Aerodynamics of Insect Flight

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Aerodynamics of Insect Flight Effects of wind gusts on a rigid flapping NACA 0012 airfoil at Re = 3000 Marcus Lundberg [email protected] Degree Project in Engineering Physics, First Cycle, SA104X at KTH Mechanics Supervisors: Luca Brandt & Walter Fornari Examiner: M˚ arten Olsson Stockholm, Sweden 2015

Transcript of Aerodynamics of Insect Flight

Page 1: Aerodynamics of Insect Flight

Aerodynamics of Insect Flight

Effects of wind gusts on a rigid flapping NACA 0012 airfoil at Re = 3000

Marcus Lundberg

[email protected]

Degree Project in Engineering Physics, First Cycle, SA104X at KTH MechanicsSupervisors: Luca Brandt & Walter Fornari

Examiner: Marten OlssonStockholm, Sweden 2015

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Abstract

Insects and small flyers operate at Reynolds numbers ranging from ap-proximately 10− 105, where viscous forces are important. Due to theirsmall size and weight, they are sensitive to small changes in the freestream during flight, such as wind gusts. First, the aerodynamics offlapping flight is briefly explained. Then the lift, drag and power us-age for a flapping NACA 0012 airfoil is simulated in ANSYS Fluentfor different oncoming wind directions. The aim of the report is to un-derstand how the pitching amplitude, the flapping frequency and theplunging amplitude can be adjusted to compensate for oncoming windgusts. The simulation is modelled as quasi-static since the time-scale ofthe flapping wings of the insect is much shorter than the time-scale ofthe wind gusts.

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Contents

1 Introduction 41.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory of flapping wing aerodynamics 52.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Wing nomenclature . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Reynolds number, Re . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Reduced frequency, k . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Strouhal number, St . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 Leading edge vortices (LEVs) . . . . . . . . . . . . . . . . . . 82.2.5 Dynamic stall . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Wing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Equations for wing motions in 2D . . . . . . . . . . . . . . . 8

2.4 Power considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Numerical simulations 103.1 Governing equations and turbulence model . . . . . . . . . . . . . . 103.2 Geometry and mesh parameters . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Grid characteristics . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Calculation of power usage . . . . . . . . . . . . . . . . . . . 163.3.3 Moving airfoil setup . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.1 Reference simulation . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Varying the reduced frequency, k . . . . . . . . . . . . . . . . 193.4.3 Varying the plunging amplitude, ha . . . . . . . . . . . . . . 213.4.4 Varying the pitching amplitude, αa . . . . . . . . . . . . . . . 23

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5.1 Wind from below, β > 0◦ . . . . . . . . . . . . . . . . . . . . 253.5.2 Wind from above, β < 0◦ . . . . . . . . . . . . . . . . . . . . 253.5.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Appendix A 274.1 User Defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . 27

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1 Introduction

The flight of birds, bats and insects has intrigued humans for many centuries.There exists almost a million species of flying insects and over 13000 warm-bloodedvertebrate species have the ability to roam the sky. Although great progress hasbeen made in aeronautical technology over the past 100 years, birds’, bats’ andinsects’ ability to manoeuvre a body efficiently through air is still very impressive.Studying their flight mechanisms is important in the development of Micro AirVehicles with flapping wings. By studying the flapping wing flight in animals, wecan improve the designs and build smaller, more efficient and more manoeuvrableMAVs. However, a consequence of the small size of insects is that they are verysensitive to disturbances in the environment, such as wind gusts.

1.1 Objectives

It is the aim of this report to first explain the basics of flapping wing flight and thenstudy the effects of wind gusts on lift, drag and thrust on insect-sized bodies, andto give a recommendation for how the wings should be aligned to counteract thegust and have optimal performance. The frequencies of typical wind gusts, aroundO(1) Hz, are low compared to the flapping frequencies of the insects, which areO(101) Hz to O(102) Hz. Therefore, in the time-scale of the flapping wings, thewind gusts can often be modelled as quasi-steady [1].

