Nonlinear Aeroelastic Analysis of Flexible High

Aspect Ratio Wings Including Correlation With

Experiment

by

Justin W. Jaworski

Department of Mechanical Engineering and Materials ScienceDuke University

Date:

Approved:

Dr. Earl H. Dowell, Chair

Dr. Donald B. Bliss

Dr. Kenneth C. Hall

Dr. Laurens E. Howle

Dr. Lawrence N. Virgin

Dissertation submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in the

Department of Mechanical Engineering and Materials Sciencein the Graduate School of Duke University

2009

Abstract(Aerospace)

Nonlinear Aeroelastic Analysis of Flexible High Aspect Ratio

Wings Including Correlation With Experiment

by

Justin W. Jaworski

Department of Mechanical Engineering and Materials ScienceDuke University

Date:

Approved:

Dr. Earl H. Dowell, Chair

Dr. Donald B. Bliss

Dr. Kenneth C. Hall

Dr. Laurens E. Howle

Dr. Lawrence N. Virgin

An abstract of a dissertation submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in the

Department of Mechanical Engineering and Materials Sciencein the Graduate School of Duke University

2009

Copyright c© 2009 by Justin W. JaworskiAll rights reserved

Abstract

A series of aeroelastic analyses is performed for a flexible high-aspect-ratio wing

representative of a high altitude long endurance (HALE) aircraft. Such aircraft are

susceptible to dynamic instabilities such as flutter, which can lead to large ampli-

tude limit cycle oscillations. These structural motions are modeled by a representa-

tive linear typical section model and by Hodges-Dowell beam theory, which includes

leading-order nonlinear elastic coupling. Aerodynamic forces are represented by the

ONERA dynamic stall model with its coefficients calibrated to CFD data versus wind

tunnel test data. Time marching computations of the coupled nonlinear beam and

ONERA system highlight a number of features relevant to the aeroelastic response

of HALE aircraft, including the influence of a tip store, the sensitivity of the flutter

boundary and limit cycle oscillations to aerodynamic CFD or test data, and the roles

of structural nonlinearity and nonlinear aerodynamic stall in the dynamic stability

of high-aspect-ratio wings.

iv

To Warren, Melissa, and Bill

v

Contents

Abstract iv

List of Tables viii

List of Figures ix

Nomenclature x

Acknowledgements xvii

1 Introduction 1

1.1 Research Questions and Outline . . . . . . . . . . . . . . . . . . . . . 2

2 Structural Models 4

2.1 Typical Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Step Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Balsa Wood Fairings . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Tip Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Combined Effects . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.6 Theoretical-Experimental Comparison . . . . . . . . . . . . . 16

2.2.7 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Hodges-Dowell Nonlinear Beam Equations . . . . . . . . . . . . . . . 21

vi

3 Aerodynamic Models 24

3.1 Slender Body Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Theodorsen Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 ONERA Dynamic Stall Model . . . . . . . . . . . . . . . . . . . . . . 28

4 Aeroelastic Formulations and Results 33

4.1 Typical Section with Theodorsen Aerodynamics . . . . . . . . . . . . 33

4.1.1 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Hodges-Dowell Nonlinear Beam & ONERA Dynamic Stall Model . . 38

4.2.1 Equation Formulation and Time-Marching Scheme . . . . . . 38

4.2.2 Flutter Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Limit Cycle Oscillations . . . . . . . . . . . . . . . . . . . . . 43

4.2.4 Time Estimates for First-Principles Aeroelastic Model . . . . . 47

5 Conclusions 52

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A Time-Domain Computational Fluid Dynamics Solver 57

A.1 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.2 Reynolds Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B ONERA Model Parameter Identification 68

B.1 Static Lift Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B.2 Unsteady Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography 74

Biography 85

vii

List of Tables

2.1 Comparison of ANSYS and experimental natural frequencies for dN =0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Typical section structural data . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Coefficients for typical section model eigenvalue analysis . . . . . . . 35

4.3 Typical section flutter comparison with HALE wing data . . . . . . . 38

4.4 Experimental wing model data . . . . . . . . . . . . . . . . . . . . . . 41

B.1 Comparison of ONERA lift parameters based on wind tunnel and CFDdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

viii

List of Figures

2.1 Schematic of typical section on linear springs . . . . . . . . . . . . . . 5

2.2 ANSYS wing model schematic . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Natural frequencies of a spar with varying step depth . . . . . . . . . 9

2.4 Natural frequency comparison of 2D and 3D spar with ribs, with vary-ing step depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Natural frequencies of a spar with balsa wood fairings, with varyingstep depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Natural frequencies of a spar with tip store, with varying step depth . 13

2.7 Combined structural effects on f1B for varying step depth . . . . . . . 14

2.8 Combined structural effects on f2B for varying step depth . . . . . . . 15

2.9 Combined structural effects on f1C for varying step depth . . . . . . . 15

2.10 Combined structural effects on f1T for varying step depth . . . . . . . 16

2.11 Flapwise mode shape comparison, with tip store . . . . . . . . . . . . 19

2.12 Chordwise mode shape comparison, with tip store . . . . . . . . . . . 20

2.13 Torsion mode shape comparison, with tip store . . . . . . . . . . . . . 20

2.14 Flapwise mode shape comparison, without tip store . . . . . . . . . . 21

2.15 Chordwise mode shape comparison, without tip store . . . . . . . . . 22

2.16 Torsion mode shape comparison, without tip store . . . . . . . . . . . 22

3.1 Schematic of ∆CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 V -g plot of HALE typical section model with steady aerodynamics . . 36

ix

4.2 V -g plot of HALE typical section model with quasi-steady aerodynamics 37

4.3 V -g plot of HALE typical section model with unsteady Theodorsenaerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Dependence of flutter speed on number of panels and aerodynamicevaluation location . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Sensitivity of flutter speed to source data for linear ONERA modelparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Effect of nonlinear beam stiffness on LCO amplitude and hysteresis . 45

4.7 Bifurcation diagram for computational and experimental LCO results 48

4.8 Time series and FFT spectrum for large-amplitude LCO at U = 40.5m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.9 Bifurcation diagram comparison of quasi-steady ONERA aerodynamicmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.1 Physical grid for NACA 0012 airfoil . . . . . . . . . . . . . . . . . . . 62

B.1 Piecewise curve fit to CFD static lift data . . . . . . . . . . . . . . . 71

B.2 Comparison of static lift curves from original ONERA model, CFDdata, and experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B.3 Lift hysteresis comparison of ONERA models with unsteady CFD data 73

x

Nomenclature

τ nondimensional time, ωαt

g gravitational constant

H(i)n Hankel function of the ith kind of order n

t time

x, x spanwise position coordinate, x/L

′ d/dx = L−1 d/dx

δ Dirac delta

δij Kronecker delta

˙ d/dt

∗ ωα d/dτ

D/Dt substantial derivative

Structural

An normalization factor

b half chord, c/2

bs spar width

c, c chord length, c/L

Cξφ torsion damping coefficient, 2mK2ξφω1T

Cξv lag damping coefficient, 2mξvω1C

Cξw flapwise damping coefficient, 2mξwω1B

xi

cSB longitudinal length of slender body

dN notch depth nondimensionalized by bs/2

E modulus of elasticity

e, e distance from elastic axis to center of gravity, positive aft; e/L

EI1 flapwise (out-of-plane) flexural rigidity

EI2 lag (chordwise, in-plane) flexural rigidity

f frequency [Hz]

GJ torsional rigidity

h, h plunge displacement, eigenvalue variable

Iv tip store moment of inertia about axis of lag displacement

Iw tip store moment of inertia about axis of flapwise displacement

Iα second moment of inertia of typical section airfoil about the elastic axis

Iφ tip store moment of inertia about wing elastic axis

K, K wing radius of gyration about spanwise elastic axis, K/L

Kh linear plunge stiffness for typical section

Kα linear torsional stiffness for typical section

M mass of tip store

m wing mass per unit span

Pz fluid momentum in xz-plane due to slender body motion

R slender body radius

rα radius of gyration for typical section airfoil about the elastic axis,√Iα/(mb2)

S slender body cross-sectional area

Sα first moment of inertia of typical section airfoil about the elastic axis

ts spar thickness

v, v lag displacement, v/L

w, w flapwise displacement, w/L

xii

xα airfoil static unbalance, Sα/(mb)

y length coordinate of slender body

yB pitch axis of slender body

za longitudinal centerline position of slender body

a¯

distance from midchord to elastic axis, positive aft

β dimensional grouping for nonlinear stiffness, (EI2 − EI1)/(mL4)

βn nth torsional eigenvalue

χn nth flapwise eigenvalue

φ geometric twist angle, φ+∫ x

0v′w′′ dx

λn nth bending eigenvalue

ν Poisson ratio

ω frequency [rad/s]

ωh plunge structural frequency for typical section model,√Kh/m

ωα torsional structural frequency for typical section model,√Kα/m

φ twist about deformed elastic axis, positive nose up

ψn nth lag eigenvalue

ρm material density

Θn nth torsional mode shape

ξv modal damping coefficient for lag bending modes

ξw modal damping coefficient for flap bending modes

ξφ modal damping coefficient for torsion modes

–V1 slender body volume

–V2 slender body volume static unbalance

–V3 slender body volume moment of inertia

Aerodynamic

xiii

∆Fv slender body aerodynamic force in lag direction

∆Fw slender body aerodynamic force in flap direction

∆Mx slender body aerodynamic moment

a a nonlinear ONERA load coefficient, a0 + a2(∆CL)2

a0L linear lift curve slope

C(k) Theodorsen function

CD coefficient of drag

CL coefficient of lift

CM coefficient of moment

Cz total ONERA aerodynamic load, Cz1 + Cz2

Czγ circulatory contribution to ONERA aerodynamic load

Cz1 linear ONERA aerodynamic load

Cz2 nonlinear ONERA aerodynamic load contribution from dynamic stall

D drag

dD/dx sectional drag on wing section

dFv/dx sectional aerodynamic force in lag direction

dFw/dx sectional aerodynamic force in flap direction

dL/dx sectional lift on wing section

dM0/dx sectional moment on wing section about the aerodynamic center

dMx/dx sectional aerodynamic moment

e a nonlinear ONERA load coefficient, e0 + e2(∆CL)2

kvz a noncirculatory linear ONERA load coefficient

L lift

M Mach number

Mx pitching moment beam aeroelastic model, positive nose up

My pitching moment for typical section

xiv

r a nonlinear ONERA load coefficient, r0 + r2(∆CL)2

Re Reynolds number based on chord

sz a noncirculatory linear ONERA load coefficient

U uniform flow velocity

wa downwash velocity

yac distance from elastic axis to aerodynamic center, positive aft

α, α angle of attack, eigenvalue variable

αz a circulatory linear ONERA load coefficient

∆CL stalled lift deficit

λz a circulatory linear ONERA load coefficient

ρ fluid density

σz a circulatory linear ONERA load coefficient

θ0 static root angle of attack

Aeroelastic

(CLγ)l ONERA circulatory lift contribution for lth panel

(CL2)l nonlinear ONERA lift contribution for lth panel

xl nondimensional spanwise location for evaluation of aerodynamic loads

∆t time step

Eij =∫ 1

0ψi θj dx

k reduced frequency, ωb/U

Kijk =∫ 1

0ψ′′i ψ

′′j θk dx

NB number of flapwise bending modes

NC number of lag bending modes

NT number of torsion bending modes

NAERO number of aerodynamic panel sections

xv

tτ = b/U

V reduced velocity, 2U∞/(ωαc)

Vj jth lag state variable

Wj jth flap state variable

∆l nondimensional spanwise length of aerodynamic panel, 1/NAERO

κ1 dimensional group, ρ∞U2c/(2mL)

κ2 dimensional group, ρ∞/(mL2)

κ3 dimensional group, ρ∞/(mK2L)

µ typical section mass ratio, m/(πρ∞b2)

φλl inflow angle evaluated at xl, φλ(x = xl)

φλ inflow angle, w/(U + v + wθ0)

Φj jth torsion state variable

Ψj =∫ 1

0ψj dx

Subscripts

∞ freestream condition

deg degrees

F value at flutter point

nB nth flapwise bending mode

nC nth lag bending mode

nT nth torsion mode

ss static stall

ub uniform beam value

xvi

Acknowledgements

First and foremost I would like to thank my advisor, Professor Earl Dowell, for

his introduction to the joy of research during my undergraduate years and for the

fruitful graduate years that followed. His skills as a researcher continue to inspire

me to a higher level of mentorship and scholarship, and to seek deeper insight into

engineering problems.

I am also grateful for the excellent support from my thesis committee: Profes-

sors Don Bliss, Kenneth Hall, Laurens Howle, and Lawrie Virgin. Their thoughtful

criticism and exacting standards have enriched greatly my thesis research and the

enjoyment and value of my graduate education.

xvii

1

Introduction

High-altitude, long-endurance (HALE) aircraft define a unique class of uninhabited

air vehicles (UAV) designed to perform missions for intelligence, surveillance, recon-

naissance, and communications purposes. HALE aircraft employ slender, flexible

wings to reduce weight and enable the high lift-to-drag ratios necessary to achieve

sustained flight for months or years. Increased wing flexibility can lead to large static

structural deformations for trimmed states, which has attracted researchers to the

effects of geometric nonlinearity on the flight dynamics [20, 78, 97] and dynamic

stability [80] of standard unswept wings or more advanced joined-wing vehicle con-

cepts [19]. Most research efforts apply the intrinsic, geometrically-exact beam model

developed by Hodges [50] augmented by intrinsic kinematic equations [78], a combi-

nation which is well-suited for large-deformation finite element analyses. Aeroelastic

analyses are then performed assuming finite state representations of classical two-

dimensional inviscid unsteady aerodynamics [82, 83] sufficient for flutter and flight

dynamic stability predictions. However, the determination of loads for post-flutter

or dynamically unstable situations, such as experienced prior to the mid-flight break

up of the Helios prototype [76], require an integrated understanding of both nonlin-

1

ear structural behavior and of the dynamic fluid loads arising from large unsteady

motions and nonlinear aerodynamic stall effects.

