48th AIAA Aerospace Sciences Meeting, 4–7 Jan 2010, Orlando, Florida

Flexible Hovering Wing Motions: Proper Orthogonal

Decomposition Analysis

T. Fitzgerald∗, M. Valdez∗, and B. Balachandran†

University of Maryland, College Park, MD, 20742, USA

A two-dimensional passive structure undergoing symmetric hovering is considered, andthe associated flow field is examined by using proper orthogonal decomposition (POD)analysis. The dominant structures and patterns observed in the fluid are hierarchicallyranked, and these structures are used to understand the peak aerodynamic efficiency re-alized for a particular choice of the wing stiffness. In ongoing work, a database of PODmode shapes is being used to characterize the flow across a range of parameters. Theoverall effort is aimed at exploiting the structures found in the POD modes and developingnonlinear phenomena based control to enhance aerodynamic performance.

I. Introduction

Flapping wings inspired by nature are of interest to micro-air vehicle design.1 However, open questionsremain on the influence of flexibility on the performance of a flapping wing. As a starting point to understandthe phenomena associated with flexible, flapping wing systems, a two-dimensional system with a discretespring element has been considered. As shown in Figure 1, a cross-section of a wing can be consideredas a profile in two dimensions. Fully coupled simulations of fluid-structure interactions are carried out byusing this two-dimensional structural system, and two different computational models are considered. Oneof them is based on the direct numerical simulation (DNS)scheme,2 and the other is based on the unsteadyvortex lattice method (UVLM).3,4 While the DNS computations provide detailed physics information at aconsiderable computational expense, the inviscid UVLM computations provide coarse information close tothe body and a good approximation to the unsteady wake.

Figure 1: Illustration of two-dimensional wing profile as a representation of an insect wing cross-section.Photo of female Villa hottentotta, Creative Commons License, accessed from http://commons.wikimedia.org/wiki/File:Villa hottentotta female.jpg

∗Graduate Research Assistant, Department of Mechanical Engineering, AIAA Student Member†Professor and Associate Chair, Department of Mechanical Engineering, 2133 Glenn L. Martin Hall, AIAA Associate FellowCopyright c© 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-1027

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Figure 2: Schematic of two-dimensional profile with discrete flexibility. (a) Rigid-link model in blue, withan aerodynamic surface in black wrapping the structure with a thickness of 10% of the chord length. (b)Same structural model with a thickness of 3% of the chord length.

In the model studied, a single measure for the chordwise deflection is used. This model is presented inFigure 2. Here, there are two rigid elements connected at point b by a torsion spring. This torsion springis assumed to be a linear function of the deflection angle α between the two rigid elements. The locationof point b is specified by the coordinates (x, y), and θ is the orientation of link B. The center of mass mi

(i ∈ A,B) of each link is located a distance ηi from the connection point b. The mass moment of inertiaof each link with respect to the hinge point b is Ii. The centerline length of the profile is l.

I.A. Description of Kinematics

To drive the system in symmetric hovering flight the quantities of (x(t), y(t), θ(t)) are prescribed in time.These quantities define the location and orientation of the top segment of the profile in Figure 2, and asa consequence, one degree of freedom is needed to describe the motions of the structural system. Thesemotions are determined by computationally investigating the fluid-structure interactions. The harmonichovering kinematics2,5 used in this study is described by

x(t) =(1− e−t/τ

) Ax

2cos(ωf t)

y(t) = 0

θ(t) = θ0 +(1− e−t/τ

)Aθ sin(ωf t+ γ)

τ = 0.8(

2πωf

).

(1)

where ωf is the forcing frequency, Ax is the length of the stroke, and γ is the phase difference between thetranslational and rotational motions. An exponential-decay factor is used to reduce the numerical transientsassociated with an impulse start. This factor was found to be beneficial in removing the startup noise in thesimulations, and this helped keep the flow fields well behaved through the initial transients. The value of τis chosen so that it takes around five periods of hovering to achieve the harmonic kinematics.

