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Flapping wing energy harvesting: aerodynamic aspects

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Abstract

Aerodynamic forces on flapping wings create forward thrust. Natural flyers like birds and insects apply these forces effectively. Extensive studies have shown that the efficiency of flapping wings can be improved by various means. Of importance is the development and control of leading edge vortices (LEV’s). Thrust production means that energy is transferred from the flapping system into the fluid. In a similar way it can be achieved that energy flows from the fluid into the flapping system, i.e. fluid energy may be harvested. Responsible for the direction of energy flow is the ratio of pitching amplitude versus amplitude of the induced incidence of the plunging motion. If this ratio is smaller than unity thrust energy is produced; if it is larger than unity energy is transferred into the flapping system. In the present paper, emphasis is placed on the detailed study of the aerodynamic effects and on some ideas of optimization of energy harvesting of a flapping system. It will be shown that similar to the thrust production mode also in the energy harvesting mode, the influence of LEV’s is of major concern. The control of these vortices by airfoil deformation is shown to be beneficial for optimizing the efficiency of energy harvesting.

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Abbreviations

a ∞ :

Speed of sound, m/s

c :

Airfoil chord, m

c P :

Power coefficient (time-dependant)

C P :

Mean power coefficient

c M :

Moment coefficient (time-dependant)

C M :

Mean pitching moment coefficient

c N :

Normal force coeff. (time-dependant)

C N :

Mean normal force coefficient

d :

Difference between the highest and lowest point reached by the airfoil, referred to chord

f :

Frequency of oscillation, Hz

f*:

Reduced frequency, fc/U∞

h :

Non-dimensional plunging amplitude referred to chord, h = z/c

Ma:

Mach number, Ma  =  U∞/a∞

P :

Mean value of section power per unit span, (Nm/s)/m

M :

Pitching moment about pitch axis per unit span, Nm/m

Re :

Reynolds number: Re = U∞c/ν

t :

Time, s

T :

Non-dimensional time, T = tU∞/c

T p :

Non-dimensional time of an oscillation period Tp = 2π/ω*

T :

Normalized time, T = T/Tp

U ∞ :

Free-stream velocity, m/s

x, z :

Horizontal and vertical coordinate, m

xD, zD :

Location of rotation axis, referred to chord

xp, zp :

Location of flex-center, referred to chord

X, Z :

Horizontal and vertical section force per unit span, N/m

X :

Mean value of horizontal section force per unit span

Z :

Mean value of normal section force per unit span

θ :

Effective incidence, θ = θh + θp

θ h :

Incidence induced by plunging motion, θh = tan−1(vh/U∞), deg

θ p :

Incidence of pitching motion, deg

V h :

Non-dimensional plunging velocity, Vh = hω* sin (ω*T)

V θ :

Non-dimensional pitching velocity, Vθ = θp0ω* sin (ω*T +Φ)

η :

Efficiency: η = P/(1/2ρU3∞d)

ν :

Kinematic viscosity, m2/s

ρ :

Density, kg/m3

Φ :

Phase shift between pitch and plunge

ψ :

Nose droop angle

Δψ :

Amplitude of nose-droop angle

ω :

Rotational frequency of airfoil oscillation, ω = 2 π f, rad/s

ω*:

Rotational frequency of airfoil oscillation, ω* = ωc/U∞ = 2πf

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  1. Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), Göttingen, Germany

    W. Geissler

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Geissler, W. Flapping wing energy harvesting: aerodynamic aspects. CEAS Aeronaut J 11, 379–389 (2020). https://doi.org/10.1007/s13272-019-00394-1

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