Research efforts by Tang & Dowell [102, 103, 104, 105, 106, 107, 108, 109]

and other researchers [32, 33, 34, 123] describe the nonlinear, unsteady aerody-

namic loads for prismatic wing sections with the semi-empirical ONERA dynamic

stall model [25, 120], which is calibrated to dynamic stall data obtained by exper-

iment [68, 69, 70]. Aeroelastic models including dynamic stall effects successfully

correlate with experimentally observed flutter and limit cycle oscillation (LCO) be-

havior, including hysteresis [108] and chaotic behavior [79]. Theoretical-experimental

comparisons pave the way for computational aeroelastic predictions for HALE air-

craft using computational fluid dynamics (CFD) solver, which at present time are

limited to either the predictions of a static aeroelastic shape [46, 77] or flutter bound-

ary [8] for three-dimensional wings due to the large associated computational cost.

The present research extends the work of Tang & Dowell [106] to include, by approx-

imation within the ONERA model framework, the nonlinear aerodynamic effects

arising from dynamic stall as computed by a CFD solver. The impact of struc-

tural geometric nonlinearity and dynamic stall are isolated and interpreted using

this approach, which provides estimates for the anticipated sensitivity of the flut-

ter boundary and limit cycle oscillation behavior for a fully-coupled, first-principles

nonlinear aeroelastic analysis.

1.1 Research Questions and Outline

The present work seeks answers to the following research questions.

• What level of modeling fidelity is necessary and sufficient for a very flexible

wing structure and its aerodynamic loads to predict accurately flutter and

limit cycle oscillation phenomena?

2

• What are the effects of geometric structural nonlinearity and fluid nonlinearity

due to aerodynamic stall on the aeroelastic behavior of slender wings?

• How sensitive are the flutter boundary and the LCO amplitude and hysteresis

metrics of HALE wings to first-principles-based aerodynamic data versus wind

tunnel data?

Following this Introduction, a range of structural models and aerodynamics models

used for aeroelastic computations are described separately in Chapters 2 and 3, re-

spectively. Chapter 4 integrates the structural and aerodynamic chapters into two

principal aeroelastic models: a linear typical section model with unsteady Theo-

dorsen aerodynamics; and the Hodges-Dowell nonlinear beam equations with the

ONERA dynamic stall lift model. Results from these aeroelastic models and dimi-

nuitive models thereof focus on the effects of flow unsteadiness, beam geometry, and

nonlinearities from structural and aerodynamic origins. Chapter 5 summarizes the

contributions of this research and suggests avenues for future work.

3

2

Structural Models

Very flexible wings may sustain large structural deformations during flight due to

loading, gusts, or fluid-structure instability. This chapter discusses a range of struc-

tural models for aeroelastic analysis to determine the level of physical fidelity to

predict successfully flutter and limit cycle oscillations of an experimental wing rep-

resentative of a HALE-type aircraft.

First, the classical typical section model on linear springs is considered to estab-

lish the simplest scenario for modeling an aeroelastic system with coalescence flutter.

Nondimensional groups for this model are later inferred from full aeroelastic wing

model data to make a posteriori flutter predictions. Second, a finite element analysis

is performed for the experimental aeroelastic wing to determine the individual effects

of its non-uniform components, i.e. spar, ribs, and tip store, on the natural frequen-

cies and mode shapes. Third, the Hodges-Dowell nonlinear beam-torsion equations

are considered to highlight the effects of elastic coupling and second-order geometric

nonlinearities. The removal of these nonlinearities reduces the Hodges-Dowell equa-

tions to Euler-Bernoulli beam theory and thus enables the systematic investigation

of structural nonlinear effects on flutter and limit cycle oscillations.

4

Mid Chord

c

+h

+α

b

a b

Elastic Axis

α

b

Mean Position Mass Center

K

x b

αK

h

Figure 2.1: Schematic of typical section on linear springs (courtesy of J.P. Thomas).

2.1 Typical Section

The typical airfoil section model [13, 29] shown in Figure 2.1 allows for the direct

study of aerodynamic interaction with a representative elastic structure constrained

to pitch, α, and plunge, h, motions only. Despite the simplicity of this model, it

continues to be a test bed for investigations of nonlinear stiffness [7, 9, 27, 26, 56, 57,

73], control surface free-play [10, 21, 28, 40, 47, 58, 73, 101, 110, 121], flutter and limit

cycle oscillations [9, 10, 27, 26, 40, 47, 58, 73, 95, 110], dynamic stall [44, 45, 105],

and stochastic [7, 88] and transonic aerodynamics [22, 28, 36, 58, 66, 73, 95]. The

present work restricts the structural model to linear stiffness behavior in pitch and

plunge motions modeled by the following set of equations.

mh+Kh h+ Sα α = −L (2.1)

Sα h+ Iα α+Kα α = My (2.2)

These equations can be recast into a nondimensional form that requires only the

5

external lift and moment forces as determined by an aerodynamic model.

(∗ ∗h/b) + (ωh/ωα)2 (h/b) + xα

∗ ∗α= − V

2

π µCL (2.3)

xα (∗ ∗h/b) + r2

α

∗ ∗α +r2

α α =2V 2

π µCM (2.4)

2.2 Finite Element Analysis

Materials of various densities and stiffnesses typically constitute the experimental

slender wing structures designed for flutter and limit cycle oscillation experiments

of HALE aircraft. Therefore, it is necessary to determine the effects of structural

nonuniformity on the natural frequencies and mode shapes of very flexible beams,

which are frequently used in geometrically nonlinear structural analyses of HALE

wings [32, 106, 107, 108, 109]. Jaworski and Dowell [54] demonstrated that such a

beam-like structure with spanwise discontinuities could be represented modally as

a uniform beam for sufficiently small spanwise discontinuities. However, determin-

ing the set of natural frequencies requires knowledge of any such discontinuities or

variations.

This section addresses the effects of these nonuniformities on the four lowest nat-

ural frequencies of a cantilevered HALE-type wing [53]. Here, the experimental wing

model used by Tang and Dowell [106, 107, 108] is analyzed using the commercial

finite element program ANSYS. As shown in Fig. 2.2, the HALE wing is composed

of a steel spar with multiple steps, periodically-spaced aluminum ribs, balsa wood

fairings to fill the space between ribs, and a tip store. Specifically, the effects ex-

amined herein are the step depth for the particular step distribution used by Tang

and Dowell [106, 107, 108]; the addition of periodically-spaced ribs; the presence of

balsa wood fairings with varying levels of rigid connection to the central spar; and

the addition of a tip store. For this study the total spar length, maximum width,

6

(a)

(b)

Figure 2.2: (a) Aeroelastic wing model; (b) Schematic of ANSYS wing model. Tipmass spanwise location indicated by ?. Dimensions in millimeters.

7

thickness, step distribution, and step width are held constant; only the step depth

varies. Step depth is defined as the chordwise dimension of the symmetrical material

removed from the spar, which is 3.17mm for the experimental model in Fig. 2.2(b).

The step depth is scaled by the spar half-width, and the natural frequencies by their

uniform beam values of the spar alone, fub. The four natural frequencies of interest

(f1B, f2B, f1C , f1T ) are tracked as the nondimensional step depth dN varies.

The physical parameters used in the finite element computations for balsa wood,

aluminum, and steel are ρm = 138.5, 2664, 7850 kgm−3; E = 2.34, 60.6, 200GPa; and

ν = 0.3, 0.33, 0.3, respectively. The densities are calculated from total mass and

dimensional data of material specimens, and the elastic moduli are calibrated to

their first flapwise resonance using classical beam theory. Standard handbook values

are assumed for the Poisson ratios.

All figures herein indicate actual finite element data with symbols unless otherwise

noted, and the curves between these symbols are interpolated using piecewise cubic

splines [67]. The results are analyzed for individual additions of ribs, fairings, and a

tip store, as well as their combined effects on natural frequencies.

2.2.1 Step Depth

The spar is modeled using two-dimensional, 8-node structural shell elements (shell-

93) [5]. The finite element meshes are generated by the “SmartSizing” free-meshing

function within ANSYS, set to the highest possible resolution.

The natural frequency results in Fig. 2.3 show that the first two out-of-plane

bending mode trends are virtually coincident, and that the torsion mode follows a

very similar trend. This similarity is anticipated because both torisonal and out-of-

plane bending stiffnesses scale as bst3. The frequency trend for the in-plane bending

mode is an almost linear function of the step depth, as expected from uniform beam

theory where the in-plane stiffness scales as b3st.

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

dN

f/f u

b

Figure 2.3: Spar natural frequencies as step depth is varied: , f1B; ♦, f2B; 4,f1C ; , f1T .

2.2.2 Ribs

The effect of adding aluminum ribs with a NACA 0012 profile [1] is evaluated using

both two-dimensional shell (SHELL93) and three-dimensional solid (SOLID45) finite

element models. As shown in Fig. 2.4, the two- and three-dimensional finite element

models are in close agreement, which suggests that three-dimensional effects at the

constraints between the ribs and spars may be neglected.

A comparison between Figs. 2.3 and 2.4 indicates that rib addition does not

change the qualitative frequency trends observed for the stepped spar, but rather

the ribs change the magnitude of the results. The bending mode results are roughly

85% of those for the spar alone, whereas the torsion mode results are reduced by

nearly 40%. Clearly, the main contribution of the ribs is an increase in torsional

inertia.

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dN

f/f u

b

Figure 2.4: Natural frequencies of spar with ribs as step depth is varied: solid line,2D; dashed line, 3D. , f1B; ♦, f2B; 4, f1C ; , f1T .

2.2.3 Balsa Wood Fairings

The addition of balsa wood fairings and the constraint conditions between the spar

and fairings are modeled by three-dimensional solid elements. Figure 2.5 shows that

the fairings dominate the torsion and out-of-plane bending modes and that the step

depth has virtually no effect. Therefore, the balsa wood addition acts effectively as

a stiffness increase for the out-of-plane bending modes and an inertial increase for

the torsion mode.

The particular distribution of balsa wood provides a modest 5% change in the

natural frequency of the in-plane bending mode for zero step depth. The in-plane

bending mode trend resembles that of the spar alone but approaches a nonzero

frequency as dN → 1; the other modes also have nonzero frequencies in this limit

because the balsa wood holds the structure together. It is also inferred from the

limit dN → 1 that the presence of the balsa wood accounts for a quarter of the

10

effective in-plane bending stiffness, recalling f 2∼EI2.

The computations for the results shown in Fig. 2.5 assume that the balsa wood

and spar are perfectly joined, i.e. the interfacial node displacements are identical.

To investigate the influence of the compatibility condition, three alternative con-

straint conditions are considered in the computational model for single- and double-

component segments of the full structure.

First, the constraint between the fairings and spar is relaxed (i.e. stiffness due to

shear flow is eliminated) along the thin edges of the spar; this effect is very small

(∼1%) for all modes. Second, the constraint along the wide edges of the spar is

relaxed instead. The in-plane bending mode is relatively unchanged, but the out-

of-plane bending and torsion values decrease by roughly 5-10%. However, the com-

parison of component results suggests that the first out-of-plane natural frequency

becomes progressively lower than the corresponding perfectly joined cases, whereas

the second out-of-plane bending and torsion natural frequencies have a relative in-

crease. The stiffening of the second bending mode for the double-component case is

thought to explain its trend difference from the first out-of-plane mode.

Third, the balsa wood fairings are connected to the spar only at its outermost

corner points using constraint equations to relate node displacements. The net ef-

fect is a limiting case where the balsa wood is solely a mass addition. The in-plane

bending and torsion results are most affected. The change in out-of-plane natural

frequencies is roughly 5-10%, though more data would be needed to deduce a trend

because the constraint locations and node lines of the bending modes factor signif-

icantly into the analysis. This was a lesser concern for the other cases because the

constraint was applied over an area instead of at a small number of points.

Overall, the effect of varying the compatibility conditions is modest for the in-

and out-of-plane bending modes, but more pronounced for the torsion mode.

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

dN

f/f

ub

Figure 2.5: Natural frequencies of 3D spar with balsa wood fairings as step depthis varied: , f1B; ♦, f2B; 4, f1C ; , f1T .

2.2.4 Tip Store

This section examines the three-dimensional spar when fitted with a tip store. The

store is modeled as two identical point masses positioned such that the effective

mass and torsional inertia match the measured values of M = 36.95 g and Iφ =

8.314× 10−5 kgm2, respectively.

The results in Fig. 2.6 show that the tip store inertia renders the torsional mode

almost insensitive to step depth. Also, the bending mode trends are similar to those

of the spar alone. For the first time, the out-of-plane bending mode curves do not

overlap because the frequency results depend on the tip store placement relative to

the nodes of the particular mode shape.

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dN

f/f

ub

Figure 2.6: Natural frequencies of 3D spar with a tip store as step depth is varied:, f1B; ♦, f2B; 4, f1C ; , f1T .

2.2.5 Combined Effects

The modeling variations analyzed herein are now combined to observe their net effect

on the natural frequencies. Figures 2.7–2.10 describe the frequency behavior of the

stepped spar as the ribs, balsa wood fairings, and tip store are added in sequence.

The trends are similar for the out-of-plane bending and torsion modes. The

addition of the ribs reduces the magnitude of the stepped spar frequency results to

varying degrees, which indicates that the ribs essentially contribute more inertia than

stiffness. Balsa wood fairings flatten the curve and increase the frequency values,

indicating that the fairing stiffness dominates and that step depth effects become

less important. The tip store affects the curves such that the final configuration has

out-of-plane bending and torsion frequencies lower than those of the spar alone. The

resulting out-of-plane bending and torsion modes are effectively independent of step

depth.

13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dN

f/f u

b

Figure 2.7: Combined effects on f1B for varying step depth: solid line, spar alone;, with ribs; , with ribs and balsa; 4, with ribs, balsa, and tip store.

The change in the in-plane bending mode frequency behavior in Fig. 2.9 is less

pronounced than for the other modes. The rib addition changes the uniform spar

frequency by 15%, a difference that diminishes as the step depth is increased. The

subsequent balsa wood addition retains the uniform spar trend but changes the

frequency for dN → 1 to a non-zero value as with the other modes. The tip store

scales down the magnitude of previous results as seen with the other modes.

The results shown in Figs. 2.7–2.10 support the hypothesis that an aeroelastic

wing could be designed by varying the tip store properties and step depth alone.

Explicitly, the out-of-plane bending modes depend primarily on the tip store mass;

the tip store mass and step depth tune the in-plane bending frequency; and tip store

torsional inertia controls the torsion frequency. The next section correlates such

computational results against experiment.