An additional set of kinematics has also been studied in order to examine how the structural responseand flow characteristics varied with respect to kinematics. The type of motions selected are based on theprescribed motion of the Dickinson Lab Robofly.6 To obtain expressions for describing these motions, a

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reconstruction of the translation speed has been parameterized from the corresponding graphs as

t = mod(t− γx, T ) (2a)

x(t) =

Bx sin(

π2ax

tT

), 0 ≤ t/T < ax

Bx, ax ≤ t/T < 12 − ax

−Bx cos(

πax

(tT − 1

2

)), 1

2 − ax ≤ t/T < 12 + ax

−Bx, else

(2b)

where mod(·, ·) is the modulo function, T is the period of the stroke, γx is a phase shift, Bx is the peaktranslational speed, and ax determines the amount of time per stroke reversal. The following relationconstrains the parameters such that the length of each stroke is Ax:

T =Ax/Bx

1/2− 2ax(1− 2/π)=

2πωf. (2c)

Next, the angular speed is constructed as

t = mod(t− γθ, T ) (2d)

θ(t) =

2Aθ

aθT

(1− cos

(2πaθ

tT

)), 0 ≤ t/T < aθ

0, aθ ≤ t/T < 12

2Aθ

aθT

[−1 + cos

(2πaθ

(tT − 1

2

))], 1

2 ≤ t/T < 12 + aθ

0, else

(2e)

where γθ is a phase shift, Aθ is the maximum rotation angle, and aθ determines the amount of the timespent during rotation near the end of stroke.

This set of kinematics provides a uniform horizontal speed and orientation for much of the stroke, andfast reversals at the end of stroke. As with the harmonic kinematics, y(t) = 0. The parameters ax andaθ have been selected from numerical fitting of the curves presented for the Robofly.6 The remainingparameters (Bx, γx, Aθ, γθ, Ax) are selected to ensure symmetric hovering and suitable scaling for the non-dimensionalization used in the numerical simulations. The curves shown in Figure 3 illustrate the alternatekinematics studied in this article.

I.B. Parameter selection

To motivate the coupled fluid-structure interaction study, the system parameters of interest are taken asthe ratio of the structure’s density to the fluid’s density, the ratio of the forcing frequency to the natural

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

Time t/T

Hor

izon

talve

loci

ty:x(t

)

−90

−45

0

45

90

Rot

atio

nal

velo

city

:θ(t)

[deg

/tim

e]

Figure 3: Translational and rotational speeds associated with the alternate kinematics used in this work.

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1/2 1/3 1/4 1/6 Rigid0

0.5

1

1.5

2

Mea

nC

L

Re = 75, Harmonic Re = 250, HarmonicRe = 75, Alternate Re = 250, Alternate

1/2 1/3 1/4 1/6 Rigid

1

1.5

2

2.5

Mea

nC

D

1/2 1/3 1/4 1/6 Rigid0

0.2

0.4

0.6

0.8

Mea

nC

L/C

D

1/2 1/3 1/4 1/6 Rigid0

0.2

0.4

0.6

0.8

1

Frequency Ratio ωf/ωn

Mea

nC

L/C

PW

(a)

(b)

(c)

(d)

1/2 1/3 1/4 1/6 Rigid

0.6

0.8

1

1.2

Mea

nC

L

Re = 75 Re = 250Re = 1000 UVLM

1/2 1/3 1/4 1/6 Rigid0.8

1

1.2

1.4

Mea

nC

D

1/2 1/3 1/4 1/6 Rigid0.5

0.6

0.7

0.8

0.9

Mea

nC

L/C

D

1/2 1/3 1/4 1/6 Rigid0.4

0.6

0.8

1

Frequency Ratio ωf/ωn

Mea

nC

L/C

PW

(e)

(f)

