comparison.Figure 1
The schematic of the wing planform.
(a)
(b)In contrast to the previous study that consider the chordwise and spanwise flexibility of the wing, the flexibility of the wing in this paper is mimicked as the passive pitching motion by connecting a rigid wing to a torsional spring; therefore, the flexibility is just lumped at the root of the wing, and the simplified wing is in reasonable agreement with the natural insect wing [23].The flapping motionθ of the wing which is defined as the angle between the axis Gx and the z-axis follows the simple sinusoidal function:
(1)θt=θm2sin2πft,where θm is the flapping amplitude and f is the flapping frequency.The passive pitching angleβ which is defined as the angle between the axis Gy and the y-axis can be decided by the equation
(2)Jβ¨+Kβ=Q,where K is the torsion stiffness, Q the pitching torque generated by the aerodynamic force, and J is the inertia moment of the wing which can be defined as
(3)J=∑iΔmiri2,where mi is the element mass and ri is the distance from the element mass to the pitching axis Gx.


## 3. Parameters of Bionic Hovering Wing

To clearly elaborate the aerodynamic characteristics of the lumped flexibility wing, three dimensionless parameters are defined, namely, the Reynolds number Re, density ratioρ∗, and frequency ratio f∗. They can be defined as
(4)Re=Urefcν,ρ∗=ρsρ,f∗=ffs,where Uref = πfθm(2b/3 + br) is the maximum of flapping velocity, ν is the fluid kinematic viscosity, ρ is the fluid density, ρs is the wing density, and fs is the wing vibration frequency which can be calculated as
(5)fs=K/J2π.Note that the fluid field characteristics are determined by Re and the flexibility of the wing is characterized byf∗ and ρ∗.The coefficient of lift, drag, and energy consumption are described as follows:(6)CL=FL0.5ρUref2S,CD=FD0.5ρUref2S,CP=QYθt•0.5ρUref3S,where FL is the lift, FD the drag, Qy the aerodynamic torque of wing about the y-axis, and θt• is the derivative of θ(t). The hovering efficiency η then can be described as
(7)η=CL¯CP¯,where CL¯ and CP¯ are the mean lift and energy coefficient, respectively.


## 4. The Coupling Program to Simulate the Interaction between Fluid and Flexible Wing

Considering the bumblebee’s flight environment in nature, the Reynolds number of the hovering wing is at the range of 102–104; therefore, the flow field of the wing is assumed to be unsteady, laminar, and incompressible with constant thermophysical properties, and the governing equations of this flow field are described by the 3D unsteady incompressible laminar N-S equations as follows:(8)∇⋅V=0,∂V∂t+V⋅∇V=−∇P+1Re∇2V,where V is the nondimensional fluid velocity and P is the nondimensional fluid pressure.To solve (8), we implement a numerical solver (Fluent 6.3) based on a finite volume method. The third-order MUSCL scheme [24] is used for spatial interpolation, and a first-order implicit algorithm is employed for time discretization. Meanwhile, the coupling between the pressure and the velocity is achieved by means of the PISO [25] algorithm. The flapping and passive pitching motion of the wing are controlled and achieved by the local remeshing dynamic mesh technique (see Fluent 6.3 Tutorial Guide).In order to solve the governing equation of the passive pitching motion of the wing, the Newmark method is applied to discretization derivative of passive pitching angle, and then (2) can be rewritten as
(9)1+γΔt2KJβt+Δt=Q¯t+ΔtJΔt2+2−12−2γ+δΔt2KJβt+−1−12+γ−δΔt2KJβt−Δt,where
(10)Q¯t+Δt=γQt+Δt+12−2γ+δQt+12−γ−δQt−Δt,where βt+Δt, βt, and βt−Δt are the pitching angle of the next step time, current moment, and last step time, respectively. Qt+Δt, Qt, and Qt−Δt are the aerodynamic torque of the next step time, current moment, and last step time, respectively. Δt is the iteration time step and γ and δ are the control parameters, which are fixed as γ = 1/6 and δ = 1/2.An iterative coupling method is developed to simulate the interaction of fluid, flapping, and passive pitching motion of wing, where the fluid field based on finite volume method and passive pitching based on Newmark method are solved separately at each time step. First, the fluid field is solved, then the aerodynamic torque on the wing is obtained, and then the pitching angle of flexibility wing can be determined under the aerodynamic torque using Newmark method, which is embedded in Fluent using the UDF. In the next time step, the fluid field is solved with the new location of the wing updated by using the dynamic mesh technique. In this way, the fluid and the flapping and passive pitching motion are coupled so that the fluid-structure interaction is taken into account properly. The details of the coupling process are illustrated in Figure2.Figure 2
The schematic of coupling method.