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dN

f/f u

b

Figure 2.8: Combined effects on f2B for varying step depth: solid line, spar alone;, with ribs; , with ribs and balsa; 4, with ribs, balsa, and tip store.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dN

f/f

ub

Figure 2.9: Combined effects on f1C for varying step depth: solid line, spar alone;, with ribs; , with ribs and balsa; 4, with ribs, balsa, and tip store.

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

dN

f/f u

b

Figure 2.10: Combined effects on f1T for varying step depth: solid line, spar alone;, with ribs; , with ribs and balsa; 4, with ribs, balsa, and tip store.

2.2.6 Theoretical-Experimental Comparison

The HALE-type wing [106, 107, 108] natural frequencies were measured using a

transducer-fitted impact hammer (Bruel & Kjær (B&K) type 8204) and an ac-

celerometer (B&K type 4374). The signal from the accelerometer is boosted by

a charge amplifier (B&K type 2635), and the transfer function between the hammer

and accelerometer is measured by the B&K pulse data acquisition system [16]. Each

transfer function is averaged linearly over five impacts at the spar tip. Every impact

is sampled at 256Hz for eight seconds, yielding a frequency resolution of 125mHz.

Table 2.1 compares the results for the three-dimensional finite element model

both with and without the tip store with observed values for the HALE-type wing.

Without the tip store, the finite element model agrees to within 15-30% of comparable

experimental values for the bending modes. Lesser agreement is observed for the

torsion mode. The full model with tip store overestimates the experimental values

16

Table 2.1: Comparison of ANSYS and experimental natural frequencies for dN = 0.5.

ANSYS ExperimentMode

w/o Store w/ Store w/o Store w/ Storef1B 4.598 (3.968)† 2.882 (2.658) 4.000 2.625f2B 28.76 (24.83) 22.37 (19.87) 22.38 17.88f1C 25.82 (22.29) 15.86 (14.93) 23.13 14.13f1T 145.3 (140.9) 25.75 (25.71) 102.9 22.88

† total mass matched to experiment for parentheses values

by roughly 10-25%; the second out-of-plane mode has the largest overestimate.

The results for the mass-corrected model are placed in parentheses for Table 2.1.

The ANSYS bending mode results agree to within 11% of experiment using the mass

correction. The torsion mode is virtually unaffected because the assumed radius of

gyration of the added mass is small.

2.2.7 Mode Shapes

This section compares the mode shapes of uniform beam theory to those from the

finite element wing model for dN =0.5, which corresponds to the experimental wing

in Ref. [106] (cf. Fig. 2.2). This comparison demonstrates how accurately the present

three-dimensional HALE wing finite element model with its many nonuniformities

can be approximated by classical beam theory.

Figures 2.11-2.13 compare the wing mode shapes with the tip store. The ANSYS

modal deflections are recorded for 18 spanwise locations at the midchord, and a least-

squares tenth-order polynomial curve fit is drawn through the data. The classical

bending modes are described by

χn(x), ψn(x) = An

[(cosλnx− coshλnx)−

(cosλn + coshλn

sinλn + sinhλn

)(sinλnx− sinhλnx)

],

(2.5)

where An is a scaling factor to normalize the mode shape, and λn are solutions to

17

the transcendental equation [52]

1 + cosλn coshλn = λn

(M

mL

)(sinλn coshλn − cosλn sinhλn). (2.6)

The torsion mode shape is Θn(x)=An sin(βnx), where βn satisfies [52, 13]

βn tan βn =mLK2

Iφ. (2.7)

To compare the mode shapes directly, the values M/mL= 0.4032 and mLK2/Iφ =

9.496×10−2 follow from the ANSYS model. Classical theory closely approximates

the resulting ANSYS mode shapes when including tip store effects, especially for

the higher-order flapwise modes in Fig. 2.11. Figures 2.14–2.16 indicate that the

first modes remain virtually coincident without the tip store, but the maximum

percentage differences between the second-, third-, and fourth-order flapwise modes

grow to 5.38%, 9.72%, and 13.5%, respectively.

Overall, the finite element mode shapes of the considered nonuniform HALE

wing model are well-described by classical uniform beam theory. The addition of

a tip store improves the approximation of the computed mode shapes by classical

theory.

2.2.8 Conclusions

A computational structural analysis is performed for a high-aspect-ratio, experimen-

tal aeroelastic wing model using the commercial finite element program ANSYS.

The computational results quantify the effects of spanwise nonuniformities such as

ribs, fairings, and a tip store on the first four wing modes. The mode shapes of the

nonuniform finite element wing model are shown to be well-approximated by classical

beam modes, which are typically assumed as trial functions for nonlinear aeroelastic

analyses of slender wings. In addition, the computed natural frequency results are

18

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

χ1(x

)

(a)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ2(x

)

(b)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ3(x

)

(c)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ4(x

)

(d)

Figure 2.11: Flapwise mode shapes for HALE wing with tip store: a) χ1; b) χ2; c)χ3; d) χ4. , ANSYS data; solid line, least-squares polynomial fit of ANSYS data;dashed line, uniform beam theory.

compared with those from experiment and shown to be in reasonable agreement.

The agreement between computational and experimental results without resorting

to empiricism supports the use of ANSYS as a design tool for aeroelastic analyses

of HALE-type wings. Also, the close agreement of the computed mode shapes and

those from classical beam theory enbales and validates the use of a more sophisticated

homogeneous and isotropic continuum beam model.

19

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

ψ1(x

)

Figure 2.12: First chordwise mode shape for HALE wing with tip store: , ANSYSdata; solid line, least-squares polynomial fit of ANSYS data; dashed line, uniformbeam theory.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

Θ1(x

)

Figure 2.13: First torsion mode shape for HALE wing with tip store: , ANSYSdata; solid line, least-squares polynomial fit of ANSYS data; dashed line, uniformbeam theory.

20

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

χ1(x

)

(a)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ2(x

)

(b)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ3(x

)

(c)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x

χ4(x

)

(d)

Figure 2.14: Flapwise mode shapes for HALE wing without tip store: a) χ1; b)χ2; c) χ3; d) χ4. , ANSYS data; solid line, least-squares polynomial fit of ANSYSdata; dashed line, uniform beam theory.

2.3 Hodges-Dowell Nonlinear Beam Equations

The Hodges-Dowell equations [51] describe the nonlinear interactions between elastic

bending and torsion motions for a slender, straight, homogeneous, and isotropic beam

without cross-sectional warping. Originally developed to investigate the importance

of nonlinearity on the aeroelastic stability and behavior of hingeless helicopter rotor

blades, these equations reduce to expressions suitable for the nonlinear analysis of

the slender wings of HALE-type aircraft. Hodges-Dowell theory features nonlinear

elastic coupling between the bending and torsion motions arising from a nonlinear

strain-displacement relationship, which enables an ordering scheme for geometrically

21

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

ψ1(x

)

Figure 2.15: First chordwise mode shape for HALE wing without tip store: ,ANSYS data; solid line, least-squares polynomial fit of ANSYS data; dashed line,uniform beam theory.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

Θ1(x

)

Figure 2.16: First torsion mode shape for HALE wing without tip store: , ANSYSdata; solid line, least-squares polynomial fit of ANSYS data; dashed line, uniformbeam theory.

22

nonlinear stiffness effects based on the assumptions that the squares of the bending

slopes, twist, thickness to length, and aspect ratio are small compared to unity.

For the present investigation, the Hodges-Dowell equations describe flap, lag, and

twist motions under gravitational loading and include second-order geometrical strain

effects.

EI2 v(IV ) + (EI2 − EI1)[φw

′′]′′ +mv + Cξ v + (Mv + Ivv′) δ(x− L) =

dFv

dx+ ∆Fv − [Mg δ(x− L) +mg] sin θ0 (2.8)

EI1w(IV ) + (EI2 − EI1)[φv

′′]′′ +mw −meφ+ Cξ w + (Mw + Iww′) δ(x− L) =

dFw

dx+ ∆Fw − [Mg δ(x− L) +mg] cos θ0 (2.9)

−GJ φ′′ + (EI2 − EI1)w′′v′′ + Iφ δ(x− L) +mK2

mφ+ Cξ φ−mew =

dMx

dx+ ∆Mx (2.10)

These continuum equations are well-suited for the presently considered experimen-

tal aeroelastic wing as supported by the finite element mode shape comparisons in

Section 2.2.7.

The right-hand sides of Eqns. 2.8–2.10 include the aerodynamic conributions

from the wing surface, dFv/dx, dFw/dx, and dMx/dx, as well as from the slender

body at the wing tip, ∆Fv, ∆Fw, and ∆Mx [106]. As a consequence of the second-

order geometrical accuracy, the projection of the deformed twist angle, φ, onto the

twisting plane of the undeformed axis is defined as the geometric angle, φ [31].

This relationship will be necessary to properly determine the angle of attack at a

prescribed spanwise location within strip-theory aerodynamic assumptions.

φ = φ+

∫ x

0

v′w′′ dx (2.11)

23

3

Aerodynamic Models

High-aspect-ratio wings achieve high aerodynamic efficiency by virtue of their large

spans. Except for the regions influenced by the vortices generated at the ends of these

wings, the flow over most of the inboard wing section remains two-dimensional. This

observation supports the assumption of strip-theory aerodynamics, where the aerody-

namic loads at a particular spanwise location are dependent strictly on the geometric

angle of attack at that location and are independent of the (three-dimensional) aero-

dynamic influence of other spanwise locations [42].

This chapter identifies a range of aerodynamic models to describe the aerody-

namic loads of the wing store and the spanwise ‘strips’ constituting the aeroelastic

wing surface. Slender body theory is first described to model the aerodynamics of the

store at the wing tip. Subsequent sections discuss the classical Theodorsen thin air-

foil theory for incompressible flow and the ONERA dynamic stall model. By design,

the semi-empirical, nonlinear, large motion ONERA model is calibrated to provide

steady and unsteady aerodynamic loads for a particular airfoil section and reduces

to an equivalent Theodorsen-like state-space model for small motions. The present

work uses a computational fluid dynamics (CFD) code to generate the aerodynamic

24

calibration data and identify a new ONERA dynamic stall model for the NACA 0012

airfoil based on first-principles information.

3.1 Slender Body Theory

A body is considered aerodynamically ‘slender’ if its crosswise dimensions such as

span and thickness are small compared to its length [61]. Specifically, slender body

theory assumes that the disturbed flow is two-dimensional in planes normal to the

flight direction [13, 55, 72], which renders the theory valid for any Mach number

so long as the flow normal to the flight direction is effectively incompressible. The

present model follows the inviscid flow derivation of Ref. [13] for a rigid body of

revolution that is restricted to pitching and plunging motions with small incidence

angles.

Consider a longitudinal centerline of the slender body defined by za(y, t), where

y is the freestream flow direction and the length coordinate of the slender body with

a pitch axis at yB.

za = −h(t)− α(t)[y − yB] (3.1)

The assumption of small incidence angles allows the generated downwash, wa, to be

expressed as wa = Dza/Dt. The fluid momentum contained in xz-planes separated

by distance dy can then be expressed as

dPz = ρ∞ S wa dy

= ρ∞ S

[dza

dt+ U

dza

dy

]dy, (3.2)

where the cross-section is circular with area distribution S(y) = πR2(y). The momen-

tum flux of displaced fluid in the xz-plane produces a lift reaction on the differential

25

length segment dy of the slender body.

dL

dy= − D

Dt

[dPz

dy

](3.3)

= ρ∞U(h+ Uα)dS

dy+ ρ∞Uα(y − yB)

dS

dy+ ρ∞(h+ 2Uα)S + ρ∞α(y − yB)S

(3.4)

Total lift and moment are calculated by direct integration of Eqn. 3.4 over the body,

noting that S(y = 0, cSB) = 0.

L =

∫ cSB

0

dL

dydy (3.5)

= ρ∞(h+ Uα) –V1 + ρ∞α –V1 (3.6)

Mx = −∫ cSB

0

(y − yB)dL

dydy (3.7)

= ρ∞U(h+ Uα) –V1 − ρ∞h –V2 − ρ∞α –V3 (3.8)

The slender body volume, volume static unbalance, and volume moment of inertia

are constant-valued coefficients defined respectively by the following integrals.

–V1 =

∫ cSB

0

S dy (3.9)

–V2 =

∫ cSB

0

(y − yB)S dy (3.10)

–V3 =

∫ cSB

0

(y − yB)2S dy (3.11)

Aerodynamic forces from the slender body relate to the wing motions of Eqns. 2.8–

2.10 through the definitions h = −w|x=L and α = φ|x=L. Also, within the small inci-

dence angle approximation a lag force appears by tipping the lift vector; viscous drag

effects that would also act in this direction are not accounted for in this framework.

26

The final expressions for the slender body aerodynamic loads follow [103, 106, 107].

∆Fw = ρ∞

[(U

˙φ− w) –V1 +

¨φ –V2

]δ(x− L) (3.12)

∆Fv = (θ0 + φ− φλ)∆Fw (3.13)

∆Mx = ρ∞

[U(Uφ− w) –V1 + w –V2 − ¨

φ –V3

]δ(x− L) (3.14)

3.2 Theodorsen Theory

Theodorsen [113] first published the solution for unsteady aerodynamic loads on a

thin airfoil in incompressible flow. The circulatory and non-circulatory contributions

to sectional lift and moment can be expressed directly by assuming simple harmonic

motion of the airfoil and wake [13].

L = πρ∞b2[h+ Uα− ba α] + 2πρ∞UbC(k)[h+ Uα+ b (1/2− a) α] (3.15)

My = πρ∞b2[bah− Ub (1/2− a) α− b2 (1/8 + a2) α]

+ 2πρ∞Ub2 (a+ 1/2)C(k)[h+ Uα + b (1/2− a) α] (3.16)

The Theodorsen function,

C(k) =H

(2)1 (k)

H(2)1 (k) + iH

(2)0 (k)

(3.17)

quantifies the dependence of the circulatory lift on the vorticity shed unsteadily into

the wake, and may be physically regarded as the lag in the development of bound

circulation due to the influence of shed wake vortices [61]. Flutter analyses using the

original Theodorsen function are valid only at the flutter point due to the assumption

of undamped harmonic motion.

Aeroelastic calculations rearrange Eqns. 3.15 and 3.16 typically into independent

load coefficients due to pitch and plunge motions, which may then be expressed in

27

their frequency-domain forms for eigenvalue computations.