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2 Theory of flapping wing aerodynamics

In this section a short theoretical background on flapping wing aerodynamics atlow Reynolds numbers is introduced.

2.1 Definitions

2.1.1 Wing nomenclature

Figure (1) shows the different parts of an asymmetric airfoil.

Figure 1: The terminology for lift producing airfoils.

The chord line is the imaginary straight line between the leading edge and thetrailing edge of the airfoil. The camber line is defined as the curve that lies halfwaybetween the upper and lower surfaces of the airfoil. The chord line and camber linecoincide for symmetric airfoils, such as the NACA 0012 airfoil, which will be usedin the simulations. The maximum distance between the camber line and the chordline is the maximum camber, which is a measure of the curvature of the airfoil.

2.1.2 Reynolds number, Re

The Reynolds number is defined as the ratio between the inertial and viscous forces.It is defined as

Re =UrefLref

ν(1)

where Uref is a reference velocity, Lref is a reference length and ν is the kinematicviscosity of the fluid. In flapping wing flight, the reference length is generally takento be either a mean chord length or the wing length R. The mean chord length isdefined as the average chord length in the spanwise direction. In forward flight, thereference velocity Uref is the forward velocity [1].

2.1.3 Reduced frequency, k

The ratio between the forward velocity and the flapping velocity is another impor-tant dimensionless parameter used to analyse the aerodynamic performance of anatural flyer [2]. It is expressed using the reduced frequency k,

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k =πfc

U, (2)

where f is the flapping frequency, c is the mean chord length and U is the forwardvelocity U .

2.1.4 Strouhal number, St

The Strouhal number St is a dimensionless parameter that describes the wing kine-matics for flying animals [2]. It is well known to govern the vortex dynamics andshedding behaviour for airfoils undergoing pitching and plunging motions. It isdefined as the stroke frequency f times a reference length Lref = 2ha, where ha isthe plunging amplitude, divided by the forward speed U .

St =2fhaU

=2khaπc

, (3)

where St is expressed in the reduced frequency using equation (2). The Strouhalnumber for natural flyers and swimmers in cruising conditions has been found tobe in the range 0.2 < St < 0.4 [2].

2.2 Aerodynamic forces

2.2.1 Lift

When an airfoil moves forward through air, the air is deflected and exerts an aero-dynamic force on the airfoil. Lift is defined as the component of this force perpen-dicular to the direction of the oncoming flow, as shown in Figure (2). The airfoilexerts a downward force on the air when it flows past. According to Newton’s thirdlaw, the air exerts an equal but opposite force on the airfoil, the lift. The directionof the air flow changes as it passes over the airfoil and curves downward. Thischange of direction in the air results in a reaction force opposite to the directionalchange.

Reynolds numbers for insects ranges between approximately 10 to 105 [3]. Elling-ton [4] observed, in a comprehensive investigation, that classical, steady state the-ories predicted insufficient forces in flapping flight, to explain the characteristicsof insect and bird flight [5]. It has been shown in experiments that the reducedfrequency increases as the mass and size of the flying animal decreases, which in-dicates that unsteady effects are important in lift and thrust generation for smallflyers [5].

Figure 2: Directions of forces. Lift L, drag D, total aerodynamic force R and angleof attack α

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2.2.2 Drag

Drag is a mechanical force generated by the contact and interaction of a solid bodyand a fluid. It is a non-conservative force which depends on the speed of the bodyrelative to the speed of the fluid. If the relative speed is zero, there is no drag. Thedrag force is the component of the aerodynamic force that is opposed to the motionof the object, as shown in Figure (2). It is a type of friction and refers to forcesacting in the opposite direction of a body’s motion through a fluid. A drag forcecan exist between a body and a fluid or between two fluid layers. It acts to reducethe relative speed between the body and the fluid [6].

2.2.3 Thrust

Generally, bird and insect flight can be separated into two types; unpowered (glidingand soaring) and powered (flapping wings). The thrust is the component of thetotal aerodynamic force parallel to the direction of the flying animal and dependson the power output of its flight muscles. Consider an airfoil with a sinusoidalplunging motion, with no pitching, flying forward through a stationary fluid. Duringthe motion of the airfoil, the effective angle of attack changes, as can be seen inFigure (3). Positive lift is generated during the downstroke since the airfoil isexposed to a flow with positive angle of attack, and vice versa on the upstroke.