(g)

(h)

Figure 4: (a)–(d) Averages of aerodynamic forces obtained for harmonic kinematics and the alternatekinematics with a 10% thickness profile. (e)–(h) Averages of aerodynamic forces obtained for a broad rangeof Reynolds numbers with harmonic kinematics and a 3% thickness profile.

frequency of the structure, and the Reynolds number. While keeping ωf fixed, varying the ratio of ωf/ωn isequivalent to varying the spring stiffness constant k for fixed inertial parameters. The structural system isdescribed by

IAα(t) + kα(t) = −IAθ(t) +mAηAx(t) sin(α(t) + θ(t)) +Qα. (3)

Choosing the density ratio so that the inertial forces are close to the aerodynamic forces provides aninteresting parametric region in which to explore the effects of flexibility. The ratio of frequencies ωf/ωn canbe viewed as a non-dimensional measure of the spring stiffness where ωf/ωn = 1/2 corresponds to a verysoft spring, and ωf/ωn = 1/6 corresponds to a highly stiff spring. The Reynolds number can be viewed asproviding a measure of the dissipation in the fluid.

The thickness of the profile is varied to be 10% or 3% of the length l in the viscous DNS studies. Thisis done to assess the effects of thickness at low to moderate Reynolds numbers for comparison with themembrane thickness implemented in the UVLM studies. In these studies, as the frequency ratio is varied,it is found that the system exhibited a discernible performance peak at a particular value of ωf/ωn for theharmonic kinematics. As depicted in Figures 4(d,h), for all Reynolds numbers, both thicknesses, and thetwo different computational fluid models, there is a peak in the efficiency at ωf/ωn = 1/3. For the alternatekinematics, this peaking trend is not observable in the associated data. Since the alternate kinematics havea much higher angular speed at the end of stroke, it appears that this motion favors stiffer springs. Overall,with the alterante kinematics, the lift and drag forces are found to be higher, but the lift to drag ratiois nearly the same as that obtained for the harmonic case. However, the power associated with the quickreversal significantly penalizes the overall energy cost.

Examining the overall trends, the Reynolds number also appears to have little impact on the grossmeasures of performance, in particular, for the alternate kinematics. These observations suggest that theremay be some underlying physics of the system shared across all Reynolds numbers and thicknesses studied.To further understand these observations, a hierarchical decomposition of the flow field is considered.

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I.C. Analysis of principal fluid field components

Proper orthogonal decomposition (POD) analysis is referred to as the Karhunen-Loeve transform, principalcomponent analysis, or singular systems analysis depending on the discipline. This decomposition can alsobe formulated in a continuous or discrete sense, and used for experimental or computational data. Here, thecontinuous formulation is applied to computational data.

The goal of the POD analysis is to decompose data into hierarchical sets of spatial basis functions (modeshapes). Here, the fluid kinematic fields are decomposed in terms of spatial modes. To this end, the fluidfield is represented as

u(x, t) =∞∑

i=1

ξi(t)φi(x). (4)

where the time-dependent coefficients may be interpreted as modal coordinates, as done in vibration analysis.To construct the modes φ(x), it is chosen to maximize the projection of the empirical data onto these modesin a L2 sense. So, the problem statement is formulated in the scalar case as7

maxφ∈L2[0,1]

⟨|(u, φ)|2

⟩‖φ‖2

.

where (·, ·) is the L2 inner product on the domain of interest Ω and 〈·〉 is an averaging operation on theensemble of data. The implication is that φ is chosen to maximize the energy contained in the originaldata. By using arguments of variational calculus imposing the constraint that each mode is normalized, theFredholm equation is recovered;7 that is,∫