CLh/b= πV −2[(

∗ ∗h/b) + 2V C(k) (

∗h/b)] (3.18)

CLα = πV −2[V∗α −a ∗ ∗

α +2C(k) (V 2α+ V (1/2− a)∗α)] (3.19)

CMh/b= πV −2/4 [a (

∗ ∗h/b) + 2 (a+ 1/2)V C(k) (

∗h/b)] (3.20)

CMα = πV −2/4[− V (1/2− a)∗α −(1/8 + a2)

∗ ∗α

+ 2 (a+ 1/2)C(k) (V 2α+ V (1/2− a)∗α)] (3.21)

3.3 ONERA Dynamic Stall Model

Early dynamic stall studies looked to experimental programs [17, 68, 70, 71] to inves-

tigate the effects of airfoil shape, Mach and Reynolds numbers, mean and unsteady

angles of attack, and reduced frequency on the aerodynamic loads in response to

prescribed harmonic motion. These data led to the development of a semi-empirical

model by Tran & Petot [120] and Dat & Tran [25] at l’Office National d’Etudes et

Recherches Aerospatiales (ONERA) designed to predict the loads on helicopter ro-

tor blades encountering large unsteady motions and dynamic stall phenomena. The

ONERA dynamic stall model consists of a set of differential equations with nonlinear

coefficients but linear operators, which enable the convenient and simultaneous up-

date of structural motion and aerodynamic forces in state-space for a fully-coupled

aeroelastic computation. Moreover, the model equations of the model can be lin-

earized about a nonlinear state for use in traditional dynamic analysis and stability

programs [81].

Despite the immediate utility of the ONERA equations for both fixed and rotating

blades, the original model had no basis in first-principles physics and had relatively

little in common with classical aerodynamic theory. Researchers sought to improve

the consistency of the ONERA model with classical theory [81, 87] and to overcome

28

its limitations when applied to extreme unsteady behavior, such as numerical insta-

bility for large angles of attack involving reversed flow [81]. Peters [81] reformulated

the model to fix this deficiency and enable the reproduction of Theodorsen [113]

and Greenberg (pulsatile freestream) [48] aerodynamics in a logical manner. Fur-

thermore, Peters [81] and Rogers [93] worked to isolate the effects of pitching and

plunging motions on dynamic stall [18, 35, 41], whose distinctions affect the circula-

tory lift and apparent mass terms.

The final form of the ONERA equations investigated here follows from Dunn [32,

33, 34] and has been used extensively in aeroelastic investigations using the NACA

0012 airfoil section [32, 102, 104, 105, 106, 107, 108, 110].

Cz = Cz1 + Cz2 (3.22)

Cz1 = tτszα+ t2τkvz¨φ+ Czγ (3.23)

tτ Czγ + λzCzγ = λz(a0zα+ tτσz˙φ) + αz(tτa0zα+ t2τσz

¨φ) (3.24)

t2τ Cz2 + atτ Cz2 + r Cz2 = −r[∆Cz + tτe

∂∆Cz

∂αα

](3.25)

The ONERA model separates the total lift (z = L) or moment (z = M) coefficient

into linear and nonlinear contributions to be solved independently; the ONERA

model describes only the unsteady lift in this work. The linear aerodynamics of

Eqns. 3.23 and 3.24 constitute a state-space representation of Theodorsen-like airfoil

aerodynamics for a particular airfoil geometry and flow condition. Effects due to

fluid viscosity, compressibility, and airfoil geometry are embedded into the constant

coefficients of each equation by the parameter identification of given aerodynamic

data.

Nonlinearity in the ONERA model arises from Eqn. 3.25 due to the dependence of

its coefficients on ∆CL [69, 84, 86], the force deficit measured between the unstalled

29

Figure 3.1: Schematic of ∆CL.

static lift coefficient and the actual static lift coefficient including stall (cf. Fig. 3.1).

a = a0 + a2(∆CL)2 (3.26)

r = r0 + r2(∆CL)2 (3.27)

e = e0 + e2(∆CL)2 (3.28)

The static lift deficiency, ∆CL, also forces the nonlinear lift equation, which would

otherwise represent a damped oscillator. Therefore, the ONERA model does not

predict any kind of nonlinear or dynamic stall behavior unless the effective angle

of attack exceeds that of static stall, α > αss, when ∆CL > 0. The right-hand

side (RHS) of Eqn. 3.25 also includes lag effects that are apparent in deep stall,

where the aerodynamic loads are sensitive to the generation and convection of vor-

tices interacting with the time-dependent motion of the structure. Debate exists

as to how these lag effects are best incorporated into the ONERA model, as dis-

cussed at length by Dunn [34]. Early forms of the model included an explicit lag

30

term [85], which has been replaced in modern versions by a dependence on the terms

r and e from Eqns. 3.27 and 3.28. The model selection depends on a balance be-

tween the retention of physical behavior in the modeling effort and the convenience

to its particular application of the ONERA model. For example, Dunn [34] chose

the form RHS=[−r∆CL + tτe(∂∆CL/∂α)α] to simplify a harmonic balance analysis.

The present RHS of Eqn. 3.25 follows from Petot & Dat [87], where the phase-lag

is expressed in direct relation to the force deficit ∆CL [34] to facilitate physical

interpretation of aerodynamic lag.

Here, the ONERA dynamic stall model determines the two-dimensional lift co-

efficients for each NACA 0012 spanwise wing section. The sectional lift, drag, and

moment values of each wing section are defined by the following.

dL

dx=

1

2ρ∞c U

2CL (3.29)

dD

dx=

1

2ρ∞c U

2CD (3.30)

dM0

dx=

1

2ρ∞c

2U2CM (3.31)

The moment about the aerodynamic center is determined by a simpler form of

the ONERA equations with a quasi-static nonlinear stall contribution for α > αss,

∆CM = −0.08 sgn(α) [103].

CM = CM1 + CM2 (3.32)

CM1 = tτsM α+ tτσM˙φ+ t2τkvM

¨φ (3.33)

CM2 = −∆CM (3.34)

Values for the unsteady moment coefficients follow from thin airfoil theory: sM ,

σM = −π/4; kvM = −3π/16. A curve fit to static experimental data [1] defines the

drag dependence on the instantaneous angle of attack.

CD = 0.008 + 1.7α2deg (3.35)

31

To be consistent with the Hodges-Dowell equations, the quasi-steady inflow angle,

φλ, and effective angle of attack, α, must incorporate the root pretwist of the wing,

the geometric twist angle, and kinematic effects of three-dimensional wing motion.

φλ ≈ w/(U + v + wθ0) (3.36)

α = θ0 + φ− φλ (3.37)

Thus, the sectional aerodynamic loads for the Hodges-Dowell equations can now be

written explicitly.

dFw

dx=dL

dx+ (θ0 − φλ)

dD

dx(3.38)

dFv

dx= −dD

dx+ (θ0 − φλ)

dL

dx(3.39)

dMx

dx=dM0

dx− yac

dFw

dx(3.40)

32

4

Aeroelastic Formulations and Results

The structural and aerodynamic models of Chapters 2 and 3 are integrated to per-

form a range of aeroelastic analyses for the HALE experimental wing. The aeroelas-

tic models investigate the effects of beam geometry modeling, structural nonlinearity

arising from elastic coupling among the degrees of freedom, and dynamic stall aero-

dynamic nonlinearity on the flutter and limit cycle oscillation behavior. Also, the

two ONERA dynamic stall models based on wind tunnel or first-principles CFD data

are compared with regard to their impact on the dynamic behavior on the aeroelastic

wing. The goal of the present chapter is to establish the role of structural and aero-

dynamic modeling fidelity on the simulated linear and nonlinear dynamic response

of a HALE wing, and to benchmark the agreement between experimental data and

simulated results based on aerodynamic data computed by a CFD solver.

4.1 Typical Section with Theodorsen Aerodynamics

4.1.1 Eigenvalue Analysis

The typical section model on linear springs is combined with Theodorsen unsteady

thin airfoil theory to perform an a posteriori flutter analysis using structural param-

33

Table 4.1: Typical section data representative of a HALE-type wing [106].

xα -0.005rα 0.3682ωh/ωα 0.7815µ 103.1

eters adapted from the experimental HALE wing, which are identified in Table 4.1

from data in Ref. [106]. Note that the plunge-to-pitch frequency ratio, ωh/ωα, is

tuned to the known second-bending/first-torsion coalescence flutter mode.

The eigenvalue forms h = h epτ and α = α epτ are substituted into Eqns. 2.3–2.4

and combined with Eqns. 3.18–3.21 to arrive at following expression. A(p)

h/(αb)

1

= 0 (4.1)

Setting the determinant of the coefficient matrix A to zero determines the aeroelastic

characteristic equation for the typical section model.

A4 p4 + A3 p

3 + A2 p2 + A1 p+ A0 = 0 (4.2)

Table 4.2 identifies these polynomial coefficients for the full unsteady Theodorsen

aerodynamic model, in addition to the standard steady and quasi-steady aerody-

namic models. The steady and quasi-steady approximations include only circulatory

lift based on the instantaneous effective angles of attack of α and (α + h/U), re-

spectively [30]. The eigenvalue solution of Eqn. 4.2 can be computed directly by a

complex root solver for the steady and quasi-steady aerodynamic approximations.

For the general unsteady case, the reduced frequency must be iterated for each value

of reduced velocity until the frequency ratio determined by these values matches the

frequency ratio of the least stable eigenvalue.

Figures 4.1–4.3 plot the real and imaginary parts of the eigenvalue parameter,

p = pR + i(ω/ωα), for each aerodynamic model over a range of reduced velocities.

34

Tab

le4.

2:C

oeffi

cien

tsfo

rty

pic

alse

ctio

nm

odel

eige

nva

lue

anal

ysi

s.

Aer

odynam

icM

odel

Coeffi

cien

tsSte

ady

Quas

i-Ste

ady

Unst

eady

A4

µ(r

2 α−x

2 α)

µ(r

2 α−x

2 α)

µ(r

2 α−x

2 α)+r2 α

+2ax

α

+(1/8

+a

2)+

1/(8µ)

A3

02V

[r2 α

+(a

+1/

2)x

α]

2VC

(k)[r2 α

+a(a

+2x

α)−

1/4−

1/(8µ)]

+V

[1/(

2µ)−x

α+

(1/2−a)]

A2

µr2 α

[1+

(ωh/ω

α)2

]µr2 α

[1+

(ωh/ω

α)2

]µr2 α

[1+

(ωh/ω

α)2

]−

2V2[x

α+

(a+

1/2)

]−

2V2[x

α+

(a+

1/2)

]−

2V2C

(k)[x

α+

(a+

1/2)−

1/(2µ)]

+r2 α

+(1/8

+a

2)(ω

h/ω

α)2

A1

02r

2 αV

2VC

(k)[r2 α

+(a

2−

1/4)

(ωh/ω

α)2

]+V

(1/2−a)(ω

h/ω

α)2

A0

(ωh/ω

α)2

[µr2 α−

2V2(a

+1/

2)]

(ωh/ω

α)2

[µr2 α−

2V2(a

+1/

2)]

(ωh/ω

α)2

[µr2 α−

2V2C

(k)(a

+1/

2)]

35

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−100%

−50%

0%

50%

100%

pR

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

V = 2 U∞/(ωαc)

ω/ω

α

Figure 4.1: V -g plot of HALE typical section model with steady aerodynamics:stable mode (dashed line); unstable mode (solid line).

Each aerodynamic model predicts a different behavior of dynamic instability for the

structural data from Table 4.1. The steady aerodynamic model predicts divergence,

whereas the quasi-steady model is dynamically unstable for all flow velocities. By

contrast, only the complete unsteady model predicts the flutter behavior observed ex-

perimentally for the HALE wing; these flutter results are compared against available

data in Table 4.3. Despite tailoring the typical section analysis to the experimental

wing and its known flutter mode, the reduced frequency and reduced velocity at the

flutter point differ from the experimental values by factors of 2.9 and 0.58, respec-

tively. The flutter frequency is in relatively good agreement with a prediction 16%

lower than experiment.

4.1.2 Conclusions

The unsteady and non-circulatory contributions to lift and moment are necessary to

predict the flutter behavior of the linear typical section representation of the HALE

36

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2−3%

−2%

−1%

0%

1%

2%

3%

pR

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.2

0.4

0.6

0.8

1

V = 2 U∞/(ωαc)

ω/ω

α

Figure 4.2: V -g plot of HALE typical section model with quasi-steady aerodynam-ics: stable mode (dashed line); unstable mode (solid line).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−3%

−2%

−1%

0%

1%

2%

3%

pR

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

V = 2 U∞/(ωαc)

ω/ω

α

Figure 4.3: V -g plot of HALE typical section model with unsteady Theodorsenaerodynamics: stable mode (dashed line); unstable mode (solid line).

37

Table 4.3: Typical section flutter comparison with HALE wing data.

Parameters Present Analysis Tang & Dowell [106]k 0.305 0.105V 2.72 4.72ω/ωα 0.829 0.989h/(αb) 1.127 + 5.852i –

experimental wing. A reduction of the aerodynamic model to steady and quasi-steady

approximations leads to drastically different instability behavior, as corroborated by

other flutter studies, e.g. Ref. [64]. However, the poor numerical agreement between

experiment and the typical section model using Theodorsen unsteady aerodynamics

suggests the need for more sophisticated structural and/or aerodynamic modeling to

represent the HALE wing configuration.

4.2 Hodges-Dowell Nonlinear Beam & ONERA Dynamic Stall Model

4.2.1 Equation Formulation and Time-Marching Scheme

The Hodges-Dowell nonlinear beam equations of Section 2.3 and the ONERA aerody-

namic model in Section 3.3 are combined to create a fully-coupled nonlinear aeroelas-

tic system. This time-marching formulation extends beyond the modeling capacity of

the typical section model to include the effects of beam geometry, nonlinear stiffness,

and nonlinear stall aerodynamics on aeroelastic stability and limit cycle oscillations.

The aeroelastic model is cast into the following set of ordinary differential equa-

tions in time by assuming modal forms of the generalized coordinates and integrating

the equations over the wing span.

