Lecture 14

Mechanisms and Efficiency of Wing Flapping

AA200B

Lecture 14

November 29, 2007

AA200B - Applied Aerodynamics II 1

AA200B - Applied Aerodynamics II Lecture 14

Introduction

Interest in the aerodynamics of flapping flight has been rekindled withconsideration of micro-air vehicles, autonomous underwater vehicles, andrecent experiments with insect models. Much of the previous and currentresearch in this area is empirical due to the complexity of the relevant flows,although some mechanisms have been identified and postulated as beingimportant to flapping performance. In these notes, we examine some ofthe basic mechanisms for efficient flapping flight with analysis suitable fordesign. The analysis starts simply with a quasi-steady look at flapping inforward flight.

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Introduction: Flapping Propulsion – Myth and Reality

An analysis of bumblebee flight in 1934 [3], concluded that according to“the laws of the resistance of the air” applied to insects, for “them flight isimpossible.” Since that time, and likely long before, people have imaginedthat certain mysteries of flapping propulsion elude aerodynamicists. Recentyears have brought a number of articles that seek to “explain” flappingflight with complex unsteady viscous effects and vortex dynamics. To besure, the hovering of insects involves very complicated low Reynolds numberflow phenomena, but some of the basic concepts can be understood quitesimply and that is the aim of this introduction.

Those studying fish and marine mammal locomotion have their ownversion of the “bumblebees cannot fly” myth:

In 1936 the British zoologist James Gray created a stir bycalculating the power that a dolphin would need to move at 20

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knots, as some were reported to do. Gray assumed that the resistanceof the moving dolphin was the same as that of a rigid model andestimated the power that the muscles of the dolphin could deliver.His conclusion, known as Grays paradox, was that the dolphin wastoo weak, by a factor of about seven, to attain such speeds. Theinescapable implication is that there are flow mechanisms at workaround the body of the moving dolphin that lower its drag by a factorof seven. – From [9].

Similarly, the mechanism for swimming propulsion is often considereda mystery, with recent issues of Nature [1] declaring that “One numberexplains animal flight.”

One number describes the beating of animal wings and tails,researchers have found. The simple rule of thumb for animallocomotion could help to design miniature flying machines.

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”We’ve described the geometry of the wingbeat,” says GrahamTaylor of the University of Oxford, UK. Swimmers and fliers frominsects to whales all cruise at the speed that lets them slip along mosteasily, he and his colleagues show.

Wings and tails create eddies as they move. These need to beleft behind, because turbulent air or water is more difficult to travelthrough. So limbs shed vortices at the bottom of their downstrokes.Flap too quickly, and you have to fight this turbulence on the way up.Too slowly, and turbulence sticks.

This concept is described in a bit more detail in some lectures byresearchers at M.I.T.:

Exploitation of Vortical Wake Dynamics by Live FishEvidence suggests that many fish exploit the natural instability of

the flow energetics to assist them in propulsion and maneuvering. By

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tuning their own kinematics, the fish is able to swim efficiently, togenerate large thrust and turning forces, and to move silently throughthe flow with minimal wasted energy. –From [10].

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MIT Center for Ocean Engineering, 2005

Fish Swimming: Thrust WakeFish Swimming: Thrust Wake

! Non-dimensional Numbers: Strouhal Number and

Reynolds Number

! Reverse Kármán vortex street = Jet Wake

! Combined body and tail motion = unsteady flow control

JET

MIT Center for Ocean Engineering, 2005

Cylinder wake: KCylinder wake: Káármrmáán vortex street, induced jet flow n vortex street, induced jet flow towards the body, causing drag forcetowards the body, causing drag force

Fish wake: reverse KFish wake: reverse Káármrmáán vortex street, induced jet n vortex street, induced jet flow away from the body, causing thrust forceflow away from the body, causing thrust force

Control volume

Control volume

Figure 1. Vorticity shedding from cylinder and fish (from Ref. 10)

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This “natural instability of the flow energetics” may be described by theunsteady Navier Stokes equations, but this hardly helps to understand themechanisms.