## 5. Simulation Setup and Method Validation

### 5.1. Simulation Setup

The computational domain has a hemispherical shape, and the radius of the hemisphere is fixed at a value of 5b as shown in Figure 3. A uniform unstructured mesh system is employed, where the hemispherical shape computational domain contains an inner domain and an outer domain. Tetrahedron cells are used to cluster to the wing in the inner domain, and the density of the tetrahedron cells are decreased from the inner domain to the outer domain. The no-slip wall boundary condition is applied at the bionic hovering wing surface. Meanwhile, pressure-outlet boundary condition which has zero static pressure is imposed on the outer boundary, and symmetry boundary condition is applied on the right computational boundary to save the simulation cost.Figure 3
The computational grid and boundary conditions.


### 5.2. Method Validation

A grid sensitivity study is carried out first to evaluate the independence of the numerical result on the mesh size. The unsteady flow fields of a flexible wing were computed under conditionsρ∗=1000, f∗=1.50, and Re = 2.347 × 103 with three different grid systems. The sizes of involved grid systems were as follows: 3.5 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.040b (system A); 2.3 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.060b (system B); and 1. 9 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.070b (system C). Each grid was computed for 6 periods, and iteration time step was fixed at 0.001T. Only the comparison of passive pitching angle β at different resolutions is given, because the aerodynamic performance of the developed lumped flexibility wing is mainly determined by β. The results in Figure 4 show that the difference of passive pitching angle β of the grid systems A and B are very small (no more than 1%). Therefore, grid system B is sufficiently dense and it is employed for the next simulations.Figure 4
The time variation of passive pitching angleβ under three different grid systems.Three different iteration time steps 0.05T, 0.001T, and 0.0005T (T = 1/f) are employed to again simulate the unsteady flow field of a flexible wing with ρ∗=1000, f∗=1.50, and Re = 2.347 × 103 to validate the time-discretization independence. The results in Figure 5 show that the difference of the passive pitching angle of the three iteration time steps are very small (no more than 2%), which indicates that the iteration time step with t=0.001T is sufficiently small, and it is employed for the next simulations.Figure 5
The time variation of passive pitching angleβ under three different iteration time steps.A typically bionic bumblebee hovering wing which was experimentally studied by Nagai et al. [22] is employed to validate the reliability of the present numerical method for simulating the unsteady flow around 3D hovering wing. The mean drag and lift coefficients are −0.00 and 0.46, respectively, in the present numerical computation, and in Nagai et al.’s experimental study, these two data are 0.00 and 0.49, respectively. The comparison of instantaneous drag and lift coefficient are also illustrated in Figure 6. It is again found that our computational results agree with the experimental results very well.Figure 6
Comparison of the drag and lift coefficient of the wing obtained by the present numerical method and literature experimental data.
(a)
Drag coefficient
(b)
Lift coefficientValidation of the reliability of the developed weak coupling method for simulating the fluid-flapping wing interaction problem had been demonstrated in our previous study. The details of the validation process can be found in [14, 26].


## 5.1. Simulation Setup

The computational domain has a hemispherical shape, and the radius of the hemisphere is fixed at a value of 5b as shown in Figure 3. A uniform unstructured mesh system is employed, where the hemispherical shape computational domain contains an inner domain and an outer domain. Tetrahedron cells are used to cluster to the wing in the inner domain, and the density of the tetrahedron cells are decreased from the inner domain to the outer domain. The no-slip wall boundary condition is applied at the bionic hovering wing surface. Meanwhile, pressure-outlet boundary condition which has zero static pressure is imposed on the outer boundary, and symmetry boundary condition is applied on the right computational boundary to save the simulation cost.Figure 3
The computational grid and boundary conditions.