Ω

R(x, s)φ(s) ds = λφ(x)

where R(x, s) = 〈u(x, t)⊗ u(s, t)〉t is the time averaged autocorrelation tensor of the field, φ are the unknownmode shapes, and λ are the associated eigenvalues. Further simplification can be made, since it can berecognized that the modes are a special superposition of the data snapshots. By using Sirovich’s “methodof snapshots”,8 it is shown that the modes φ can be approximated as a finite sum over the known data inthe form

φi(x) =M∑

k=1

ψik

u(x, tk)v(x, tk)p(x, tk)

. (5)

Making use of scaling arguments, the pressure components can be neglected in the substitution back intothe Fredholm equation.9 The simplified results become an algebraic eigenvalue problem of size M

[C]ψ = λψ (6)

whereClk =

1M

∫Ω

u(x, tl) · u(x, tk) dV. (7)

From the construction of [C], it is noted that it is symmetric; that is, [C] = [C]T . Hence, only the upper orlower triangular part needs to be constructed. So the process to construct the POD set can be described asa sequence of the following steps:

1. Generate velocity field data at equal time intervals

2. Construct each element of (7) by integrating over the domain

3. Solve the RM×M algebraic eigenvalue problem of (6) to get the set of eigenvalues λk and the asso-ciated eigenvectors ψk

4. Back substitute ψk into (5) to get a truncated set of POD eigenfunctions φi.

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The set of spatial mode shapes, φi, are ranked by using the eigenvalues λi. This gives a quantitativemeasure of the importance of each mode as constructed from the data. For most of the examples consideredapproximately 99% of the energy in the snapshots are contained in the modes associated with the first 10leading eigenvalues. This means that one can consider just a few leading terms in the modal expansion. Anoteworthy aspect of this POD analysis, is that φ must be divergence free. Since the mode shapes are aweighted superposition of divergence free data, then, φ must also be divergence free (incompressible). Soany use of these modes faithfully preserves the incompressibility of the flow field.

II. Results and Discussion

Next, the results obtained from various POD computations are presented and discussed. It is noted thatthese figures represent the hierarchical spatial structures of the flow field. The center joint of the profilemoves through the region x = ±1.4, with y = 0 for all cases except for Re = 250 where y = 1.0.

II.A. Profile Thickness of 10%

In Figure 5, the vorticity contours obtained for the first several modes are shown for Re = 75 and the (10%)thickness case. All of these modes are orthonormalized, so that the maximal value of the curl for each ofthem is one. This provides a consistent way to compare the features of the mode shape relative to peakvalues across different modes. This works since the modes are scaled by ξi(t) when placed back in a truncatedform of (4). From this figure, it is observed that all mode one results contain a large pair of vortices. Thiscorresponds to a downward jet of fluid, and the production of lift on the profile. Furthermore, the vortexpair concentration is the highest and proximity of this vortex pair is closest to the bottom of the airfoillocation for ωf/ωn = 1/3. This frequency ratio also corresponds to the best aerodynamic performance forthe chosen harmonic input kinematics. For ωf/ωn ∈ 1/4, 1/6; that is, for springs stiffer than those forωf/ωn = 1/3, this vortex pair moves down and its peak is not as pronouced. It is possible that the locationand intensity of this pair is an indicator for favorable performance of a specific configuration.

Also as expected, the scales of the coherent structures decrease as the mode number increases. Thisfollows intuition gained from mode shape characteristics observed during vibrations of structures. This alsoshows how large the dissipation is for Re = 75. For modes four and greater, almost all vorticity is confinedto the region containing the body. Far away from the body, there are almost no structures in the wake.

In contrast to the well organized structures seen at Re = 75, the results obtained for Re = 250 andpresented in Figure 6 have a different organization. It is interesting to note patterns similar to those observedin the low Reynolds number case; that is, mode 1 corresponds to a downward jet and mode two correspondsto an end of stroke vortex pair. In this flow regime, another interesting observation is that for spring valuesof intermediate stiffness (i.e., ωf/ωn ∈ 1/3, 1/4), the flow appears to be more spatially regular than thatobserved in the stiff case of ωf/ωn = 1/6. Low dissipation is discernible, as rich structures can be seenin the wakes of the higher modes. The asymmetry introduced in the kinematics by the exponential decayterm is not damped out, as revealed by wake slant towards the left observed in mode one corresponding toωf/ωn = 1/6.