Figure 3: The effective angle of attack varies during the plunging cycle [1].

Cross-sections of the wings further from the body will move faster than partscloser to the body during upstrokes and downstrokes. Consequently, the flow di-rection hitting the wing during forward flight will vary along the wing span. Birdwings are flexible and can twist to make sure the angle of attack is correct alongthe entire wing span. Since the vertical speed of the wings is highest at the tips,they are angled more forward during down-stroke than the inner parts of the wing.This lets the bird generate a forward propulsive force without any loss of altitude[7]. Birds and insects can pitch their wings to get the optimal angle of attack. Liftand thrust is generated during the down-stroke. During the upstroke, the wing ispitched so that the effective angle of attack is zero, which results in the smallestdrag force possible and as little negative lift as possible. Birds often partially foldtheir wings during the upstroke to reduce the surface area, which reduces the drageven more [7].

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2.2.4 Leading edge vortices (LEVs)

When the angle of attack or speed of an airfoil is changed, a corresponding amountof vorticity is deposited in the wake. It takes time for the bound vortex to reachits steady state strength when an airfoil is accelerated quickly [1]. The LEV istrapped by the airflow and remains trapped to the upper surface of the wing forseveral chord-lengths of forward flight, shown in Figure (4). When air flows aroundthe leading edge, it flows over the trapped vortex and is pulled in by the lowerpressure generated by the vortex, which in turn generates lift. This mechanism wasfirst discovered by Ellington et al. [4], when they studied the mechanics of forwardflight in bumblebees. The lift enhancing LEV is a main feature during the plungingmotion of the stroke.

Figure 4: Leading edge vortex formation in flapping flight.

2.2.5 Dynamic stall

The rapid change in angle of attack when the airfoil switches stroke direction shedsa strong vortex from the leading edge. This vortex then moves along the surfaceof the airfoil, causing a reduction in pressure and increase in lift. However, whenthe vortex passes the trailing edge, the lift is dramatically reduced. This non-linearunsteady aerodynamic effect is called dynamic stall [1].

2.3 Wing model

2.3.1 Equations for wing motions in 2D

Free flight wing kinematics measurements of many insects using high speed videoshowed that the translational velocity and pitching of the wings varies approxi-mately as simple harmonic functions [8]. The airfoil’s instantaneous location and

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incidence can be uniquely defined using its translational and rotational coordi-nates [1] [5],

h(t) = ha sin(2πft+ φ) (4)

α(t) = α0 + αa sin(2πft), (5)

where h(t) is the instantaneous plunging amplitude, ha is the maximum plungingamplitude, normalized by the chord length, f is the plunging frequency, α(t) is theinstantaneous angle of attack, α0 is the initial pitching angle, αa is the pitchingamplitude and φ is the phase difference between plunging and pitching motion.

Figure 5: Snapshots of wing profile motion during the upstroke for advanced, syn-chronized and delayed rotation.

The phase difference between the pitching and plunging motion has a big impacton the generation of thrust and lift. It was found by Kramer et al. that an advancedwing rotation produces a mean thrust coefficient that is 72% higher than the meanthrust coefficient of an airfoil with delayed rotation [1]. A symmetric rotation, usedin the simulations in this report, produced a 65% higher mean thrust coefficientcompared to the delayed rotation.

2.4 Power considerations

Most birds and other flying animals have well developed flight muscles to be able togenerate the power required for flight. The strength of these muscles imposes limitson the flight modes of the animal. In birds, the flight muscles constitute around17% of the total weight of the animals [1]. The maximum flapping frequency isa physical limitation for flapping animals and birds heavier than around 12 to 15kg are not able to maintain a flapping frequency high enough to sustain horizontalpowered flight [1].

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3 Numerical simulations

The symmetric NACA 0012 airfoil, undergoing plunging and pitching motion, isstudied numerically for different values of plunging amplitude h0, reduced fre-quency k, pitching amplitude αa and oncoming freestream angle β.