## 5.2. Method Validation

A grid sensitivity study is carried out first to evaluate the independence of the numerical result on the mesh size. The unsteady flow fields of a flexible wing were computed under conditionsρ∗=1000, f∗=1.50, and Re = 2.347 × 103 with three different grid systems. The sizes of involved grid systems were as follows: 3.5 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.040b (system A); 2.3 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.060b (system B); and 1. 9 × 105 tetrahedron cells around the wing, the first layer cell spacing above the wing 0.070b (system C). Each grid was computed for 6 periods, and iteration time step was fixed at 0.001T. Only the comparison of passive pitching angle β at different resolutions is given, because the aerodynamic performance of the developed lumped flexibility wing is mainly determined by β. The results in Figure 4 show that the difference of passive pitching angle β of the grid systems A and B are very small (no more than 1%). Therefore, grid system B is sufficiently dense and it is employed for the next simulations.Figure 4
The time variation of passive pitching angleβ under three different grid systems.Three different iteration time steps 0.05T, 0.001T, and 0.0005T (T = 1/f) are employed to again simulate the unsteady flow field of a flexible wing with ρ∗=1000, f∗=1.50, and Re = 2.347 × 103 to validate the time-discretization independence. The results in Figure 5 show that the difference of the passive pitching angle of the three iteration time steps are very small (no more than 2%), which indicates that the iteration time step with t=0.001T is sufficiently small, and it is employed for the next simulations.Figure 5
The time variation of passive pitching angleβ under three different iteration time steps.A typically bionic bumblebee hovering wing which was experimentally studied by Nagai et al. [22] is employed to validate the reliability of the present numerical method for simulating the unsteady flow around 3D hovering wing. The mean drag and lift coefficients are −0.00 and 0.46, respectively, in the present numerical computation, and in Nagai et al.’s experimental study, these two data are 0.00 and 0.49, respectively. The comparison of instantaneous drag and lift coefficient are also illustrated in Figure 6. It is again found that our computational results agree with the experimental results very well.Figure 6
Comparison of the drag and lift coefficient of the wing obtained by the present numerical method and literature experimental data.
(a)
Drag coefficient
(b)
Lift coefficientValidation of the reliability of the developed weak coupling method for simulating the fluid-flapping wing interaction problem had been demonstrated in our previous study. The details of the validation process can be found in [14, 26].


## 6. Results and Discussion

To study the aerodynamic performance of the three-dimensional (3D) lumped flexibility bionic hovering wing, the results of the 3D rigid flapping wing need to be employed for comparison. The 3D rigid and lumped flexibility bionic hovering wing are performed withθm = π/3, f=120 Hz, and Re = 2347, and the parameters of the lumped flexibility wing are characterized by frequency ratio f∗ and density ratio ρ∗. The details of f∗ as well as ρ∗ on the lift and hovering efficiency of the wing will be analyzed in the following.