38

NC∑p=1

δip(Vp + 2 ξv ω1C Vp + ω2

pCVp

)+ β

NB∑q=1

NT∑r=1

KiqrWqΦr

+Vp ψi(x = 1)

[(M

mL

)ψp(x = 1) +

(IvmL2

)ψ′p(x = 1)

]

= κ1

NAERO∑l=1

∆l ψi(x = xl) [−CDl + (θ0 − φλl)CLl]

+ κ2ψi(x = 1)∆Fv(x = 1)−( gL

sin θ0

) [(M

mL

)ψi(x = 1) + Ψi

](4.3)

NB∑q=1

δiq(Wq + 2 ξw ω1BWq + ω2

qBWq

)+ β

NC∑p=1

NT∑r=1

KpirVpΦr − e

NT∑r=1

EirΦr

+Wq ψi(x = 1)

[(M

mL

)ψq(x = 1) +

(IwmL2

)ψ′q(x = 1)

]

= κ1

NAERO∑l=1

∆l ψi(x = xl) [CLl + (θ0 − φλl)CDl]

+ κ2 ψi(x = 1)∆Fw(x = 1)−( gL

cos θ0

) [(M

mL

)ψi(x = 1) + Ψi

](4.4)

NT∑r=1

δir(Φr + 2 ξφ ω1T Φr + ω2

rT Φr

)+

β

K2

NC∑p=1

NB∑q=1

KpqiVpWq

− e

K2

NB∑q=1

EqiWq +

(Iφ

mK2L

)Θi(x = 1)

NT∑r=1

ΦrΘr(x = 1)

= κ1c

K2

NAERO∑l=1

∆lΘi(x = xl) CMl − (yac/c) [CLl + (θ0 − φλl)CDl]

+ κ3 Θi(x = 1)∆Mx(x = 1) (4.5)

Let q be a state vector defined as

q =Vp, Vp, Wq,Wq, Φr,Φr, (CL2)l, (CL2)l, (CLγ)l

T

, (4.6)

39

such that Eqns. 4.3–4.5 can be expressed as a state-space matrix equation to which

a time-marching scheme can be readily applied. The present work employs the

standard fourth-order Runge-Kutta time-marching scheme [15, 59, 60, 91, 94] using

a fixed time step interval.

[A] q + [B]q = F0 + FN (4.7)

Coefficient matrices [A] and [B] depend on the flow velocity and structural parame-

ters, and the force vectors F0 and FN represent respectively the gravity effects and

the nonlinear forces due to structural and aerodynamic stall nonlinearities [106].

All simulations performed herein use the experimentally obtained data from Ta-

ble 4.4 with a time step of ∆t = 8192−1 s assuming ten spanwise aerodynamic strips,

NAERO = 10. The numbers of assumed modes for the flap, lag, and twist directions

are NC = 1, NB = 4, and NT = 1, respectively. All comparisons between simulation

and experiment assume a representative root angle of θ0 = 1.

4.2.2 Flutter Prediction

Setting the force vectors F0 and FN of Eqn. 4.7 to zero determines the strictly linear

flutter boundary, which does not include the effects of initial conditions, static defor-

mation, or nonlinearity. Figure 4.4 illustrates the sensitivity of the flutter boundary

to both the number of specified aerodynamic panels and the spanwise location on

each of the panels at which the aerodynamic loads are evaluated. As the number of

panels increases, the evaluation location has a weaker influence on the flutter speed.

However, Fig. 4.4 also suggests that an accurate flutter prediction is possible for

only a few panels with a proper choice of evaluation location. The flutter speed

using the 65%-span location agrees well for a number of panels greater than or equal

to two, suggesting that at least two panels are required to describe appropriately the

second-bending flap motion of the flutter and LCO response of the present HALE

40

Table 4.4: Experimental wing model data (adapted from Ref. [106]).

Property Value

WingSpan L 0.4508 mChord c 0.0508 mMass per unit length m 0.2351 kg/mMoment of inertia (50% chord) mK2

m 2.056×10−5 kgmSpanwise elastic axis 50% chordCenter of gravity 49% chordFlap bending rigidity EI1 0.4186 N m2

Lag bending rigidity EI2 18.44 Nm2

Torsional rigidity GJ 0.9539 N m2

Flap structural modal damping ξw 0.02Lag structural modal damping ξv 0.025Torsional structural modal damping ξφ 0.031First flap natural frequency ω1B 3.675 HzSecond flap natural frequency ω2B 23.03 HzThird flap natural frequency ω3B 64.50 HzFourth flap natural frequency ω4B 126.3 HzFirst lag natural frequency ω1C 24.39 HzFirst torsional natural frequency ω1T 119.5 Hz

Slender BodyRadius R 4.762×10−2 mChord length cSB 0.1406 mMass M 0.0417 kgMoment of inertia Iw 0.9753×10−4 kgm2

Moment of inertia Iv 0.3783×10−4 kgm2

Moment of inertia Iφ 0.9753×10−4 kgm2

Volume –V1 1.001×10−5 m3

Volume static unbalance –V2 0.0 m4

Volume moment of inertia –V3 1.64×10−8 m5

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

22

24

26

28

30

32

34

36

38

40

Normalized Evaluation Location on Panel

UF

[m/s]

12451020

Figure 4.4: Dependence of flutter speed on the number of panels and the spanwiseaerodynamic evaluation location on each panel for the HALE wing configuration.Assumes the ONERA lift model coefficients from Tang & Dowell [106].

model. Also, the agreement of flutter speeds at the 65%-span location of each panel

is the equivalent ‘3/4 span rule’ typically used to reduce beam flutter problems to

equivalent typical sections.

The observed second-bending/first-torsion coalescence flutter mode is predicted

using the linear beam model with linear unsteady aerodynamics. Also, the converged

flutter speed of UF = 34.5m/s in Fig. 4.4 is in good agreement with the experimental

value of UF = 33.5m/s. Therefore, the simplest model required to make a useful

flutter prediction for a HALE-type wing must include beam geometry and fully

unsteady, linear aerodynamics.

Figure 4.5 compares the source data used for ONERA lift model identification

with respect to their impact on the flutter speed. Despite the close numerical agree-

ment between the ONERA models based on wind tunnel or CFD unsteady data (cf.

42

2.8 3 3.2 3.4 3.6 3.8 4 4.234

35

36

37

38

39

40

a0L · αL

UF

Wind Tunnel DataCFD Data

Figure 4.5: Sensitivity of flutter speed to source data for the linear ONERA liftmodel and to the product of parameters a0L and αL.

Table B.1), the flutter speed is notably sensitive to the product of the two parameters

with the largest variation: the static lift curve slope, a0L; and the unsteady ONERA

parameter αL. The product of these terms varies from 2.83 to 3.45 to 4.2 for the

linear ONERA models considered by Dunn [32, 33, 34], Tang & Dowell [106], and the

CFD-based model identified in the present work. The sensitivity to these parameters

yields a flutter speed variation of up to 14% relative to the observed flutter speed.

The cumulative effect of the differences between the rest of the other linear ONERA

coefficients in Table B.1 amounts to a much smaller variation in flutter speed of

nearly a percent.

4.2.3 Limit Cycle Oscillations

Flutter can lead to an exponential growth of unsteady motion into a limit cycle os-

cillation, which by definition must embody a nonlinear feature in the structure, the

aerodynamic flow, or both. The sensitivity of the computed limit cycle oscillation

43

amplitude and hysteresis to these nonlinearities and to the source data of the ON-

ERA dynamic stall model is compared against experimental results from Ref. [106].

Each simulated limit cycle analysis begins at a sufficiently low flow velocity such

that the wing achieves a static steady state, after which the velocity is increased by

∆U = 0.1m/s. The aeroelastic model simulates 200 seconds of time at each velocity,

and the root-mean-squared (rms) amplitude of the unsteady motion is determined

by data from the final 10 seconds. The flow velocity increases until a critical veloc-

ity is reached, beyond which a numerical or possibly a physical divergence occurs

in the theoretical model, which is supported by observations for the present aeroe-

lastic model in previous researches, e.g. [106, 109]. The simulated flow velocity is

then decreased incrementally until the limit cycle oscillation disappears and a static

aeroelastic state is recovered.

Figure 4.6 investigates the effect of structural nonlinearity arising from elastic

coupling between the the flap, lag, and twist motions. The aeroelastic model used

by Tang & Dowell [106] including both structural and aerodynamic stall nonlineari-

ties is in relatively good agreement with experiment with respect to LCO amplitude

and hysteresis. Without the structural nonlinearity, the limit cycle amplitude curve

is reversible with respect to flow velocity. Therefore, the existence of hysteresis is

a direct consequence of structural nonlinearity arising from elastic coupling. Also,

the effect of structural nonlinearity increases the limit cycle amplitude at the flut-

ter point. However, elastic coupling leads to an anticipated stiffening effect and

comparatively lower LCO amplitudes at higher flow velocities.

Figure 4.7 shows the impact of the aerodynamic model on the flutter and limit

cycle oscillation behavior of the HALE wing. The ONERA model used by Tang &

Dowell [106] agrees well with experiment with respect to flutter speed, hysteresis

bandwidth, and LCO amplitude; note that the flutter speed for the statically de-

formed wing is only slightly lower than the speed predicted by the strictly linear

44

30 32 34 36 38 40 420

5

10

15

20

25

Flow Velocity, U [m/s]

Mid

span

rms

Am

plitu

de

[mm

]

Without Nonlinear StiffnessWith Nonlinear StiffnessExperiment

Figure 4.6: Comparison of computational results with and without nonlinear beamstiffness, including experimental data from Ref. [106]. Solid/dashed lines denoteincreasing/decreasing flow velocity.

calculation. When the linear ONERA lift coefficients based on wind tunnel data are

replaced with those identified by CFD data in Table B.1, the flutter speed increases

from 34.3m/s to 39.8m/s and the predicted LCO amplitude is roughly 20% larger.

Thus, the linear aerodynamic coefficients are significant for quantitative accuracy of

both flutter speed and LCO amplitude. The non-smooth bifurcation curve for this

aeroelastic response is due to the assumption of a sinusoidal response at a single fre-

quency in the rms amplitude calculation. Figure 4.8 depicts the unsteady oscillation

at U = 40.5m/s and its frequency spectrum, which includes low-frequency content

that effects the rms amplitude computation.

Conversely, an ONERA model is also constructed from the linear ONERA coef-

ficients from Ref. [106] and the nonlinear aerodynamic terms and static stall curve

45

identified by CFD. Because the nonlinear ONERA coefficients of the wind tunnel and

CFD models are virtually identical, a comparison of the nonlinearity in these two

models is effectively a comparison of their static lift curves. Recall that these static

lift curves prescribe the forcing function for the nonlinear component of the ONERA

lift model. The new prediction for LCO amplitude also increases but with a lesser

sensitivity of the amplitude to the flow velocity. The hysteresis bandwidth doubles

for the new nonlinear model in comparison to the original aeroelastic computation,

as it did for the new linear coefficients. Therefore, the aerodynamic nonlinearity

due to dynamic stall, which for the ONERA model also depends on its linear coeffi-

cients, and the structural geometric nonlinearity are both important in quantifying

hysteresis behavior.

Both modified aerodynamic models, identified as Case 1 and Case 2 in Fig. 4.7,

hinder the ability to simulate limit cycle oscillations at flow velocities much greater

than the flutter speed. When using the ONERA aerodynamic model with both linear

and nonlinear parts identified by CFD data, the simulation diverges at the flutter

speed. The sensitivity of the aeroelastic model based on the Hodges-Dowell equations

and ONERA dynamic stall aerodynamics is noted in the literature [106, 109], where

small parametric variations in the flow scenario can lead to divergent behavior that

may be rooted in either numerical or physical instability. It should also be noted that

the limit cycle calculations shown in Figs. 4.6 and 4.7 are strictly divergent when the

aerodynamic nonlinearity is removed.

The divergent nature of the nonlinear aeroelastic system is further investigated by

considering a quasi-steady approximation of the aerodynamic model. For this model

the aerodynamic lift is computed from the static lift curve and the effective angle

of attack, which includes the instantaneous incidence angle and the quasi-steady in-

flow angle. Figure 4.9 compares the bifurcation curves of the ONERA models based

on wind tunnel of CFD static lift data. The flutter speeds and resulting LCO am-

46

plitudes are much smaller than the results predicted by unsteady aerodynamics and

experimental observation. Hysteresis is also observed for quasi-steady aerodynamics;

however, the bandwidth is narrower, noting the difference in scales in comparison

with Fig. 4.7. The bifurcation curves for the quasi-steady cases can be extended to

greater flow velocities beyond the flutter point than for the comparable unsteady

computations, up to 48.9 and 50.0m/s for the CFD- and wind-tunnel-based models,

respectively. When the aerodynamic nonlinearity due to stall is removed, both quasi-

steady model simulations diverge as also observed for the unsteady cases. Therefore,

the present aeroelastic model is exquisitely sensitive to input parameter combinations

and suggests that a stable limit cycle oscillation is not possible without the presence

of the aerodynamic nonlinearity associated with dynamic stall. Moreover, the aeroe-

lastic simulations indicate that unsteadiness in the aerodynamic model is essential

to the accurate flutter and LCO prediction of HALE wings, but this unsteadiness

may also be the source of the divergent computational behavior for post-flutter cal-

culations.

4.2.4 Time Estimates for First-Principles Aeroelastic Model

Application of the ONERA dynamic stall model to aeroelastic analyses enables rapid

time-marching and stability computations. For example, the simulated bifurcation

curves for Figs. 4.6 and 4.7 require computational times of less than half an hour

on a single processor. However, as with other reduced-order models, the ONERA

nonlinear lift model is limited by the data used for its parameter identification and

by the ability of the model equations to approximate this data appropriately over

a range of flow situations. Also, the ONERA model represents only the integrated

load on a particular airfoil section and does not provide any information about the

physics of the unsteady flow field. Thus, a natural extension of the present work is

to couple the Hodges-Dowell nonlinear beam equations instead with a CFD solver

47

3032

3436

3840

42051015202530

Flo

wV

eloci

ty,U

[m/s]

MidspanrmsAmplitude[mm]

E

xper

imen

t:U

↑E

xper

imen

t:U

↓T

ang

&D

owel

l(2

001)

:U

↑

Tan

g&

Dow

ell(2

001)

:U

↓C

ase

1:U

↑C

ase

1:U

↓C

ase

2:U

↑C

ase

2:U

↓

Fig

ure

4.7

:B

ifurc

atio

ndia

gram

com

par

ison

offu

lly

non

linea

rco

mputa

tion

alre

sult

san

dex

per

imen

tal

dat

afr

omR

ef.