II.B. Profile Thickness of 3%

To investigate the effects of simple geometric changes, the thickness of the profile was changed from 10%to 3%. This more closely matches the case of the membrane which is of interest both in terms of relatingto insect wings, as well as to engineered structures. Since the UVLM formulation is based on a panelmethod, no aerodynamic surface is needed to encase the structural body, and hence, it naturally allowsfor a membrane model. In Figure 7, the characteristic flow fields obtained for this thin profile are shownfor viscous DNS studies and harmonic kinematics. On examining both the distribution and intensity ofthe coherent structures, the results are seen to be only slightly different from those obtained for the 10%thickness cases. This trend matches that seen in the mean lift and mean drag results shown in Figure 4.Athough, the instantaneous time variations are thickness sensitive, in an average sense, the results do notshow much variation respect to a thickness change.

Similar coherent structures are observed in the UVLM POD modes shown in Figure 8. Since the UVLMcomputations are inviscid, the vorticity contours cannot be used, and instead, the velocity component con-tours are investigated. In contrast to the DNS results, the observed scales are larger. However, as seen

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−2

0

Mode

1

ωf/ωn = 1/3 ωf/ωn = 1/4 ωf/ωn = 1/6

−2

0

Mode

2

−2

0

Mode

3

−2 −1 0 1 2

−2

0

Mode

4

−2 −1 0 1 2 −2 −1 0 1 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 5: Vorticity contours POD for Re = 75. Normalization is maxx

|∇ × φi(x)| = 1.

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−2

0

Mode

1

ωf/ωn = 1/3 ωf/ωn = 1/4 ωf/ωn = 1/6

−2

0

Mode

2

−2

0

Mode

3

−2 −1 0 1 2

−2

0

Mode

4

−2 −1 0 1 2 −2 −1 0 1 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 6: Vorticity contours POD for Re = 250. Normalization is maxx

|∇ × φi(x)| = 1.

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−2

0R

e=

75

ωf/ωn = 1/3 ωf/ωn = 1/4 ωf/ωn = 1/6

−2 −1 0 1 2

−2

0

Re

=25

0

−2 −1 0 1 2 −2 −1 0 1 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 7: Vorticity contours POD of Mode 1. Normalization is maxx

|∇ × φi(x)| = 1.

before, the higher modes appear less organized than the low Reynolds number modes. As expected, theprominent jet structure is also visible in the first mode shape for all of the considered stiffness values. Theincrease in the length scales of flow features is most likely attributable to the coarse approximations of theunsteady vortex lattice method formulation. Since this scheme is computationally much less expensive thana direct numerical simulation for the same configuration, this scheme can serve as an useful design aid giventhat the averaged results appear to match the DNS results well.

To understand the flow field characteristics in the time domain, projections can be carried onto the basisof mode shapes obtained from the POD process. Placing the mode shapes in context, the time-varyingcoefficients ξi(t) can be computed from (4) by using the orthogonality of the mode shapes; that is,

ξi(t) =

∫Ω

u(x, t) · φi(x) dV∫Ω

φi(x) · φi(x) dV. (8)

The first eight ξi(t) obtained for two sample configurations are shown in Figure 9. Although this figurehas many details, several features can be discerned. First, mode one is persistent and it has a much highermagnitude than the others. Since the mean flow is not removed before performing the POD computations,this is captured in the first mode, and this could be termed a DC offset. All the configurations shared thisfeature. Second, there are noticeable differences between the results obtained for the low and high Reynoldsnumber cases. For Re = 75, the coefficients are periodic and the magnitude of each high ξi is smaller thanobtained for a previous mode. So the spatial motions are primarily captured by the first several modes. Thisfollows from the observation that a high dissipation retards the formation of small scale structures in theflow. Therefore, the low Reynolds number case is a good candidate for developing a reduced-order modelthrough a truncated modal expansion.