### 6.1. Varying Frequency Ratio

To study the effect of frequency ratiof∗ on the aerodynamic performance of the 3D lumped flexibility bionic hovering wing, we fix the density ratio ρ∗=1000, and the frequency ratio with f∗=0.00, 0.40, 0.50, 0.67, 1.00, 1.50, and 2.00 are considered. Note that f∗=0.00 is represented as rigid wing, and as f∗ increases, the flexibility of the wing increases.Figure7 shows the performance of the 3D lumped flexibility wing with different f∗. It is seen from Figure 7(a) that the mean lift coefficients first increase as f∗ increases when f∗ is smaller than 1.00 and reach a maximum at f∗=1.00; however, they drop sharply when f∗ crosses over 1.00, which indicates that the flexibility wing with f∗=1.00 has the best lift generation performance for the wing considered in this section.Figure 7
The performance of the 3D lumped flexibility wing with differentf∗.
(a)
Mean lift coefficients
(b)
Mean drag coefficients
(c)
Mean energy coefficients
(d)
Hovering efficienciesIt is clear in Figure7(b) that the mean drag coefficients increase as f∗ increases when f∗ is smaller than 0.67 and reach a maximum value 0.00 at f∗=0.67; however, they drop sharply when f∗ crosses over 0.67. It is also found in this figure that the mean drag coefficients have negative values expect for the wing with f∗=0.67, which indicates that the thrust are generated for the other flexible wings.It is obvious in Figure7(c) that the minimum mean energy coefficient is obtained when the flexible wing with f∗=1.00, and in comparison with the rigid wing, the mean energy coefficient of this wing is decreased by 36.27%. However, for the other flexible wing, the frequency ratio f∗ has little effect on the mean energy coefficient; they have almost identical values.It is seen from Figure7(d) that a similar variation trend of hovering efficiencies and lift coefficients with frequency ratio f∗ are observed. The energy efficiencies increase with the frequency ratio when the ratio is smaller than 1.00 and reach a maximum at f∗=1.00, which indicates that the best hovering performance of the wing is achieved when the flexible wing with f∗=1.00. However, they drop sharply when f∗ crosses over 1.00 and even a negative value is observed, which indicates that the hovering performance is deteriorated. This conclusion is different with the results obtained in our previous 2D lumped flexible wing study [27], at which we concluded that the best performance of the wing is achieved when the wing with frequency ratio f∗ is close to but less than 1.00.To analyze the mechanism of how the flexibility enhances the hovering performance of the lumped flexibility bionic wing details, Figure8 shows the time variation of lift coefficients, drag coefficients, and energy coefficients of the above considered wings. It is obvious in this figure that after six flapping cycle simulations, periodic lift, drag, and energy coefficients have been established. It is also observed in this figure that the flexible wing with f∗=1.00 not only has smaller drag and energy coefficient amplitude than the other flexible wings but also has smoother time variation trend (with smaller curve slope), while for the lift coefficient, it has a larger amplitude, which is the reason why the flexible wing with f∗=1.00 has a larger mean lift and smaller energy coefficient as shown in Figure 7.Figure 8