An article in Scientific American [9] describes this idea in more detail:

Any object in a flow, whether it is a wire in the wind or a swimmingswordfish, creates a trail of spinning vortices. The wire obstructs theflow and leaves a wake, whereas the tail of a fish pushes waterbackward, establishing what is more properly known as a jet–a columnof moving fluid that includes thrust-producing vortices. We becameconvinced that these jet vortices play the central role in the generationof thrust, and we argued that their optimal formation would increaseefficiency tremendously.

From previous studies we had done on the vortices produced by awire in a stream of air, we were well acquainted with a fluid-dynamic

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parameter known as the Strouhal number. It is the product of thefrequency of vortex formation behind an object in a flow and the widthof the wake, divided by the speed of the flow. What the numberindicates, compactly, is how often vortices are created in the wake andhow close they are. Interestingly, the ratio remains constant at about0.2 for a variety of flow conditions and object shapes.

Although the Strouhal number was invented to describe the wakesbehind flow obstructions, the similarities between wakes and jets aresuch that we realized we could use the number to describe jets. Fora swimming fish, we defined the Strouhal number as the product ofthe frequency of tail swishing and the width of the jet, divided by thespeed of the fish.

By analyzing data from flapping foils, we found that thrust-inducing vortices form optimally when the Strouhal number liesbetween 0.25 and 0.35. We anticipated that efficiency should beat a maximum for these values. Some preliminary experiments at the

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M.I.T. testing tank confirmed that the efficiency of a flapping foildoes indeed peak when the Strouhal number is in this range.

There is nothing wrong with this approach to understanding flappingpropulsion, but it is a kind of far-field analysis of a phenomenon that mayalso be understood in the near field. In the following sections we attemptto describe flapping propulsion in this way, relying on the more conventionalapproaches to aircraft performance analysis.

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Basic Mechanism

How flapping is just like gust soaring In a previous chapter, we consideredhow animals might extract energy from gusts and propel themselves byvarying the lift of their wings in phase with the ambient air motion. Figure2 is the picture we used to understand how thrust might be generated fromvertical gusts. If, however, we think of figure 2 as a top view of a fish tailand the “gusts” are caused by the tail motion, the situation is completelyequivalent.

Figure 2. Forces on flapping wing (e.g. fish tail viewed from above)

If a 2D wing is moved through a fluid with a velocity of v and the system

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is moving forward at speed U, it generates a forward thrust given by:

T =Lv

U

If the lift and lateral velocity are properly phased, the average thrust isgiven by:

T =L0v0

2U

This is the most basic (2D, quasi-steady) way to think of thrustproduction due to wing flapping. Many effects are left out here andcan be added in increasingly complex expressions for the average thrust.

The added lift on the wing also generates some unsteady drag, but forlow frequencies the majority of the extra drag is due to 3D quasi-steady

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effects. If we include these effects:

T =L0v0

2U− L2

0

2qπb2

The average power required to move the wing is given by:

P =L0v0

2

so the net efficiency is:

η = 1− UL0

qv0πb2= 1− 4T

ρv20πb2

Which suggests that for a given thrust, we should move the wing as quicklyas possible.

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However, if the wing is not articulated, moving the wing very quicklywill produce an angle of attack that is too large and the wing will stall. Ifthe wing incidence can be changed by, say 30 deg, and the maximum angleof attack is about 15 deg then the maximum value for v0 is equal to U(since the total angle is 30 + 15 = 45 deg).

If the motion is generated by oscillating the aft part of the vehicle withan amplitude, a, the Strouhal number based on overall tail amplitude (2a)is:

St =2af

U=

2aω

2πU=

v0

πUSo, if v0 = U , the Strouhal number is 0.318. If v0 is restricted by stalling:

St =1π

tan (αmax + ∆i)

With an incidence change restricted to 20 deg, the Strouhal number is0.21. These are just in the range suggested for efficient swimming. (No

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mysterious unsteady vortex interaction is needed to explain this, it’s justkinematics.)

The Strouhal number is just one dimensionless frequency measure.Another is the classical reduced frequency, k, which is defined as: k =ωc2U = π c

aSt. For a wing with a motion amplitude of 5 times the chordlength at a Strouhal number of 0.3, the reduced frequency is about 0.2,meaning that unsteady effects are not insignificant, but that the quasi-steadyapproximation should be quite good for initial design purposes.