In contrast to the orderly periodic patterns seen at Re = 75, the results obtained for Re = 250 are muchless clearer. It appears that some modes are nearly periodic, and that the high modes have large amplitudes.This may mean that one requires more modes to capture the key features of the flow.

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Re = 250

−2 −1 0 1 2

−5

−4

−3

−2

−1

0

1

Ver

tica

lVel

oci

tyu

Re = 75

−2 −1 0 1 2

−5

−4

−3

−2

−1

0

1

Hor

izto

nal

Vel

oci

tyw

UVLM

−2 −1 0 1 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 8: Comparisons of velocity contours POD obtained from viscous DNS and inviscid UVLM studiesfor ωf/ωn = 1/3 and harmonic kinematics. Normalizations are max

x|φx

i (x)| = 1 and maxx

|φyi (x)| = 1,

respectively.

6 7 8 9 10 11 12−2

−1

0

1

2

3

4

Period t/T

Modal

Coeffi

cien

tξ i

(t)

Mode 1 Mode 2 Mode 3 Mode 4Mode 5 Mode 6 Mode 7 Mode 8

(a) Re = 75, ωf /ωn = 1/3

6 7 8 9 10 11 12−2

−1

0

1

2

3

4

Period t/T

Modal

Coeffi

cien

tξ i

(t)

Mode 1 Mode 2 Mode 3 Mode 4Mode 5 Mode 6 Mode 7 Mode 8

(b) Re = 250, ωf /ωn = 1/4

Figure 9: Time varying modal coefficients ξi(t) computed a posteriori for the harmonic kinematics cases. Itis noted that for the low Reynolds number case, the magnitudes quickly diminish as the mode number isincreased whereas more information is carried in the modes of the high Reynolds number case.

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−2

0

Re

=75

ωf/ωn = 1/3 ωf/ωn = 1/4 ωf/ωn = 1/6

−2 −1 0 1 2

−2

0

Re

=25

0

−2 −1 0 1 2 −2 −1 0 1 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 10: Vorticity contours POD of Mode 1. Normalizaltion is maxx

|∇ × φi(x)| = 1.

II.C. Alternate Kinematics

For the 10% thick profile, the alternate kinematics was employed across the same configuration space asbefore. The POD results are noticeably different, as shown in Figure 10. A similar pattern is seen in thatthere is a large pair of vortices just under the region the body passes through. Again, this indicates adownward jet of fluid resulting in generation of lift on the profile. However, the shape and location of thisvortex pair has shifted. The vortex pair is closer to the body than that observed in the case of harmonickinematics, and the shape seen in the wake bulges out slightly. Although the same patterns seen from theharmonic kinematics are observed for the alternate kinematics, more configurations need to be tested forhigher values of ωf/ωn.

The computed time varying modal coefficients show charactertics similar to that observed for the har-monic kinematics case. First, for low Reynolds numbers, there is a decay in the modal amplitudes, althoughthe overall amplitudes are higher than those observed in the harmonic kinematics case. Second, for Re = 250,the amplitudes again do not decrease as the mode number increases.

III. Concluding Remarks

Analyses of flow fields associated with a two-dimensional flexible flapping wing profile have been presentedin this article. These analyses show the promise of the construction of spatial modes to characterize the flowfield. The enhanced aerodynamic performance peaks appear to be directly related to the location and relativeintensity of the vortex pair located under the profile in the first POD mode. This feature is prominent atRe = 75 for the ωf/ωn = 1/3 case, and one can view this state as a desired realization for a control scheme.