3.1 Governing equations and turbulence model

The fluid flow around an insect wing is adequately described by the incompress-ible two-dimensional Navier-Stokes equation (without gravity) together with thecontinuity equation for incompressible flow [3].

∂u

∂t+ u · ∇u = −1

ρ∇p+ ν∇2u, (6)

∇ · u = 0. (7)

ANSYS Fluent (version 15.0) is used to solve these non-linear equations numer-ically using the Spalart-Allmaras turbulence model, which is based on the ReynoldsAveraged Navier-Stokes (RANS) model available in Fluent. It models the extradissipation produced by the turbulent fluctuations and was used in all the simu-lations. It is a one equation model that solves a transport equation for kinematiceddy (turbulent) viscosity without calculating the length scale related to the shearlayer thickness [9].

3.2 Geometry and mesh parameters

The airfoil geometry is modelled in ANSYS DesignModeler using coordinates down-loaded from the on-line database Airfoil Tools1. It is centered in a circular domainwith a diameter of d = 20c, where c = 1 is the chord length. The airfoil is translatedso that the origin is located a distance c/4 to the right of the leading edge. Theairfoil is then rotated to give it a positive mean angle of attack α0 = 5◦. The origin’slocation inside the airfoil will be the center of rotation for the pitching motion.

1http://airfoiltools.com/airfoil/details?airfoil=n0012-il, Retrieved 2015-03-29

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Figure 6: Simulation geometry

The simulation grid is created using the meshing software included in ANSYSWorkbench. The grid density is higher near the airfoil to capture more detail there.

Figure 7: Simulation grid with a mean angle of attack α0 = 5◦.

3.2.1 Grid characteristics

Mesh Size

Level Cells Faces Nodes Partitions

0 24736 37380 12644 1

1 cell zone, 4 face zones.

Domain Extents:

x-coordinate: min (m) = -1.000000e+01, max (m) = 1.000000e+01

y-coordinate: min (m) = -1.000000e+01, max (m) = 1.000000e+01

Volume statistics:

minimum volume (m3): 2.224370e-06

maximum volume (m3): 4.570552e-02

total volume (m3): 3.140571e+02

Face area statistics:

minimum face area (m2): 1.351579e-03

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maximum face area (m2): 3.907141e-01

3.2.2 Validation

A validation is performed on the simulation mesh to verify that is produces accurateresults. Figure (8) is a plot of the lift and drag coefficients as functions of the angleof attack for a stationary NACA 0012 airfoil.

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Angle of attack α [degrees]

Coe

ffici

ent v

alue

s [ −

]

c

L

cd

Figure 8: Plot of lift and drag coefficients versus angle of attack.

In Figure (8) it can be seen that around α ≈ 20◦ the airfoil reaches the stallangle where the flow separates from the top surface of the airfoil, which results inreduced lift. The drag is increasing even when the stall angle has been passed. Thisis the expected behaviour of the lift and drag coefficients [10].

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3.3 Simulation model

L

U

(t)

D

Upstroke motion

y^

x^z^

Figure 9: Definition of angles and coordinate system used in the simulation.

The free stream angle β is defined as in Figure (9), positive when the free streamis hitting the airfoil from below.

In this simulation Re = 3000 will be used, the value in forward flight of bum-blebees (Bombus terrestris) [1]. Dynamic scaling, conserving the dimensionlessparameters Re and k, ensures that the underlying fluid phenomena are conserved[3]. Since the simulation depends only on the Reynolds number and the reducedfrequency, the airfoil chord is scaled to be unit length, c = 1, and the freestream ve-locity is set to U = 1. The viscosity is calculated using the formula for the Reynoldsnumber, equation (1).