Time variation of the lift coefficients, drag coefficients, and energy coefficients of the 3D lumped flexibility wing with differentf∗.
(a)
Lift coefficients
(b)
Drag coefficients
(c)
Energy coefficientsFigure9 plots the time variation of passive pitching angle β of the flexible wing with f∗=1.00 and 1.50, and the time variation of flapping motion θ is also presented for comparison. It is obvious in this figure that f∗ influences β significantly. For the flexible wing with f∗=1.00, the peak negative angle takes place at a phase angle of 0.86π, while for flapping angle θ, the peak negative angle happens at a phase angle of 1.01π; moreover, the passive pitching angle β of the flexible wing with f∗=1.00 also has a larger amplitude than the flapping angle θ. While for the flexible wing with f∗=1.50, the passive pitching angle has visibly smaller amplitude than the flapping angle θ, and the variation cycle of β is two times that of the flapping motion θ. Those findings could be useful to explain the phenomenon why the flexible wing with f∗=1.00 and 1.50 has different lift, drag, and energy coefficients as shown in Figure 8.Figure 9
Time variation of passive pitching angle of flexible wings withf∗=1.00 and 1.50 and flapping angle θ.In order to investigate why the flexible wing withf∗=1.00 and 1.50 possesses different performance with the rigid wing, Figures 10 and 11 show the vortex of the three considered wings at t = 11.30T and 11.70T; for each time, three different wing span positions are investigated (z = 3.11c, 4.70c, and 6.29c). Obviously, it is seen in Figure 10 that similar vortex patterns near the wing surface are observed for the rigid wing and flexible wing with f∗=1.00 and 1.50; however, for the rigid wing and flexible wing with f∗=1.50, a visibly separated vortex is observed. It is also found in this figure that different vortex patterns are observed at wake of the three considered wings; a jet (in the negative Y direction) is observed for the flexible wing with f∗=1.00, while it is not for the rigid wing and flexible wing with f∗=1.50.Figure 10
The vortex of the rigid and flexible wing withf∗=1.00 and 1.50 at t = 11.30T and z = 3.11c, 4.70c, and 6.29c.Figure 11
The vortex of the rigid and flexible wing withf∗=1.00 and 1.50 at t=11.70T and z = 3.11c, 4.70c, and 6.29c.In Figure11, a comparison with the rigid wing and flexible wing with f∗=1.50 is seen, in addition to the jet in the negative Y direction; the wake capture mechanism which is considered as the reason for the flapping wing having high-lift generating capacity is also observed for the flexible wing with f∗=1.00, which is the reason why the lumped flexible wing with f∗=1.00 has better hovering performance than the rigid wing and flexible wing with f∗=1.50. It is also found in this figure that the flexible wing with f∗=1.50 has the weakest vortex near the wing surface, which leads the wing to have the worst aerodynamic performance.