The simulations computed thus far have shown the importance of modes, and the information theycontain. Further work is to be carried out to examine data at higher Reynolds numbers. In order to see theeffects of the kinematics on the overall performance, alternative kinematic descriptions of hovering have beenintroduced. The effects of quicker stroke reversals appear to heavily influence the choice of spring stiffness.This work can serve as a basis for understanding how the aerodynamic performance changes, as well as howthe structures of the fluid vary if for instance the rotational speed is increased at stroke reversal. In recent

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8 9 10 11 12 13 14

−3

−2

−1

0

1

2

3

4

5

6

Period t/T

Modal

Coeffi

cien

tξ i

(t)

Mode 1 Mode 2 Mode 3 Mode 4Mode 5 Mode 6 Mode 7 Mode 8

(a) Re = 75, ωf /ωn = 1/3

8 9 10 11 12 13 14

−3

−2

−1

0

1

2

3

4

5

6

Period t/T

Modal

Coeffi

cien

tξ i

(t)

Mode 1 Mode 2 Mode 3 Mode 4Mode 5 Mode 6 Mode 7 Mode 8

(b) Re = 250, ωf /ωn = 1/4

Figure 11: Time varying modal coefficients ξi(t) computed a posteriori for the alternate kinematics cases.It is noted that for the low Reynolds number case, the magnitudes quickly diminish as the mode number isincreased whereas more information is carried in the modes of the high Reynolds number case.

work, invariants from data have been computed for control purposes.10 If this work can be extended tosystems of high dimensionality with dissipation , then it may be conceivable that the POD modes can beconsidered as the system invariants that is one interested in for constructing a control scheme.

Acknowledgments

The authors gratefully acknowledge the support received through ARO Grant No. W911NF0610369. Weare thankful to Mr. M. Vanella for thoughtful discussions and the DNS code used in this work.

References

1Pai, P. F., Chernova, D. K., and Palazotto, A. N., “Nonlinear Modeling and Vibration Characterization of MAV FlappingWings,” 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs,California, May 2009.

2Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E., and Balachandran, B., “Influence of flexibility on the aerodynamicperformance of a hovering wing,” Journal of Experimental Biology, Vol. 212, No. 1, 2009, pp. 95–105.

3Valdez, M., Preidikman, S., and Massa, J. C., “Aerodinamica De Flujos Bidemensionales E. Inestacionaries DominadosPor Vorticidad,” Mecanica Computacional , Vol. 25, November 2006, pp. 2333–2357, In Spanish.

4Valdez, M., Preidikman, S., Massa, J. C., Balachandran, B., and Mook, D. T., “Aerodynamics of 2D Unsteady VorticityDominated Flows,” 10th Pan American Congress of Applied Mechanics, Vol. 12, Cancun, Mexico, January 7–11 2008, pp.280–283.

5Wang, Z. J., Birch, J. M., and Dickinson, M. H., “Unsteady forces and flows in low Reynolds number hovering flight:two-dimensional computations vs robotic wing experiments,” Journal of Experimental Biology, Vol. 207, 2004, pp. 449–460.

6Dickinson, M. H., Lehmann, F.-O., and Sane, S. P., “Wing Rotation and the Aerodynamic Basis of Insect Flight,”Science, Vol. 284, No. 5422, 1999, pp. 1954–1960.

7Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry,Cambridge University Press, 1996.

8Sirovich, L., “Turbulence and the dynamics of coherent structures. Part I: Coherent Structures,” Quarterly of AppliedMathematics, Vol. 45, 1987, pp. 561–571.

9O’Donnell, B. J. and Helenbrook, B. T., “Proper orthogonal decomposition and incompressible flow: An application toparticle modeling,” Computers and Fluids, Vol. 36, No. 7, 2007, pp. 1174–1186.

10Schmidt, M. and Lipson, H., “Distilling Free-Form Natural Laws from Experimental Data,” Science, Vol. 324, No. 5923,April 2009, pp. 81–85.

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