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Nonlinear Effects

Of course, with v0 ≈ U we need to include viscous drag incrementsassociated with higher dynamic pressure. When this is done, some algebraconfirms that the maximum efficiency occurs at high L/D and at v0 ≈ U .Thus, the Strouhal number may be limited to values less than 1

π = .318due to kinematics, but otherwise should approach this value. (As an

approximation η = CL1−ε/v1+εv with ε = CD/CL. Even with low L/D values,

the optimal v0/U is only somewhat larger, confirming the significance ofthe 0.3 Strouhal number using simple performance arguments.)

To see how this result comes about, consider the thrust produced byflapping:

Fx = L sin γ −D cos γ =Lv√

v2 + U2− DU√

v2 + U2

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.

= 0.5ρCLv√

v2 + U2 − 0.5ρCDU√

v2 + U2

.

Cx =√

v2 + 1 (CLv − CD)

.

Cz =√

v2 + 1 (CL + vCD)

. The efficiency of the system is the average value of Cx (with the zero liftdrag coefficient removed if the wing must be there for other reasons) dividedby the average value of Czv. This is easily computed in a spreadsheet withresults shown in figure 3. Additional nonlinear effects not considered in theabove include nonplanar wake motion and non-quadratic profile drag polars,which could be important for certain designs.

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Efficiency vs. v/U

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.5 1 1.5 2

v/U

Eff

icie

ncy

Figure 3. Efficiency of an oscillating wing based on quasi-steady aerodynamicsbut with nonlinear effects of viscous drag. CD0 = 0.01, CL0 = 1, AR = 5

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Unsteady Effects

In 1936 I. E. Garrick published NACA Report 567, ”Propulsion of aFlapping and Oscillating Airfoil”, in which he extended Theodorsen’s 2Dsimple harmonic results to include the horizontal force as well as the verticalforce of a pitching and plunging thin airfoil. He did this in two ways (andshowed that the answers are the same): by using thin airfoil theory toestimate the leading edge suction force and by computing the energy leftin the wake due to the shed vorticity. The details are provided in Garrick’spaper (posted on the class website), but may be summarized as follows.

If an airfoil is pitching, θ(t), about an axis located at ac/2 behind themid-chord and moving vertically with the displacement of the rotation axisgiven by h(t), the we can write:θ(t) = θ0e

iωt+φ0

h(t) = h0eiωt+φ2

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From Theodorsen (with h positive downward and Fz positive upward):

Fz = ρb2(πUθ + πh− πbaθ) + 2πρUbC(k)Q

M = −ρb2

[πU(

12− a)bθ + πb2(

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+ a2)θ − πabh

]with:

Q = Uθ + h + b(12− a)θ

From Garrick, assuming small angles:

Fx = πρbS2 + θFz

with:

S =√

2C(k)Q−√

12

bθ

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The work done to maintain the pitching and plunging is:

W = Fzh + Mθ

So the efficiency over one cycle is:

η =FxU

¯Fzh + M θ

In the simple case of pure plunging motion (θ = 0, Q = h), thisbecomes:

η =πρbS2U

¯Fzh

=πρb2 ¯(C(k)Q)2U

¯2πρUbC(k)Qh=

F 2 + G2

F

Here, F and G are the real and imaginary components of C(k).

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The result for this case is plotted below. The case for pitching andplunging depends on the relative phases and amplitudes and is left as anexercise.

7 PROPULSION OF A FLAPPING AND OSCILLATING AIRFOIL

Finally from (13) the average propulsive force is - Px=~PbP2[a1h02+ (az+bz)d+ (a3+c3> Po2 + 2 (a4 + b4) q h O -k2 + 6.5) POh,

$2 (a6 + bf3 %POI (29)

In order that equations (24) and (29) agree we must have that

A ~ = C Z ~

+ bz

&=as+ c5

.&=a6 + be +e6

Each of these relations may be reduced to an identity, e. g., consider AI and al. From (15)) (23)) and (26)

a1 = F2+ G2

In order that Al=al the following relation must hold

To show that this is true note that

F2+G2= (F+iG) (F-iG)=- Ji2+ Yi2 D

(cf. reference 5, p. 8) and from a well-known property of the Bessel functions,

Hence equation (31) follows. By the use of the relation (31) and the definitions of

the various T's given in the appendix, it can be veri- fied that the remaining relations in (30) are also iden- tities.