### 6.2. Varying Density Ratio

To study the effect of density ratioρ∗ on the aerodynamic performance of the 3D lumped flexibility bionic hovering wing, we fix the frequency ratio f∗=1.00 which has the best hovering performance of the studied wing in the last section, and the density ratio with ρ∗=750, 1000, 1250, 1500, 1750, and 2000 are considered.Figure12 plots the performance of the 3D lumped flexibility wing with different ρ∗. Three interesting phenomena can be concluded from this figure. Firstly, the density ratio has little effect on the mean drag coefficients, they have almost identical value at zero. Secondly, the mean energy coefficients of the wing monotonously decrease as ρ∗ increases. Thirdly, the maximum mean lift coefficient as well as hovering efficiency is achieved when the wing has ρ∗=1750, which indicates that the flexible wing with ρ∗=1750 has better hovering performance than the other considered wing in this section. According to the study on the properties of insect wings by Combes and Daniel [28], the density ratio ρ∗ of insect wing is typically of the order O(103), which is consist with our findings.Figure 12
The performance of the 3D lumped flexibility wing with differentρ∗.
(a)
Mean lift coefficients
(b)
Mean drag coefficients
(c)
Mean energy coefficients
(d)
Hovering efficienciesIn order to analyze the mechanism of how the density ratio influences aerodynamic performance of the 3D lumped flexibility bionic hovering wing, three typically density ratiosρ∗=750, 1750, and 200 are considered.Figure13 shows the time variation of lift coefficients, drag coefficients, and energy coefficients of the above three considered wings. Although a similar time variation trend of the lift coefficients are generated, there are phase differences between these three flexible wings, and the amplitude of the lift coefficient of the flexible wing with ρ∗=1750 is slightly larger than that of the flexible wing with ρ∗=750; on the contrary, the flexible wing with ρ∗=2000 results in the flexible wing with ρ∗=1750 to have larger mean lift coefficient and the flexible wing with ρ∗=2000 to have smaller mean lift coefficient, as shown in Figure 12(a). Obviously, the drag coefficient of the flexible wing with ρ∗=1750 and 2000 have almost identical values; however, the flexible wing with ρ∗=750 has larger amplitude almost during the whole flapping cycle; moreover, it also has rougher time variation trend, which indicates that the flexible wing with ρ∗=1750 and 2000 have better hovering stability. For the energy coefficient, the flexible wing with ρ∗=2000 has slightly smaller value than the flexible wing with ρ∗=1500 almost during the whole flapping cycle, while the flexible wing with ρ∗=750 has a larger value than the flexible wing with ρ∗=1500, which results in the mean energy coefficient of the flexible wing decreasing with ρ∗ increasing, as shown in Figure 12(c).Figure 13
Time variation of the lift coefficients, drag coefficients, and energy coefficients of the 3D lumped flexibility wing withρ∗=750, 1750, and 2000.
(a)
Lift coefficients
(b)
Drag coefficients
(c)
Energy coefficientsFigure14 plots the time variation of passive pitching angle β of the above three relation wings. It is clear in this figure that there are phase differences between these three angles, and the amplitude of β increases with the increase in ρ∗, which may influence the vortex generated around the wing.Figure 14
Time variation of passive pitching angle of flexible wings withρ∗=750, 1750, and 2000.Figures15 and 16 show the vortex of the flexible wing with ρ∗=750, 1750, and 2000 at t = 0.30T and 0.70T, and for each of time, again three different wing span positions are investigated (z = 3.11c, 4.70c, and 6.29c). It is seen from Figure 15 that similar vortex patterns near the wing surface are observed for the three considered wings, from root to tip; the strength of the vortex is increased; however, the separate vortex of the flexible wing with ρ∗=1750 is closer to the wing surface than the other two flexible wings, which indicates that the flexible wing with ρ∗=1750 has an enhancement delay stall mechanism.Figure 15
The vortex of the flexible wing withρ∗=750, 1750, and 2000 at t = 0.30T, z = 3.11c, 4.70c, and 6.29c.Figure 16
The vortex of the flexible wing withρ∗=750, 1750, and 2000 at t = 0.70T, z = 3.11c, 4.70c, and 6.29c.It is obvious in Figure16 that the strongest vortex is generated around the flexible wing with ρ∗=1750. Moreover, complex vortex patterns are observed under the wake of the flexible wing with ρ∗=750, and the weakest vortex is generated around the flexible wing with ρ∗=2000, which indicates that the flexible wing with ρ∗=750 needs more energy to perform the wing hovering, and the flexible wing with ρ∗=2000 generates the smallest aerodynamic force, as shown in Figure 13.