ν =Uc

Re=

1

3000. (8)

Bumblebees have a flap frequency of around 150 Hz during forward flight, meanchord length c ≈ 0.002 m and average forward flight speed of about 3 m/s [1][4].Inserting these parameters into equation (2) results in a reduced frequency k ≈ 0.3.Substituting the reduced frequency k from equation (2) into the equations for plungeheight, (4), and angle of rotation, (5), yields

h(t) = ha sin(2kt+ φ) (9)

α(t) = α0 + αa sin(2kt). (10)

The phase difference φ = 90◦ is chosen to be constant. Although changes in φresults in variation in performance, it has been found that φ = 90◦ is a good choicefor high thrust and efficiency [11]. The airfoil’s position and pitch during upstrokeis shown in Figure (10).

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Figure 10: Plunging and pitching with phase difference φ = 90◦ and α0 = 0◦. Thefigure shows snapshots of the airfoil during upstroke. For this φ and α0, the airfoilis horizontal in the min and max positions.

The thrust coefficient CT is the negative part of the drag coefficient CD, whichis calculated in Fluent by integrating the total pressure and viscous forces alongthe boundary of the airfoil. These coefficients are convenient for comparing theefficiency of different parameters for the oscillating airfoil.

Φ

L

Figure 11: Front view of flapping wing with stroke amplitude Φ and length L.The maximum plunge amplitude is reached when the wing has a stroke amplitudeΦmax = π.

The amplitude parameter span in the simulation is calculated using the strokeamplitude for bumblebees, which is Φ = 2.1 rad [1]. The following relations can beseen in Figure (11),

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{href = L sin

(Φ2

)hmax = L.

(11)

which gives a maximum amplitude increase

hmax

href=

1

sin(

Φ2

) ≈ 1.15. (12)

The amplitude span used in the simulation is therefore 0.7 ≤ ha/href ≤ 1.15. Theoncoming wind angle parameter is confined to the interval −20◦ < β < 20◦ sincethis is approximately the stall angle found in Figure (8).

3.3.1 Boundary conditions

The boundary conditions for the different parts of the domain are as follows; Theleft side of the circular outer boundary is the inlet where the inlet velocity is definedwith magnitude and direction. The direction of the inlet flow is the wind directionparameter. The right side of the circular boundary is set to pressure-outlet. Thereis a no-slip condition on the wing profile.

3.3.2 Calculation of power usage

The instantaneous power exerted by the fluid on the wing profile during steadyforward flapping flight is

Pfluid = F · v + MA · ω, (13)

where F = FLift + FDrag is the total aerodynamic force, v is the velocity of theairfoil, Ma is the moment with respect to the center of rotation of the airfoil and ωis the angular velocity of the airfoil. The power exerted by the insect on the fluidis obtained by introducing a negative sign,

Pinsect = −Pfluid. (14)

The moment is calculated in Fluent using a moment coefficient monitor whichgives the moment with respect to the origin for each time step. Since the momentneeded in equation (13) is with respect to the center of rotation of the airfoil, themoment is moved using the formula

MA = MO + RAO × F, (15)

where O is the origin, the center of the domain, and RAO is the vector from theorigin to the center of the airfoil. The simulation starts with the airfoil in its topposition which means that the position of the airfoil is given by

RAO = −ha[1− sin

(2kt+

π

2

)]y. (16)

The velocity v and angular velocity ω are given by the time derivatives ofequations (9) and (10).

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v = 2kha sin(

2kt+π

2

)y, (17)

ω = 2kαa sin(2kt)z. (18)

The lift force, drag force and moment are defined as

FL = qACL(− sinβx+ cosβy), (19)

FD = qACD(cosβx+ sinβy), (20)

MO = qAc · CM z, (21)

where β is the freestream angle, q = 12ρU

2 is the dynamic pressure, A is the planformarea of the wing, c is the chord length and CM is the moment coefficient.

3.3.3 Moving airfoil setup

The motion of the airfoil is defined using a User Defined Function (UDF) in aseparate file, which can be found in appendix A. However, since Fluent uses theUDF to set the translational and angular velocity of a boundary, the time derivativesof equations (9) and (10) are used.

3.4 Results

3.4.1 Reference simulation

The reference simulation is done with freestream velocity coming in with zero an-gle, which means that the insect is flying in direct headwind. The wind velocityincluding the wind gust is normalized to 1 in the simulation.