It msy be of interest to consider the special cases of one degree of freedom. Let the motion of the wing consist only of the vertical motion 6 at right angles to the direction of flight, i. e., flapping motion. The propelling force is then

(32) Fx= rpbp'ht ( F2 + G2)

Consider in this case the ratio

P,v-energy of propulsion w total energy --

F2+G2 F

=--- (33)

This function, shown in figure 3, represents the theo- retical efficiency of the flapping motion (unity= 100 per-

0 This result agrees with the formula of von KBrmln and Burgers (reference 2, p. 306). The expressions of reference 2 denoted by

bl=l+Ai-A AI ( Q - S ) + A A2 (P-C) ba=Az--X A1 ( Q - 8 - A AI (P-C)

reduce in our notation simply to 2F and ZQ, respectively.

sent). It is observed that a propelling force exists in the entire range of l/k, the egciency being 50 percent Cor infinitely rapid oscillations and 100 percent for Lnfinitely slow flapping.

.80

.60

Pz V

T O

.zo ' ' " I ' * ' ' * " ' * s

0 2 4 6 8 f0 fZ I4 f 6 f 8 20 l/k

FIGURE 3.-The ratio-of energy of propulsion to the energy required to maintain the

P Z V oscillations (5) as a function of l /k for the case of pure flapping.

For the special case of angular oscillations about a alone (h=O, p=O) the horizontal force is

- p, = Tpbp2b2a02{ (P + G2) [; + (a - a ) ]

+;(; - a ) -;- (+ a);] (34)

FIGURE 4.-The ratio of the energy dissipated in the wake to the energy required to

maintain the oscillations (a%) as a function of l /k for the case of pure angular

oscillations about x=a.

- _ In figure 4 there is shown the ratio E/W, in this case for several positions of the axis of rotation. These curves give the ratio of the energy per unit time released in the wake to the work per unit time required to maintain the oscillations. In the range of values O<E/V<l, 'P, is positive and denotes a thrust or propelling force; for other values it is negative and denotes a drag force.

LAN GLEY MEMORIAL AERONAUTICAL LABORATORY, NATIONAL ADVISORY COMMITTEE E OR AERONAUTICS,

LANGLEY FIELD, VA., May 4, 1936.

Note that even for relatively low frequency motion (k = 0.2) substantialdecreases in efficiency (> 20%) appear.

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3D Considerations

Additional considerations apply in 3D. For a useful discussion of someof these including the effects of large amplitude motion, see Ref. [11].Even for small motions the 3-dimensionality of the flow makes 2D results oflimited use. Here we mention two fundamental issues.

The first is that, just as in steady flow, the wing sheds vorticity thattrails downstream, induces downwash on the wing, and creates drag. Evenwhen there is no average lift on the wing, induced drag is created by the 3Dunsteady trailing vorticity. However, in this case the quasi-steady analysisis pessimistic, since the time-varying vorticity leads to reduced downwashcompared with the constant strength sheet. As the frequency increases, theextra induced drag caused by flapping is actually reduced, while the lossesdue to transverse vorticity increase. (Note that in the linear theory, theseeffects of these vorticity components may superimposed.)

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The second, very important 3D consideration is associated with optimalloadings for flapping flight. This problem was addressed in a simple, quasi-steady analysis by R.T. Jones [13], and with a more refined study describedin Ref. [11]. To illustrate some of these ideas, consider the question ofoptimal loading in the linearized, quasi-steady case. Just as described in thediscussion of minimum induced drag, we can use the method of restrictedvariations to determine the optimal load distribution for flapping.

Consider arbitrary, small simple flapping motion where the local motionof the wing is given by wf(y), and for which the additional loading duringthe flapping cycle produces a downwash distribution of ww(y). Assume alsothat we add a small increment to the loading at a station y1 and y2. In thiscase, the change in total thrust is given by:

δT = ρwf1δΓ1 + ρwf2δΓ2 − ρww1δΓ1 − ρww2δΓ2

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The change in power required to flap the wings is:

δP = ρUδΓ1wf1 + ρUδΓ2wf2

If we hold the power constant (δP = 0) and seek the optimal loading,then δT = 0. Substituting the second expression into the first and settingit to 0 shows that for arbitrary (but small) flapping motions, the optimalloading should be that associated with a downwash distribution ww thatis proportional to the local motion due to flapping, wf . So for simpleplunging, the optimal loading is elliptic, while for rotation about a centralhinge, the optimal downwash is linear. This then would allow us to computethe lift distribution and then solve for the required wing twist.