## 6.1. Varying Frequency Ratio

To study the effect of frequency ratiof∗ on the aerodynamic performance of the 3D lumped flexibility bionic hovering wing, we fix the density ratio ρ∗=1000, and the frequency ratio with f∗=0.00, 0.40, 0.50, 0.67, 1.00, 1.50, and 2.00 are considered. Note that f∗=0.00 is represented as rigid wing, and as f∗ increases, the flexibility of the wing increases.Figure7 shows the performance of the 3D lumped flexibility wing with different f∗. It is seen from Figure 7(a) that the mean lift coefficients first increase as f∗ increases when f∗ is smaller than 1.00 and reach a maximum at f∗=1.00; however, they drop sharply when f∗ crosses over 1.00, which indicates that the flexibility wing with f∗=1.00 has the best lift generation performance for the wing considered in this section.Figure 7
The performance of the 3D lumped flexibility wing with differentf∗.
(a)
Mean lift coefficients
(b)
Mean drag coefficients
(c)
Mean energy coefficients
(d)
Hovering efficienciesIt is clear in Figure7(b) that the mean drag coefficients increase as f∗ increases when f∗ is smaller than 0.67 and reach a maximum value 0.00 at f∗=0.67; however, they drop sharply when f∗ crosses over 0.67. It is also found in this figure that the mean drag coefficients have negative values expect for the wing with f∗=0.67, which indicates that the thrust are generated for the other flexible wings.It is obvious in Figure7(c) that the minimum mean energy coefficient is obtained when the flexible wing with f∗=1.00, and in comparison with the rigid wing, the mean energy coefficient of this wing is decreased by 36.27%. However, for the other flexible wing, the frequency ratio f∗ has little effect on the mean energy coefficient; they have almost identical values.It is seen from Figure7(d) that a similar variation trend of hovering efficiencies and lift coefficients with frequency ratio f∗ are observed. The energy efficiencies increase with the frequency ratio when the ratio is smaller than 1.00 and reach a maximum at f∗=1.00, which indicates that the best hovering performance of the wing is achieved when the flexible wing with f∗=1.00. However, they drop sharply when f∗ crosses over 1.00 and even a negative value is observed, which indicates that the hovering performance is deteriorated. This conclusion is different with the results obtained in our previous 2D lumped flexible wing study [27], at which we concluded that the best performance of the wing is achieved when the wing with frequency ratio f∗ is close to but less than 1.00.To analyze the mechanism of how the flexibility enhances the hovering performance of the lumped flexibility bionic wing details, Figure8 shows the time variation of lift coefficients, drag coefficients, and energy coefficients of the above considered wings. It is obvious in this figure that after six flapping cycle simulations, periodic lift, drag, and energy coefficients have been established. It is also observed in this figure that the flexible wing with f∗=1.00 not only has smaller drag and energy coefficient amplitude than the other flexible wings but also has smoother time variation trend (with smaller curve slope), while for the lift coefficient, it has a larger amplitude, which is the reason why the flexible wing with f∗=1.00 has a larger mean lift and smaller energy coefficient as shown in Figure 7.Figure 8
Time variation of the lift coefficients, drag coefficients, and energy coefficients of the 3D lumped flexibility wing with differentf∗.
(a)
Lift coefficients
(b)
Drag coefficients
(c)
Energy coefficientsFigure9 plots the time variation of passive pitching angle β of the flexible wing with f∗=1.00 and 1.50, and the time variation of flapping motion θ is also presented for comparison. It is obvious in this figure that f∗ influences β significantly. For the flexible wing with f∗=1.00, the peak negative angle takes place at a phase angle of 0.86π, while for flapping angle θ, the peak negative angle happens at a phase angle of 1.01π; moreover, the passive pitching angle β of the flexible wing with f∗=1.00 also has a larger amplitude than the flapping angle θ. While for the flexible wing with f∗=1.50, the passive pitching angle has visibly smaller amplitude than the flapping angle θ, and the variation cycle of β is two times that of the flapping motion θ. Those findings could be useful to explain the phenomenon why the flexible wing with f∗=1.00 and 1.50 has different lift, drag, and energy coefficients as shown in Figure 8.Figure 9
Time variation of passive pitching angle of flexible wings withf∗=1.00 and 1.50 and flapping angle θ.In order to investigate why the flexible wing withf∗=1.00 and 1.50 possesses different performance with the rigid wing, Figures 10 and 11 show the vortex of the three considered wings at t = 11.30T and 11.70T; for each time, three different wing span positions are investigated (z = 3.11c, 4.70c, and 6.29c). Obviously, it is seen in Figure 10 that similar vortex patterns near the wing surface are observed for the rigid wing and flexible wing with f∗=1.00 and 1.50; however, for the rigid wing and flexible wing with f∗=1.50, a visibly separated vortex is observed. It is also found in this figure that different vortex patterns are observed at wake of the three considered wings; a jet (in the negative Y direction) is observed for the flexible wing with f∗=1.00, while it is not for the rigid wing and flexible wing with f∗=1.50.Figure 10
The vortex of the rigid and flexible wing withf∗=1.00 and 1.50 at t = 11.30T and z = 3.11c, 4.70c, and 6.29c.Figure 11
The vortex of the rigid and flexible wing withf∗=1.00 and 1.50 at t=11.70T and z = 3.11c, 4.70c, and 6.29c.In Figure11, a comparison with the rigid wing and flexible wing with f∗=1.50 is seen, in addition to the jet in the negative Y direction; the wake capture mechanism which is considered as the reason for the flapping wing having high-lift generating capacity is also observed for the flexible wing with f∗=1.00, which is the reason why the lumped flexible wing with f∗=1.00 has better hovering performance than the rigid wing and flexible wing with f∗=1.50. It is also found in this figure that the flexible wing with f∗=1.50 has the weakest vortex near the wing surface, which leads the wing to have the worst aerodynamic performance.