Parameter Value

k 0.3 [rad/s]Re 3000 [ - ]ha 1 [-]αa 15 [◦]α0 5 [◦]β 0 [◦]St 0.19 [ - ]

Table 1: Parameters used in the reference simulation. β is the oncoming freestreamangle. A timestep dt = 0.1 s is used. These parameters generate positive thrust andlift.

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Figure 12: Plot of velocity magnitude for the reference simulation which shows aperiodic shedding of vortices.

The simulation is run for 10 periods to remove transients from the initial values.The instantaneous lift and thrust coefficients during the tenth period is shown inFigure (13).

0 0.2 0.4 0.6 0.8 10.4

0.2

0

0.2

0.4

0.6

Time (t/T)

CT

Thrust coefficient

0 0.2 0.4 0.6 0.8 12

1

0

1

2

3

Time (t/T)

CL

Lift coefficient

Figure 13: Thrust and lift coefficients over one period, T . The airfoil starts at thetop position and begins the downstroke at t/T = 0. Thrust and lift generation ishigher during the downstroke than during the upstroke.

Parameter Value

CT 0.125

CL 0.572

Table 2: Lift and thrust coefficients averaged over four periods. The values indicatethat the airfoil produces positive lift and thrust during a reference case flap cycle.

It is informative to look at the instantaneous power during a flap cycle fordifferent β. There are physical limitations to how much power an insect can producein its muscles.

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0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

Time t/T

Rel

ativ

e po

wer

P(t

/T)

/ Pre

fMax

Reference caseAirfoil position

Figure 14: Power usage during one flapping cycle with period T in the reference case,normalized with the maximum instantaneous power. The dashed red line representsthe position of the airfoil and shows that the airfoil starts off from its topmostposition at t/T = 0. The position is not to scale in this plot.

Figure (14) indicates that the insect needs to spend more energy during thedownstroke, where the effective angle of attack of the airfoil is higher. The angleof attack is lower during the upstroke and the insect needs less energy to push thewing through the air.

3.4.2 Varying the reduced frequency, k

In this section, the effect on mean lift and thrust coefficients resulting from varyingthe reduced frequency is studied when keeping amplitude and pitch constant. Thelift and drag coefficients are calculated in the directions of the lift and drag forcesas defined in Figure (9) and CT = −CD.

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0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Reduced frequency k [rad/s]

CT [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 15: Thrust coefficients for different β and reduced frequencies k.

It can be seen in Figure (15) that the thrust generally increases with increasingk for β ≤ 0◦. However, for β > 0◦ the thrust coefficient appears to have a minimumaround k = 0.4 rad.

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Reduced frequency k [rad/s]

CL [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 16: Lift coefficient as a function of the reduced frequency k for different β.

As expected, Figure (16) shows that the lift coefficient depends strongly on theoncoming freestream angle since CL increases for larger β. The mean angle ofattack α0 = 5◦ results in a positive lift coefficient for β = 0◦. With β ≥ 0, CL

generally increases. However, for β = 20◦, the lift coefficient reaches a maximumat k = 0.4 rad.

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0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

4

5

6

Time t/T

Rel

ativ

e po

wer

P(t

/T)

/ Pre

fMax

k = 0.2k = 0.3 (Reference case)k = 0.4k = 0.5k = 0.6Airfoil position

Figure 17: Instantaneous power needed to push the airfoil through the air duringone flapping cycle. The airfoil position curve is only for visualization purposes andis not to scale.

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Reduced frequency k [rad]

Rel

ativ

e po

wer

P(t

/T)

/ Pre

fMax

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 18: Maximum power usage as a function of k. The plot shows the max-imum instantaneous power needed during one flap cycle. It is a measure of howmuch power the insect maximally needs to exert during a flap cycle compared tothe reference case. These peaks occur during the downstrokes, which can be seen inFigure (17).

It is clear in Figure (18) that the maximum power the insect needs to exertduring a flapping cycle increases rapidly with increasing frequency. If the insectwere to double its flapping frequency when facing direct headwind (β = 0◦), it

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would require a 5 times increase in maximum power during the downstroke. Sincethe insect will not be able to exert the necessary extra power for extended periodsof time, the results indicate that the frequency change must be small.