[106

].C

ases

1/2

den

ote

ON

ER

Aae

rodynam

icm

odel

susi

ng

linea

rpar

amet

ers

bas

edon

CFD

/win

dtu

nnel

dat

aan

da

non

linea

rst

atic

lift

curv

efr

omw

ind

tunnel

/CFD

resu

lts.

48

0 5 10 15 20−50

−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

Mid

span

Am

plitu

de

[mm

]

(a)

0 5 10 15 20 25 300

5

10

15

20

25

Frequency [Hz]

Am

plitu

de

[mm

]

(b)

Figure 4.8: Limit cycle oscillations at U = 40.5 m/s for aeroelastic model usinglinear ONERA coefficients based on CFD data and a nonlinear static lift curve fromRef. [106]: (a) time series; (b) frequency spectrum.

49

20 20.5 21 21.5 22 22.5 230

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Flow Velocity, U [m/s]

Mid

span

rms

Am

plitu

de

[mm

]

Wind TunnelCFD

Figure 4.9: Bifurcation diagram comparison of quasi-steady ONERA aerodynamicmodels using static lift curves based on wind tunnel or CFD data.

to create a complete first-principles aeroelastic model to gain greater insight into the

nonlinear dynamics of HALE wings.

The tradeoff for this improvement in the aerodynamic model is a large increase in

the computational cost required to determine a stable limit cycle oscillation, as well as

the long transients of the fluid-structure system near the flutter point. Using the CFD

code OVERFLOW, a parametrically-converged period of LCO for a two-dimensional

section of the experimental HALE wing requires 26.5 hours of computational time

for a limit cycle frequency of fLCO = 21.5Hz at a reduced frequency of k = 0.1. The

present work and Ref. [106] suggest that nearly 15 simulated seconds are necessary

to capture the transition from flutter to a stable limit cycle at the flutter speed,

which would take nearly a year of computational time to compute a single point

on the bifurcation diagram. By using results from the present work to prescribe

50

initial conditions for a limit cycle, the required simulated time could be reduced

to one or two seconds to capture a stable limit cycle oscillation using 47.5 days of

computational time for each value of the flow velocity.

These computational costs may be reduced by roughly an order of magnitude

using the harmonic balance technique developed by Hall et al. [49], which was recently

extended to convert existing time-domain codes to the frequency domain [23, 24,

114]. A first-principles aeroelastic model in the frequency domain would inform

the companion time-marching analysis to facilitate rapid agreement between the

two models. Thomas et al. [115, 116] have demonstrated the ability of a harmonic

balance CFD solver to predict accurate limit cycle oscillations for the linear typical

section in nonlinear transonic flow. An extension of this work to nonlinear dynamic

stall aerodynamics at low subsonic Mach numbers for a nonlinear beam model would

create the first fully nonlinear aeroelastic model for the HALE configuration based

entirely on first-principles physics.

51

5

Conclusions

The objective of this research is to perform predictive aeroelastic analyses for the

flutter and limit cycle oscillation behaviors of flexible wing representative of HALE

aircraft using a range of structural and aerodynamic models. These models include

the effects of beam geometry, nonlinear structural stiffness due to elastic coupling

between the degrees of freedom, and nonlinear aerodynamics associated with dynamic

stall. A new ONERA dynamic stall model is identified from steady and unsteady

data computed by the CFD code OVERFLOW, which is compared to the standard

model based on wind tunnel data. The range of structural and aerodynamic models

considered enables a gradual increase in the sophistication of the aeroelastic model to

determine which modeling features are essential to the prediction of flutter and LCO

behaviors for HALE wings via correlation with the experimental data of Ref. [106].

The present work also establishes the roles of structural and aerodynamic nonlinearity

on the aeroelastic system and the impact of aerodynamics based on first-principles

rather than wind tunnel data.

The non-uniform design of the HALE experimental wing is first analyzed us-

ing finite element analysis to quantify the effects of periodically-spaced elements of

52

various materials, material removal, and the presence of a tip store on the natural fre-

quencies and mode shapes. The natural frequencies of interest are affected by these

non-uniformities, and these results are reduced to design curves for the construction

of similar HALE aeroelastic wings. However, the mode shapes are well-approximated

by uniform beam modes, whose agreement improves with the addition of the tip store.

This result demonstrates the appropriate use of uniform beam modes to approximate

nonuniform wings with a beam-like structure, a result which is typically assumed to

be true in aeroelastic analyses similar to the present work. Moreover, the valid-

ity of continuous beam modes enables their use as trial functions for the Galerkin

projections of continuum beam models, such as the Hodges-Dowell nonlinear beam

equations, to describe the HALE wing structure for aeroelastic analyses.

A typical section aeroelastic model is first considered to make a posteriori es-

timates of the flutter speed using representative structural data and flutter mode

information from experiment. This system describes the lift and moment with stan-

dard steady, quasi-steady, and full unsteady Theodorsen aerodynamics models. Each

aerodynamic model predicts a different instability behavior. Only the unsteady

Theodorsen aerodynamic model is able to identify correctly the flutter instability

but with noticable disagreement between the predicted flutter speed and mode re-

sults and the experimental results. An accurate predictive model for HALE wing

flutter is found to require the additional feature of beam geometry, which is modeled

by the linear beam reduction of the Hodges-Dowell equations in conjuction with the

Theodorsen-like linear aerodynamics of the ONERA aerodynamic model.

Flutter instability leads to limit cycle oscillations for the cantilevered aeroelas-

tic wing considered in this work. The nonlinear aeroelastic model, comprised of the

Hodges-Dowell nonlinear beam equations and the ONERA dynamic stall model, sug-

gests that the aerodynamic nonlinearity due to dynamic stall is essential to predict

a stable limit cycle oscllation. Both the structural and aerodynamic nonlinearities

53

effect the hysteresis bandwidth of the limit cycle motions, but the hysteresis vanishes

when the structural nonlinearity is removed from the aeroelastic model. The nonlin-

ear elastic coupling is identifed as the key feature necessary to model the hysteresis

of HALE wing LCO and is the mechanism by which the aerodynamic nonlinearity

effects this hysteresis bandwidth.

The ONERA dynamic stall models identified by first-principles CFD data or wind

tunnel data compare the influence of aerodynamic source information on both flutter

and LCO. The standard model based on wind tunnel data is in excellent agreement

with experiment with respect to flutter speed, LCO amplitude, and hysteresis band-

width [106]. When the linear coefficients of this model are changed to those of the

CFD model, the flutter speed increases by 16% and the predicted LCO amplitude

increases by roughly 20%. A more complex frequency spectrum is observed for this

model, which includes many frequencies lower than the predominant single limit

cycle frequency found in other simulations. The dynamic stall contribution of the

CFD-based model is found to increase the LCO amplitude and hysteresis bandwidth

similarly as compared to the wind tunnel model. The close agreement between the

nonlinear ONERA coefficients for the wind tunnel and CFD models indicates that

the principal difference between dynamic stall behavior of the two models is due to

the static stall characteristics that drive the nonlinear lift dynamics.

The aeroelastic model of the Hodges-Dowell nonlinear beam equations and the

semi-empirical ONERA dynamic stall model based on CFD data is intended to

approximate and anticipate the effects of nonlinear, first-principles, computational

aeroelastic modeling for a simple HALE wing configuration. The present work may

used as a benchmark for more complex geometrical configurations and complete first-

principles analyses where the aerodynamic loads are computed by a CFD solver.

54

5.1 Future Work

The present work invites a number of natural research extensions to improve the

fidelity of the aeroelastic computation and the realism of the modeling effort for

modern HALE wing configurations.

First, the structural model may be modified to include the rigid body modes,

which are known to be important to the gust response of slender wings. A compan-

ion wind tunnel test with a cantilevered wing mounted on a tunable spring would

account for the symmetric aeroelastic effects of a HALE wing in the limit of zero

spring stiffness, which is an attractive alternative to a much more difficult free-flying

experimental setup.

Second, flow field measurements using techniques such as particle image velocime-

try (PIV) may be used to capture the dynamic interaction of the elastic wing with

vortex generation and dynamic stall events. This investigation would extend the

modest amount of published work regarding the distinction between pitching and

plunging motions for large unsteady motions, for which much more is known than

about the flow dynamics for a wing structure in a limit cycle oscillation. Experimen-

tal LCO flow field measurements would also provide benchmark data for aeroelastic

analyses using a CFD code.

And third, the present analysis may be extended to a fully-coupled aeroelastic

analysis using the existing nonlinear beam model with a computational fluid dynam-

ics solver. This may be implemented with a time domain [37, 62] or frequency domain

code [23, 24, 49, 115, 116], the latter of which may offer a significant reduction in

computational cost by avoiding the long transients of aeroelastic simulations near

the flutter point. It should be noted that the flutter speed observed for the experi-

mental wing by Tang & Dowell [106] occurs at Re = 1.15× 105, which is an order of

magnitude smaller than the Reynolds numbers associated with the data used for the

55

ONERA model parameter identification. The Reynolds number of the experiment

is sufficiently low such that transitional flow effects may become important, and a

suitable CFD code may elucidate these effects on the dynamic stability of flexible

wings to provide further insight into the complex fluid-structure interactions that

may be encountered by HALE aircraft.

56

Appendix A

Time-Domain Computational Fluid DynamicsSolver

The computational fluid dynamics code OVERFLOW 2.1 was developed by Nichols

and Buning [74] at NASA Langley Research Center to enable high-fidelity, unsteady

aerodynamic modeling for complex geometries using Chimera overset grid techniques.

OVERFLOW performs implicit time-domain analyses and employs a suite of dis-

cretization schemes and turbulence models to suit the needs of a particular flow

situation. Custer [23] reviewed recently the development of OVERFLOW and ex-

tended its capabilities to nonlinear frequency-domain analysis, including the use of

overset grids for sufficiently small motions [24, 114].

The present work uses OVERFLOW in its original time-domain form to com-

pute unsteady aerodynamic loads for prescribed motion at large angles of attack.

This Appendix summarizes the fundamental equations and assumptions made in

the computational model to identify a new ONERA dynamic stall model based on

first-principles aerodynamic data obtained from OVERFLOW.

57

A.1 Fluid Equations

The compressible, three-dimensional Navier-Stokes equations are presented in strong

form as a result of conserving mass, momentum, and energy for an arbitrary volume

of a calorically-perfect, Newtonian gas.

The mass conservation expression for a fluid, or continuity equation,

Dρ

Dt+ ρ(∇ ·V) = 0, (A.1)

relates the time rate of change of the density of a fluid particle to the divergence of the

velocity field, where this divergence can be interpreted physically as the volumetric

time rate of change of a moving fluid element [4].

Cauchy’s equation of motion represents the conservation of momentum for any

continuum substance subjected to an arbitrary distribution of body forces and surface

tractions [2].

ρDV

Dt= ρ f +∇ ·Π (A.2)

The properties of a Newtonian fluid are invoked to establish a constituitive relation-

ship for the stress tensor, Π. By definition, a fluid cannot sustain a shear stress at

rest [39], and although fluids can resist shear, they are unable to withstand a defor-

mation [96]. Furthermore, a Newtonian fluid assumes a linear relationship between

stress and the rate-of-strain. These simplifying assumptions allow for the stress ten-

sor to be separated into a hydrostatic pressure (normal stress) term and a deviatoric

stress component to account strictly for viscous shear [11].

Πij = −p δij + τij (A.3)

The isotropic fluid assumption [6, 65] simplifies the deviatoric stress tensor, τ , and

further reduces the constituitive description for a Newtonian fluid to two independent

58

fluid coefficients: the dynamic viscosity, µ, and the second (dilitational) coefficient

of viscosity, λ [43].

τij = µ (ui,j + uj,i) + λ δij(∇ ·V) (A.4)

The quantities µ and (λ + 2µ/3) must be non-negative to satisfy the second law of

thermodynamics [43]. Stokes [100] assumed the relationship (λ+ 2µ/3) = 0 to force

the mechanical pressure to be equal to the thermodynamic pressure, which differ for

nonequilibrium thermodynamic processes. However, the Stokes hypothesis holds for

most flow investigations of aeronautical interest. The result of the hypothesis on the

fluid equations is exact for inviscid flows and incompressible flows, and valid within

boundary layer approximations where the normal viscous stresses are much smaller

in comparison to the shear stresses [43]. The application of Stokes’s hypothesis to

Eqn. A.4 enables Eqn. A.2 to be rewritten as the famous Navier-Stokes equation [111].

ρDV

Dt= ρ f −∇p +

∂

∂xj

[µ

(∂ui

∂xj

+∂uj

∂xi

)− 2

3δijµ

∂uk

∂xk

](A.5)

Energy conservation can be expressed simply by direct application of the first law

of thermodynamics to a fixed infinitesimal control volume [111].

∂Et

∂t+∇ · (EtV) =

∂Q

∂t−∇ · q + ρ f ·V +∇ · (Π ·V) (A.6)

The total energy per unit volume is expressed as Et = ρ[e + (V · V)/2]. The two

terms on the left-hand side of Eqn. A.6 denote respectively the time rate of change

of total fluid energy per unit volume, and the convection of heat energy away from

the control volume. The first term on the right-hand side (RHS) represents heat

generation within the fluid, for example, from electrical power dissipation [12]. The

second RHS term describes the conduction of heat of away from the control surface,

which can be expressed in terms of temperature by Fourier’s law, q = −k∇T . The

remaining two terms denote the increase in total energy due to body forces and

59

surface forces, respectively. A final energy equation follows by incorporating Fourier’s

law, Eqn. A.4, and Stokes’s hypothesis into Eqn. A.6.

ρDe

Dt=∂Q

∂t+∇ · (k∇T )− p(∇ ·V) + Φ (A.7)

The viscous dissipation function, Φ, is defined as

Φ = ∇ · (τ ·V)− (∇ · τ ) ·V. (A.8)

Lastly, an equation of state is required to render the fluid equations well-posed

with respect to the number of equations and unknown variables. A calorically-perfect

gas obeys the expression

p = ρRT, (A.9)

with constant-valued specific heats. The calorically-perfect gas assumption allows

Eqn. A.9 to be written alternatively as p = (γ − 1)ρe or T = (γ − 1)e/R, where

γ ≡ cp/cv and cp = γR/(γ − 1).