## 6.2. Varying Density Ratio

To study the effect of density ratioρ∗ on the aerodynamic performance of the 3D lumped flexibility bionic hovering wing, we fix the frequency ratio f∗=1.00 which has the best hovering performance of the studied wing in the last section, and the density ratio with ρ∗=750, 1000, 1250, 1500, 1750, and 2000 are considered.Figure12 plots the performance of the 3D lumped flexibility wing with different ρ∗. Three interesting phenomena can be concluded from this figure. Firstly, the density ratio has little effect on the mean drag coefficients, they have almost identical value at zero. Secondly, the mean energy coefficients of the wing monotonously decrease as ρ∗ increases. Thirdly, the maximum mean lift coefficient as well as hovering efficiency is achieved when the wing has ρ∗=1750, which indicates that the flexible wing with ρ∗=1750 has better hovering performance than the other considered wing in this section. According to the study on the properties of insect wings by Combes and Daniel [28], the density ratio ρ∗ of insect wing is typically of the order O(103), which is consist with our findings.Figure 12
The performance of the 3D lumped flexibility wing with differentρ∗.
(a)
Mean lift coefficients
(b)
Mean drag coefficients
(c)
Mean energy coefficients
(d)
Hovering efficienciesIn order to analyze the mechanism of how the density ratio influences aerodynamic performance of the 3D lumped flexibility bionic hovering wing, three typically density ratiosρ∗=750, 1750, and 200 are considered.Figure13 shows the time variation of lift coefficients, drag coefficients, and energy coefficients of the above three considered wings. Although a similar time variation trend of the lift coefficients are generated, there are phase differences between these three flexible wings, and the amplitude of the lift coefficient of the flexible wing with ρ∗=1750 is slightly larger than that of the flexible wing with ρ∗=750; on the contrary, the flexible wing with ρ∗=2000 results in the flexible wing with ρ∗=1750 to have larger mean lift coefficient and the flexible wing with ρ∗=2000 to have smaller mean lift coefficient, as shown in Figure 12(a). Obviously, the drag coefficient of the flexible wing with ρ∗=1750 and 2000 have almost identical values; however, the flexible wing with ρ∗=750 has larger amplitude almost during the whole flapping cycle; moreover, it also has rougher time variation trend, which indicates that the flexible wing with ρ∗=1750 and 2000 have better hovering stability. For the energy coefficient, the flexible wing with ρ∗=2000 has slightly smaller value than the flexible wing with ρ∗=1500 almost during the whole flapping cycle, while the flexible wing with ρ∗=750 has a larger value than the flexible wing with ρ∗=1500, which results in the mean energy coefficient of the flexible wing decreasing with ρ∗ increasing, as shown in Figure 12(c).Figure 13
Time variation of the lift coefficients, drag coefficients, and energy coefficients of the 3D lumped flexibility wing withρ∗=750, 1750, and 2000.
(a)
Lift coefficients
(b)
Drag coefficients
(c)
Energy coefficientsFigure14 plots the time variation of passive pitching angle β of the above three relation wings. It is clear in this figure that there are phase differences between these three angles, and the amplitude of β increases with the increase in ρ∗, which may influence the vortex generated around the wing.Figure 14
Time variation of passive pitching angle of flexible wings withρ∗=750, 1750, and 2000.Figures15 and 16 show the vortex of the flexible wing with ρ∗=750, 1750, and 2000 at t = 0.30T and 0.70T, and for each of time, again three different wing span positions are investigated (z = 3.11c, 4.70c, and 6.29c). It is seen from Figure 15 that similar vortex patterns near the wing surface are observed for the three considered wings, from root to tip; the strength of the vortex is increased; however, the separate vortex of the flexible wing with ρ∗=1750 is closer to the wing surface than the other two flexible wings, which indicates that the flexible wing with ρ∗=1750 has an enhancement delay stall mechanism.Figure 15
The vortex of the flexible wing withρ∗=750, 1750, and 2000 at t = 0.30T, z = 3.11c, 4.70c, and 6.29c.Figure 16
The vortex of the flexible wing withρ∗=750, 1750, and 2000 at t = 0.70T, z = 3.11c, 4.70c, and 6.29c.It is obvious in Figure16 that the strongest vortex is generated around the flexible wing with ρ∗=1750. Moreover, complex vortex patterns are observed under the wake of the flexible wing with ρ∗=750, and the weakest vortex is generated around the flexible wing with ρ∗=2000, which indicates that the flexible wing with ρ∗=750 needs more energy to perform the wing hovering, and the flexible wing with ρ∗=2000 generates the smallest aerodynamic force, as shown in Figure 13.


## 7. Conclusion and Future Work

In this paper, a novel three-dimensional bionic wing which has lumped flexibility at the root is constructed; a numerical experiment is carried out to investigate the flexibility effects on the hovering performance of this novel bionic wing, where the incompressible Navier-Stokes (N-S) equations coupled with passive pitching motion by aerodynamic torque is solved. The flow field and aerodynamic forces are analyzed for a different frequency ratiof∗ and density ratio ρ∗ wings, and the results show that the f∗ and ρ∗ influence the hovering characteristics of the wing greatly. For the wing with different frequency ratio f∗, the best performance of the wing is achieved when the wing flaps at resonance (f∗=1.00). For the wing with different density ratio ρ∗, it is found that the flexible wing with ρ∗=1750 has the best hovering performance, which agrees with the insect’s density ratio in nature. In addition, delay stall and wake capture mechanic are observed around the wing with appropriate frequency ratio f∗ and density ratio ρ∗ (f∗=1.00, ρ∗=1750), therefore leading to better performance characteristics.The results presented in this paper are based on lumped flexibility wing. Future work will simplify the bumblebee’s wing as a 3D flexible plate.



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*Source: 1314258-2018-04-10.xml*