3.4.3 Varying the plunging amplitude, ha

In this set of simulations the plunging amplitude ha is varied while keeping thereduced frequency and pitching amplitude constant.

0.7 0.8 0.9 1 1.1

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Plunging amplitude ha/h

ref [ − ]

CT [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 19: Thrust coefficient as a function of plunging amplitude for different β.

The thrust coefficient is strongly influenced by the freestream angle and variesapproximately linearly with the plunging amplitude, as can be seen in Figure (19).|CT | increases with increasing plunging amplitude. However, positive average thrustis only generated for β = −10◦ and β = 0◦.

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0.7 0.8 0.9 1 1.1

−1.5

−1

−0.5

0

0.5

1

1.5

2

Plunging amplitude ha/h

ref [ − ]

CL [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 20: Lift coefficient as function of plunging amplitude for different β.

Figure (20) shows that there is an approximately linear relation between theplunging amplitude and CL. Wind coming from above, corresponding to negativeβ, pushes the airfoil downwards, reducing the lift. The positive lift for β = 0◦ is aresult of the positive mean angle of attack α0.

0.7 0.8 0.9 1 1.1

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Plunging amplitude ha/h

ref [ − ]

Rel

ativ

e po

wer

P(t

/T)

/ Pre

fMax

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 21: Power exerted by the insect as a function of plunging amplitude.

The power usage in Figure (21) increases with increasing amplitude. The slopeof the curves increase slightly with increasing β.

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3.4.4 Varying the pitching amplitude, αa

In this section the effect of the pitch amplitude αa on lift, thrust and power isstudied while other parameters are kept constant.

10 12 14 16 18 20

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Pitching amplitude αa [degrees]

CT [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 22: Thrust coefficient as a function of pitching amplitude for different β.

Although the changes in thrust coefficient are small when varying the pitchingamplitude, shown in Figure (22), a general trend is that the magnitude of thethrust force decreases with increasing αa. An increased pitching amplitude resultsin a lower effective angle of attack on both upstroke and downstroke.

10 12 14 16 18 20

−1.5

−1

−0.5

0

0.5

1

1.5

2

Pitching amplitude αa [degrees]

CL [

− ]

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 23: Lift coefficient as a function of pitch amplitude for different β.

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As for the thrust coefficient, an increase of pitching amplitude leads to a decreasein |CL|, as shown in Figure (23).

10 12 14 16 18 20

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pitching amplitude αa [degrees]

Rel

ativ

e po

wer

P(t

/T)

/ Pre

fMax

β = 20o

β = 10o

β = 0o

β = −10o

β = −20o

Reference

Figure 24: Power exerted by the insect as a function of pitching amplitude.

The effects of varying the pitching amplitude can be seen in Figure (24). When thepitching amplitude is increased, the maximum power during a flap cycle tends todecrease slightly for negative β and increase slightly for positive β.

3.5 Conclusion

According to the results of the simulations, the ideal situation for the insect is tofly in a horizontal freestream, β = 0◦, since CT and CL always are positive whenvarying the parameters. If this is not the case, the following qualitative rules canbe used to adjust the reduced frequency, plunge amplitude and pitch amplitude tocompensate for the oncoming wind gust while keeping the maximum power neededas low as possible for different β.

3.5.1 Wind from below, β > 0◦

The thrust coefficient was found to have a minimum around k = 0.4 for windgusts coming from below. It was also found that CL increases with increasing k.However, increasing k leads to a rapid increase in the maximum power usage duringthe downstroke. Since the wind comes from below, it pushes the insect upwards,which is a big contribution to the lift force. Therefore, if k is decreased, the thrustis increased and lift is reduced. Reducing k will reduce the maximum power usageof the insect.

An increased plunging amplitude was found to decrease the thrust, increasethe lift and increase the power usage. This indicates that the plunging amplitudeshould be decreased as well, to counteract the extra lift from the wind coming frombelow, generate extra thrust and further reduce the power usage.