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Initial Conclusions

The conclusion is that a well-designed flapping propulsion system maybe better or worse than a propeller in terms of efficiency, depending onthe constraints on the many parameters. The disadvantages of flappingpropulsion for fish and AUV’s include:

• Low frequencies require high gear ratios and heavy or complextransmissions

• A dorsal fin may be required to counteract the periodic sideforce

• Time-dependent incidence changes are required, but might be achievedwith passive hydroelastic tailoring

The advantages of flapping propulsion include:

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• Elimination of a rotary bearing may be useful at high pressures

• The fin area may be needed for lateral stability/control as well aspropulsion

• Small angular deflections may reduce possibility of fouling

When the flapping wing is also used for lift, as in birds and insects, thesurfaces must be located near the c.g. and cannot be translated so easily.In this case a flapping motion in which the plunging rate varies over thespan is generally used. The actual motion varies greatly from v-like motionsto much more complex motions, but may be analyzed in an analogousmanner. The paper in these notes entitled “Optimal Wing Flapping” showshow linear quasi-steady theory is applied in this case.

If we consider flapping for propulsion in forward flight, the reduced

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frequency is related to the Strouhal number by:

k =ωc

2U=

πfc

U=

fA

U

πc

A= St

πc

A

So with flapping amplitude of about πc, the two frequency parametersare equal. With Strouhal numbers in the range of 0.3, the quasi-steadyassumption may reasonable when the amplitude is larger than about 3chords, but not if it is much smaller. The paper in these notes “UnsteadyWing Flapping” addresses some of the issues in these cases.

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References

1. John Whitfield, One number explains animal flight, Science Update,Nature, 16 October 2003

2. Taylor, G. K., Nudds, R. L., Thomas, A. L. R. Flying and swimminganimals cruise at a Strouhal number tuned for high power efficiency. Nature,425, 707-711, 2003.

3. August Magnan, Le Vol Des Insects, Hermann and Cle, Paris, 1934,and Can bumblebees fly: an internet discussion.

4. Michael H. Dickinson, Fritz-Olaf Lehmann, Sanjay P. Sane, WingRotation and the Aerodynamic Basis of Insect Flight, 18 JUNE 1999 VOL284 SCIENCE, www.sciencemag.org

5. Unsteady aerodynamic forces of a flapping wing, Jiang Hao Wu and

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Mao Sun, The Journal of Experimental Biology 207, 1137-1150 Publishedby The Company of Biologists 2004 doi:10.1242/jeb.00868.

6. Magic number revealed for flying and swimming,18:00 15 October03, NewScientist.com news service

7. Michael Dickinson, Solving the Mystery of Insect Flight: Insects use acombination of aerodynamic effects to remain aloft, ScientificAmerican.com,June 17, 2001

8. Robert Dudley, BIOMECHANICS: Enhanced: UnsteadyAerodynamics, 18 JUNE 1999 VOL 284 SCIENCE.

9. Triantafyllou, Michael S., Triantafyllou, George S., An efficientswimming machine, Scientific American; Mar95, Vol. 272 Issue 3, p64.

10. Alexandra H. Techet, Fish Swimming, Lecture 1, Biologicaland Medical Engineering, Center for Ocean Engineering, Department of

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Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,MA, Spring 2005

[11] Garrick, I. E., A Review of Unsteady Aerodynamics of PotentialFlows, Applied Mechanics Review, Vol. 5, No. 3, March 1952, pp. 89-91.

[12] Hall, K.C., and Hall, S.R., “Minimum Induced Power Requirementsfor Flapping Flight,” J. Fluid Mech. (1996), vol. 323, pp. 285-315,Cambridge University Press.

[13] Jones, R. T. 1980 Wing flapping with minimum energy. Aero. J.84, 214-217.

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