The fluid equations from Eqns. A.1, A.5, A.7, and A.9 combine into a compact

vector form that enables the direct application of finite differencing scheme. Thus, the

compressible Navier-Stokes equations in a Cartesian coordinate system are expressed

as [111]

dU

dt+dE

dx+dF

dy+dG

dz= 0, (A.10)

60

where

U =

ρρuρvρwEt

(A.11)

E =

ρu

ρu2 + p− τxx

ρuv − τxy

ρuw − τxz

(Et + p)u− uτxx − vτxy − wτxz + qx

(A.12)

F =

ρv

ρuv − τxy

ρv2 + p− τyy

ρvw − τyz

(Et + p)v − uτxy − vτyy − wτyz + qy

(A.13)

G =

ρw

ρuw − τxz

ρuv − τyz

ρw2 + p− τzz

(Et + p)w − uτxz − vτyz − wτzz + qz

(A.14)

The Cartesian vector form of the Navier-Stokes equations must be converted to a

generalized coordinate system that will allow for the computation of aerodynamic

flows on complex grids. Figure A.1 shows the computational grid for the NACA

0012 airfoil considered in this work, which was designed specifically for unsteady

aerodynamic analyses by clustering high-resolution regions at the leading and trailing

edge region where large gradients are expected. The physical domain of Fig. A.1 can

be transformed to a regular computational domain in a generalized coordinate system

(ξ,η,ζ).

ξ = ξ(x, y, z) (A.15)

η = η(x, y, z) (A.16)

ζ = ζ(x, y, z) (A.17)

61

(a) (b)

Figure A.1: Physical C-grid for NACA 0012 airfoil with 401×75 resolution and19c outer radius: a) far view; b) near view.

Eqns. A.10–A.14 can be rewritten in the generalized coordinate system of the com-

putational domain by the chain rule. The Jacobian for this transformation is defined

as

J =∂(ξ, η, ζ)

∂(x, y, z)=

∣∣∣∣∣∣ξx ξy ξzηx ηy ηz

ζx ζy ζz

∣∣∣∣∣∣ , (A.18)

where the transformation metrics are computed numerically [38]. The transformed

fluid equations retain the original vector form of Eqn. A.10 with modest changes to

the vector expressions [111].

dU

dt+dE

dx+dF

dy+dG

dz= 0, (A.19)

62

U =1

JU (A.20)

E =1

J(ξxE + ξyF + ξzG) (A.21)

F =1

J(ηxE + ηyF + ηzG) (A.22)

G =1

J(ζxE + ζyF + ζzG) (A.23)

These equations enable the direct application of finite-differencing schemes for fixed

grid geometries. Deforming grid geometries require further consideration of the trans-

formation metrics by the geometric conservation law [117, 118] to be consistent and

prevent errors in the solution. All unsteady aerodynamic computations were per-

formed using solid body rotation and translation of the physical grid.

In the present work, the temporal and spatial derviatives are represented by

the Symmetric Successive Over-Relaxation (SSOR) Unfactored Method [75] and the

HLLC upwind method [119], respectively, which have proven to be robust algorithms

within OVERFLOW for large unsteady motions over a range of reduced frequencies.

A.2 Reynolds Averaging

The use of direct simulations of the Navier-Stokes equations for practical engineering

flows is precluded by the phenomenon of turbulence. Turbulence is inherently three-

dimensional and is characterized by a broad and continuous spectum of length scales.

Larger turbulent eddies break down and transfer their kinetic energy to smaller

eddies in a cascading process that continues until the energy smallest eddy dissipates

into heat through the action of molecular viscosity [122]. The size of the smallest

eddy sets the minimum length scale for a physical grid for direct Navier-Stokes

(DNS) simulations. The smallest eddy is much larger than the molecular length

scale [112] but is extremely small compared to the greater computational grid and

63

solid bodies of interest. However, most aerodynamic flows of engineering interest

do not require the turbulent structure to be fully resolved to provide useful and

accurate results. Computations for high Reynolds number flows typical of aerospace

applications focus on the impact of turbulent flow on the force distribution over

a solid body. Comparatively rapid numerical solutions rely on averaged forms of

the Navier-Stokes equations to reduce the computational dependence on predicting

turbulent structures in favor of approximating the gross effects of turbulence on the

flow field.

Reynolds [92] approached turbulence by time-averaging the Navier-Stokes equa-

tions to eliminate all unsteadiness under the premise that all unsteadiness is due to

turbulence [38]. For a statistically steady flow, every variable can be expressed as a

time-averaged value and a time-dependent fluctuation about that value.

uj(x, t) = uj(x) + u′j(x, t) (A.24)

uj(x) ≡ lim∆t→∞

1

∆t

∫ t0+∆t

t0

uj(x, t) dt (A.25)

This averaging concept can be extended to unsteady flows by instead considering

an ensemble average of the desired variable, where the amount of ensemble data is

sufficiently large to eliminate the effects of the fluctuations [38].

uj(x, t) = limN→∞

1

N

N∑n=1

uj(x, t) (A.26)

Both methods in Eqns. A.25 and A.26 are referred to as Reynolds averaging, which

when applied to the fluid equations results in the Reynolds-Averaged Navier-Stokes

(RANS) equations [111].

∂ρ

∂t+

∂

∂xj

(ρuj + ρ′u′j) = 0 (A.27)

64

∂

∂t(ρui+ρ′u′i)+

∂

∂xj

(ρuiuj+uiρ′u′j) = − ∂p

∂xi

+∂

∂xj

(τij−ujρ′u′i−ρu′iu′j−ρ′u′iu′j) (A.28)

∂

∂t(cpρT + cpρ′T ′) +

∂

∂xj

(ρcpT uj) =

∂p

∂t+ uj

∂p

∂xj

+ u′j∂p′

∂xj

+∂

∂xj

(k∂T

∂xj

− ρcpT ′u′j − cpρ′T ′u′j

)+ Φ (A.29)

Note that the internal heat generation has been dropped from Eqn. A.29, which is

neglected for most fluid analyses of aerodynamic systems. The ensemble-averaged

deviatoric stress and viscous dissipation tensors are defined by Eqns. A.30 and A.31.

τij = µ

[(∂ui

∂xj

+∂uj

∂xi

)− 2

3δij∂uk

∂xk

](A.30)

Φ = τij∂ui

∂xj

+ τ ′ij∂u′i∂xj

(A.31)

The averaging process to formulate the RANS equations generates additional un-

knowns dependent on turbulent fluctuations for which there are no additional equa-

tions. Boussinesq [14] first approximated these Reynolds stress terms by introducing

the concept of turbulent eddy viscosity, which enables the effects of turbulence to be

modeled by a single laminar-like molecular viscosity.

− ρu′iu′j = µT

(∂ui

∂xj

+∂uj

∂xi

)− 2

3δij

(µT∂uk

∂xk

+ ρk

)(A.32)

The Boussinesq eddy viscosity assumption reduces the closure problem for the RANS

equations to a calculation of an effective turbulent viscosity, µT , to be determined

by a turbulence model.

A.3 Turbulence Modeling

Prandtl [89] developed the first model for the turbulent eddy viscosity based on his

mixing-length hypothesis, which used a scaling argument to relate mathematically

65

the momentum transfer of a turbulent eddy to an equivalent viscosity (µT ) or diffu-

sivity (νT = µT/ρ). The mixing length itself is unknown and varies from one type of

flow to another [12], thus relying on experimental data to calibrate the turbulence

model for each flow situation. Algebraic or zero-equation turbulence models such

as the mixing-length model are computationally attractive as they do not require

additional transport equations to be solved along with the conservation equations

for mass, momentum, and energy [122]. Prandtl later developed a physically more

realistic mathematical turbulence model where the turbulent stresses depended on

the kinetic energy of the turbulent fluctuations [90]. The addition of a transport

equation for the turbulent kinetic energy accounts for the fact that eddy viscosity is

dependent upon flow history. Turbulence models of this type are referred to as one-

equation models, which like the algebraic models are inherently incomplete because

they do not predict the turbulent length scale. In other words, something must be

known a priori about the flow other than initial and boundary conditions to ob-

tain a solution [122]. However, modern zero- and one-equation turbulence models

have proven to be both popular and useful for computational simulations of many

engineering applications.

For the present work, the Spalart-Allmaras one-equation turbulence model [98]

was selected for use in computational simulations with OVERFLOW. This selection

was based on good agreement between dynamic stall computations and available

experimental data, which was corroborated by computational-experimental compar-

isons in the literature, e.g. Ref. [99]. However, it should be noted that the Spalart-

Allmaras and other turbulence models are calibrated with steady flow data, and their

application to unsteady flow phenomena has been moderately successful although not

warranted rigorously by the model assumptions.

The Spalart-Allmaras turbulence model without trip line specification obeys the

66

transport equation,

Dν

Dt= σ−1∇ · [(ν + νT )∇ν] + cb2|∇ν|2+ cb1Sν − cw1fw

(ν

d

)2

, (A.33)

where the kinematic eddy viscosity, νT , is related to the Spalart-Allmaras variable,

ν, by νT = νfv1. Here it is important to note the distinction between the kine-

matic eddy viscosity, a theoretical construction from the Boussinesq approximation,

and the molecular viscosity, which is a fluid property [12]. The closure coefficients

and auxiliary relations necessary to complete the turbulence model are given by

Eqns. A.34–A.35 and Eqns. A.36–A.38, respectively.

cb1 = 0.1355, cb2 = 0.622, cv1 = 7.1, σ = 2/3 (A.34)

cw1 = cb1/κ2 + (1 + cb2)/σ, cw2 = 0.3, cw3 = 2, κ = 0.41 (A.35)

fv1 = χ3/(χ3 + c3v1), fv2 = 1− χ/(1 + χfv1), fw = g[

1+c6w3

g6+c6w3

]1/6(A.36)

χ = νν, g = r + cw2(r

6 − r), r = ν/(Sκ2d2) (A.37)

S = S + fv2ν/(κ2d2), S =

√2ΩijΩij, Ωij = 1

2( ∂ui

∂xj− ∂uj

∂xi) (A.38)

The tensor Ω is the fluid rotation tensor, and d is the distance to the closest surface.

67

Appendix B

ONERA Model Parameter Identification

Parameter identification of the ONERA model coefficients began largely as a trial-

and-error effort to determine which values for the assumed model form were well-

behaved and embodied the gross features of lift hysteresis loops (cf. Refs. [25, 85,

120]). McAlister et al. [69] systematized the identification process by linearizing

the ONERA equations about a steady-state angle of attack, at which the flow field

could be either linear or nonlinear. A large set of data for small-amplitude, har-

monic pitching motions about the quarter-chord location, performed over a range

of reduced frequencies and mean angles of attack, was then used both to identify

the ONERA coefficients computationally and to refine the formulation of the model.

Petot [86] later discovered that fewer experimental data sets were necessary to iden-

tify the model coefficients when using large-amplitude hysteresis loops. This method

determines the linear and nonlinear coefficients together and is most efficient for

data sets where the airfoil goes in and out of dynamic stall over a period of mo-

tion [3]. Alternatively, one could use aerodynamic data sets that are either only

stalled or unstalled to determine independently the nonlinear and linear coefficients,

68

respectively.

The present work employs Petot’s method [86] to create a new set of ONERA

parameter coefficients based on first-principles aerodynamic data computed by a com-

putational fluid dynamics (CFD) code. The CFD code OVERFLOW described in

Appendix A computes the aerodynamic forces on an isolated NACA 0012 section un-

dergoing large-amplitude motion representative of the experimentally-observed limit

cycle oscillation of the HALE aeroelastic wing [106]. From these CFD data and pa-

rameter identification, the salient features of the aerodynamic data are highlighted

and interpreted with respect to the sensitivity of the ONERA model to its provided

data, and to the changes in the flutter (linear) and LCO (nonlinear) behaviors of the

aeroelastic model.

B.1 Static Lift Curve

Parameter identification of the ONERA model begins with the static lift curve, which

determines the forcing function ∆CL for the dynamic stall aerodynamics. Static lift

CFD computations were performed at M = 0.3 and Re = 3.66×106 to be consistent

with the experimental data used to identify the original ONERA coefficients for

the NACA 0012 section [70]. These computational results were approximated by a

piecewise continuous curve with linear, parabolic, and exponential decay segments,

which are compared against the original CFD data in Fig. B.1.

(CL)CFD =

6.2486α 0 ≤ αdeg < 11.18

−62.43 (α− 0.24514)2 + 1.3755 11.18 ≤ αdeg ≤ 18.01

0.75715 e−11.412 (α−0.31435) + 0.31931 αdeg > 18.01(B.1)

Tang & Dowell [106] used a piecewise linear curve with discontinuous slopes to de-

scribe the static lift curve for aeroelastic computations with the ONERA model

69

originally based on wind tunnel test data.

(CL)TD =

2πα 0 ≤ αdeg < 10

1.096− 0.313 [αdeg − 10] αdeg > 10(B.2)

Both static lift curves agree well in the unstalled portion, with slope values near

the thin airfoil result of 2π. Abbott & von Doenhoff [1] corroborate this result

experimentally for the NACA 0012 section, whose additional lift due to its thickness

is balanced by the lift reduction due to boundary layer effects on the shed vorticity

necessary to satisfy the Kutta condition [63].

The static lift curves of Eqns. B.1 and B.2 also agree well with the inception

of stall at αss = 11.2 and 10, respectively, but differ for stalled flow. Figure B.2

compares the force deficit for Eqns. B.1 and B.2 against the discrete CFD steady

data and experimental data from Ref. [1], which was measured at Re = 3.00 × 106

and corrected for M ≈ 0. The CFD and experimental data suggest that ∆CL is small

and nonzero within the linear portion of the lift curve, where ∆CL = 0 is an assumed

feature of the approximate static lift curves. Also, the CFD and experimental data

indicate a smooth transition to stalled flow for the NACA 0012 section for the given

conditions. The better agreement between the discrete data and the CFD curve fit

suggests that the new curve fit is an improved and more realistic nonlinear forcing

function for the ONERA aerodynamic model.

B.2 Unsteady Parameters

The unsteady model coefficients are determined from a single hysteresis loop for

prescribed pitching motion α = 12 sin(kτ) about the quarter-chord for k = 0.1.

This scenario represents the most severe condition at a spanwise location on the

aeroelastic wing in Ref. [106]. To create a new set of coefficients, a lift hysteresis

loop is simulated numerically for the same prescribed motion using the existing

70

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

α [deg]

CL

Figure B.1: Piecewise curve fit to CFD static lift data: curve fit (solid line); data,.