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The thrust increases, the lift decreases and the power usage is marginally in-creased when the pitching amplitude is increased. It is therefore a good choice toincrease the pitching amplitude.

3.5.2 Wind from above, β < 0◦

Wind coming from above pushes the insect downwards. To counteract this, the liftand thrust force should be increased to improve the manoeuvrability of the flyer.This could be accomplished by increasing the frequency and increasing the pitchingamplitude. The increase in k will result in an increase in CT , a small decrease inCL and an increase in power usage.

For β ≈ −10◦, an increase in plunging amplitude and decrease in pitchingamplitude will result in a marginally increased thrust, unchanged lift and increasedpower usage.

When β ≈ −20◦, a decrease in plunging amplitude and increase in pitchingamplitude will result in a marginally increased thrust and lift.

3.5.3 Final remarks

There are more ways for a flyer, natural or artificial, to compensate for incomingwind gusts than was studied in this report. The flyer can for example turn in the airand change its direction to counteract the wind gust, instead of being locked in onedirection, as is assumed in the simulations. This would be particularly helpful whenthe wind comes from above, since changing the flapping parameters only marginallyincreases the thrust and lift.

The results may differ for finite 3D wings where 3D effects are accounted for,such as tip vortices and the clap-and-fling mechanism [1].

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References

[1] Wei Shyy, Hikaru Aono, Chang kwon Kang, and Hao Liu. An Introduction toFlapping Wing Aerodynamics. Cambridge University Press (CUP), 2013.

[2] Graham K. Taylor, Robert L. Nudds, and Adrian L. R. Thomas. Flying andswimming animals cruise at a strouhal number tuned for high power efficiency.Nature, 425(6959):707–711, oct 2003.

[3] S. P. Sane. The aerodynamics of insect flight. Journal of Experimental Biology,206(23):4191–4208, dec 2003.

[4] R. Dudley and C. P. Ellington. Mechanics of forward flight in bumblebees.Journal of Experimental Biology, 148:19–52, jun 1990.

[5] J. Tang, D. Viieru, and W. Shyy. Effects of reynolds number and flappingkinematics on hovering aerodynamics. AIAA Journal, 46(4):967–976, apr 2008.

[6] Steven Vogel. Life in Moving Fluids. Princeton University Press, 1994.

[7] Joel Guerrero. Numerical simulation of the unsteady aerodynamics of flappingflight. Unpublished doctoral thesis, University of Genoa, 2009.

[8] Y. Sudhakar and S. Vengadesan. Flight force production by flapping insectwings in inclined stroke plane kinematics. Computers & Fluids, 39(4):683–695,apr 2010.

[9] Nadeem Akbar Najar, D. Dandotiya, and Farooq Ahmad Najar. Comparativeanalysis of k-ε and spalart-allmaras turbulence models for compressible flowthrough a convergent-divergent nozzle. The International Journal Of Engi-neering And Science (IJES), 2:8–17, 2013.

[10] NASA turbulence modeling resource. http://turbmodels.larc.nasa.gov/

naca0012_val.html. Accessed: 2015-04-27.

[11] F.S. Hover, Ø. Haugsdal, and M.S. Triantafyllou. Effect of angle of attackprofiles in flapping foil propulsion. Journal of Fluids and Structures, 19(1):37–47, jan 2004.

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4 Appendix A

4.1 User Defined Functions

Below is the UDF c code file which defines the vertical translational velocity and theangular velocity of the airfoil. The parameters are passed in from Fluent, wherethey are defined for each simulation case using a journal file. The journal file,generated by a Python script, contains all text commands needed to set up and runall simulations automatically.

#include "udf.h"

DEFINE_CG_MOTION(profilemotion, dt, vel, omega, time, dtime)

{

real t = time;

real k = RP_Get_Real("frequency");

real ampl = RP_Get_Real("amplitude");

real pitch = RP_Get_Real("pitchrange")*3.141592/180.0;

vel[1] = ampl*2*k*cos(2*k*t+3.141592/2);

omega[2] = pitch*2*k*cos(2*k*t);

}

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