ONERA coefficients [32, 33, 34]. A residual based on the mean-square difference

between the simulated lift and the discrete, unsteady CFD data determines the

error of the initial set of coefficients. MATLAB nonlinear optimization tools update

the unsteady ONERA coefficients and iterate the residual procedure until a set of

parameters converges.

Table B.1 compares the new set of ONERA coefficients against the original NACA

0012 coefficients used by Dunn [32, 33, 34], which are labeled as ‘wind tunnel’ results

to distinguish the impact of the data source on the model parameters. The sets

of parameters determined from wind tunnel and CFD data are strikingly similar,

particularly the nonlinear terms. Figure B.3 compares the lift hysteresis loops for

α = 12 sin(kτ), k = 0.1 using both sets of parameters and includes discrete lift

data from the CFD simulation. The elliptical loops of the CFD simulation and cor-

responding ONERA model suggest that the flow is weakly nonlinear, which may be

due to the unsteady airfoil motion maintaining attached flow. Therefore, the param-

71

−20 −15 −10 −5 0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

α [deg]

∆C

L

CFD Curve FitTang & Dowell (2001)CFD DataExperiment

Figure B.2: Comparison of static lift curves from original ONERA model (Tang &Dowell (2001), Ref. [106]), CFD data, and experiment [1].

eter identification effects principally the linear aerodynamic terms. Because most

spanwise sections of the wing undergo smaller unsteady motions than the prescribed

CFD airfoil motion, the CFD data and reported experimental data [106] suggest that

linear aerodynamics may be sufficient to model LCO of the actual HALE aeroelastic

wing.

72

Table B.1: Comparison of ONERA lift parameters based on wind tunnel and CFDdata.

Data SourceParameter

Wind Tunnel [32, 33, 34] CFDa0L 5.9† 6.22sL 0.09 (180/π) 0.092 (180/π)kvL π/2 (0.504)πλL 0.15 0.146αL 0.55† 0.455σL 5.9 5.75a0 0.25 0.261a2 0.4 0.400r0 0.2 0.213r2 0.23 0.233e0 0.0 1.34×10−4

e2 -2.7 -2.74

† a0L = 6.28 and αL = 0.67 for computations in Ref. [106].

−12 −9 −6 −3 0 3 6 9 12−1.2

−0.8

−0.4

0

0.4

0.8

1.2

α [deg]

CL

Original ONERANew ONERACFD

Figure B.3: Lift hysteresis loops for two sets of ONERA model parameters forα = 12 sin(kτ), k = 0.1: ONERA model from wind tunnel data (dashed line);ONERA model from CFD data (solid line); CFD data, .

73

Bibliography

[1] I. H. Abbott and A. E. von Doenhoff. Theory of Wing Sections. McGraw-HillBook Company, Inc., New York, 1 edition, 1949.

[2] D. J. Acheson. Elementary Fluid Dynamics. Oxford Applied Mathematics andComputing Science Series. Oxford University Press, Oxford, UK, 1990.

[3] L. A. Ahaus and D. A. Peters. Private communication. 21 January 2009.

[4] J. D. Anderson. Fundamentals of Aerodynamics. McGraw-Hill Book Company,Inc., New York, 3 edition, 2001.

[5] ANSYS, Inc. ANSYS Release 10.0 Online Documentation, 2005.

[6] R. Aris. Vectors, Tensors, and the Basic Equations of Fluid Mechanics.Prentice-Hall, Englewood Cliffs, NJ, 1962.

[7] P. J. Attar and P. Vedula. Direct quadrature method of moments solutionof Fokker-Planck equations in aeroelasticity. AIAA Journal, 47(5):1219–1227,May 2009.

[8] K. J. Badcock, M. A. Woodgate, and B. E. Richards. Direct aeroelastic bi-furcation analysis of a symmetric wing based on Euler equations. Journal ofAircraft, 42(3):731–737, May–June 2005.

[9] J. S. Bae and I. Lee. Limit cycle oscillation of missle control fin with structuralnon-linearity. Journal of Sound and Vibration, 269:669–687, 2004.

[10] D. H. Baldelli, R. C. Lind, and M. Brenner. Robust aeroelastic match-pointsolutions using describing function method. Journal of Aircraft, 42(6):1597–1605, November–December 2005.

[11] G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge UniversityPress, London, 1967.

74

[12] A. Bejan. Convection Heat Transfer. John Wiley & Sons, New York, 3 edition,2004.

[13] R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. Aeroelasticity. Dover Pub-lications, Inc., New York, 1996.

[14] J. Boussinesq. Theorie de l’Ecoulement Tourbillant. Memoires presentes pardivers savants a l’Academie Royale des Sciences, 23:46–50, 1877.

[15] W. E. Boyce and R. C. DiPrima. Elementary Differential Equations andBoundary Value Problems. John Wiley & Sons, New York, 6 edition, 1997.

[16] Bruel & Kjær North America, Inc. PULSE User Manual, 2005.

[17] L. W. Carr, K. W. McAlister, and W. J. McCroskey. Analysis of the devel-opment of dynamic stall based on oscillating airfoil experiments. TechnicalReport TN D–8382, NASA, September 1977.

[18] F. O. Carta. A comparison of the pitching and plunging response of an oscil-lating airfoil. Technical Report CR 3172, NASA, October 1979.

[19] C. E. S. Cesnik and W. Su. Nonlinear aeroelastic modeling and analysis of fullyflexible aircraft. 46th AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics and Materials Conference, 7, April 2005.

[20] C-S. Chang, D. H. Hodges, and M. J. Patil. Flight dynamics of highly flexibleaircraft. Journal of Aircraft, 45(2):538–545, March–April 2008.

[21] M. D. Conner, D. M. Tang, E. H. Dowell, and L. N. Virgin. Nonlinear behav-ior of a typical airfoil section with control surface freeplay: A numerical andexperimental study. Journal of Fluids and Structures, 11:89–109, 1997.

[22] G. L. Crouse and J. G. Leishman. Transonic aeroelasticity analysis usingstate-space unsteady aerodynamic modeling. Journal of Aircraft, 29(1):153–160, January–February 1992.

[23] C. H. Custer. A Nonlinear Harmonic Balance Solver for an Implicit CFDCode: OVERFLOW 2. PhD thesis, Duke University, September 2009.

[24] C. H. Custer, J. P. Thomas, E. H. Dowell, and K. C. Hall. A nonlinear harmonicbalance method for the CFD code OVERFLOW 2. In International Forumon Aeroelasticity and Structural Dynamics (IFASD), Seattle, WA, June 2009.Paper 2009-050.

75

[25] D. Dat and C. T. Tran. Investigation of the stall flutter of an airfoil with asemi-empirical model of 2-D flow. Vertica, 7(2):73–86, 1983.

[26] D. Dessi and F. Mastroddi. Limit-cycle stability via singular perturbation andwing-flap flutter. Journal of Fluids and Structures, 19:765–783, 2004.

[27] D. Dessi, F. Mastroddi, and L. Morino. A fifth-order multiple-scale solutionfor Hopf bifurcations. Computers and Structures, 82:2723–2731, 2004.

[28] L. Djayapertapa, C. B. Allen, and S. P. Fiddes. Two-dimensional transonicaeroservoelastic computations in the time domain. International Journal forNumerical Methods in Engineering, 52:1355–1377, 2001.

[29] E. H. Dowell, R. Clark, D. Cox, H. C. Curtiss, Jr., J. W. Edwards, K. C.Hall, D. A. Peters, R. Scanlan, E. Simiu, F. Sisto, and T. W. Strganac. AModern Course in Aeroelasticity. Solid Mechanics and its Applications. KluwerAcademic Publishers, Dordrecht, The Netherlands, 4 edition, 2004.

[30] E. H. Dowell and D. Tang. Dynamics of Very High Dimensional Systems.World Scientific Publishing, Singapore, 2003.

[31] E. H. Dowell, J. Traybar, and D. H. Hodges. An experimental-theoreticalcorrelation study of non-linear bending and torsion deformations of a cantileverbeam. Journal of Sound and Vibration, 50(4):533–544, 1977.

[32] P. Dunn and J. Dugundji. Nonlinear stall flutter and divergence analysis ofcantilevered graphite/epoxy wings. AIAA Journal, 30(1):153–162, January1992.

[33] P. E. Dunn. Stall flutter of graphite/epoxy wings with bending–torsion cou-pling. Master’s thesis, Massachusetts Institute of Technology, May 1989.

[34] P. E. Dunn. Nonlinear Stall Flutter of Graphite/Epoxy Wings with Bending–Torsion Coupling. PhD thesis, Massachusetts Institute of Technology, February1992.

[35] L. E. Ericsson and J. P. Reding. The difference between the effects of pitchand plunge on dynamic airfoil stall. In 9th European Rotorcraft Forum, Stresa,Italy, September 1983.

[36] C. Farhat, P. Geuzaine, and G. Brown. Application of a three-field nonlinearfluid-structure formulation to the prediction of the aeroelastic parameters ofan F–16 fighter. Computers and Fluids, 32, 2003.

76

[37] C. Farhat and M. Lesoinne. Two efficient staggered algorithms for the serial andparallel solution of three-dimensional nonlinear transient aeroelastic problems.Computer Methods in Applied Mechanics and Engineering, 182:499–515, 2000.

[38] J. H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics.Springer-Verlag, Berlin, 3 edition, 2002.

[39] R. W. Fox and A. T. McDonald. Introduction to Fluid Mechanics. John Wiley& Sons, New York, 5 edition, 1998.

[40] K. D. Frampton and R. L. Clark. Experiments on control of limit-cycle os-cillations in a typical section. Journal of Guidance Control and Dynamics,23(5):956–960, September–October 2000.

[41] T. Fukushima and L. U. Dadone. Comparison of dynamic stall phenomena forpitching and vertical translation motions. Technical Report CR 2693, NASA,July 1977.

[42] Y. C. Fung. An Introduction to the Theory of Aeroelasticity. Dover Publica-tions, Inc., New York, 1993.

[43] M. Gad-el-Hak. Stokes’ hypothesis for a Newtonian, isotropic fluid. Journal ofFluids Engineering, 117:3–5, 1995.

[44] U. Galvanetto, J. Peiro, and C. Chantharasenawong. Remarks on the nonlineardynamics of a typical aerofoil section in dynamic stall. Aeronautical Journal,111(1125):731–739, 2007.

[45] U. Galvanetto, J. Peiro, and C. Chantharasenawong. An assessment of someeffects of the nonsmoothness of the Leishman-Beddoes dynamic stall modelon the nonlinear dynamics of a typical aerofoil section. Journal of Fluids andStructures, 24:151–163, 2008.

[46] J. A. Garcia. Numerical investigation of nonlinear aeroelastic effects on flexiblehigh-aspect-ratio wings. Journal of Aircraft, 42(4), July–August 2005.

[47] J. T. Gordon, E. E. Meyer, and R. L. Minogue. Nonlinear stability analysis ofcontrol surface flutter with free-play effects. Journal of Aircraft, 45(6):1904–1916, November–December 2008.

[48] J. M. Greenberg. Airfoil in sinusoidal motion in a pulsating streams. TechnicalReport TN 1326, NACA, June 1947.

77

[49] K. C. Hall, J. P. Thomas, and W. S. Clark. Computation of unsteady non-linear flows in cascades using a harmonic balance technique. AIAA Journal,40(5):879–886, May 2002.

[50] D. H. Hodges. A mixed variational formulation based on exact intrinsic equa-tions for dynamics of moving beams. International Journal of Solids Structures,26(11):1253–1273, 1990.

[51] D. H. Hodges and E. H. Dowell. Nonlinear equations of motion for the elasticbending and torsion of twisted nonuniform rotor blades. Technical Report TND–7818, NASA, December 1974.

[52] J. W. Jaworski. Ritz Analysis for Nonuniform Beam Models of High AspectRatio Wings. Master’s thesis, Duke University, May 2006.

[53] J. W. Jaworski and E. H. Dowell. Comparison of theoretical structural modelswith experiment for a high-aspect-ratio aeroelastic wing. Journal of Aircraft,46(2):708–713, March–April 2008.

[54] J. W. Jaworski and E. H. Dowell. Free vibration of a cantilevered beam withmultiple steps: Comparison of several theoretical methods with experiment.Journal of Sound and Vibration, 312:713–725, 2008.

[55] R. T. Jones. Properties of low aspect-ratio pointed wings at speeds below andabove the speed of sound. Technical Report 835, NACA, 1946.

[56] J. Ko, T. Strganac, J. L. Junkins, M. R. Akella, and A. J. Kurdila. Structuredmodel reference adaptive control for a wing section with structural nonlinearity.Journal of Vibration and Control, 8:553–573, 2002.

[57] J. Ko, T. Strganac, and A. J. Kurdila. Adaptive feedback linearization forthe control of a typical wing section with structural nonlinearity. NonlinearDynamics, 18:289–301, 1999.

[58] K. A. Kousen and O. O. Bendiksen. Limit-cycle phenomena in computationaltransonic aeroelasticity. Journal of Aircraft, 31(6):1257–1263, November–December 1994.

[59] E. Kreysig. Advanced Engineering Mathematics. John Wiley & Sons, NewYork, 9 edition, 2005.

[60] W. Kutta. Beitrag zur naherungsweisen Integration totaler Differentialgle-ichungen. Zeitschrift fur Mathematische Physik, 46:435–453, 1901.

78

[61] M. Landahl and H. Ashley. Aerodynamics of Wings and Bodies. Dover Publi-cations, Inc., New York, 1985.

[62] M. Lesoinne and C. Farhat. Higher-order subiteration-free staggered algorithmfor nonlinear transient aeroelastic problems. AIAA Journal, 36(9):1754–1757,September 1998.

[63] M. J. Lighthill. Higher Approximations in Aerodynamic Theory. PrincetonAeronautical Paperbacks. Princeton University Press, New Jersey, 1960.

[64] S. Liska and E. H. Dowell. Continuum aeroelastic model for a folding-wingconfiguration. AIAA Journal, 47(10):2350–2358, October 2009.

[65] R. R. Long. Mechanics of Solids and Fluids. Prentice-Hall, Englewood Cliffs,NJ, 1961.

[66] A. N. Marques and J. L. F. Azevedo. Application of CFD–based unsteady forcesfor efficient aeroelastic stability analyses. Journal of Aircraft, 44(5):1499–1512,September–October 2007.

[67] The Mathworks, Inc. MATLAB User Manual, 2005.

[68] K. W. McAlister, L. W. Carr, and W. J. McCroskey. Dynamic stall experimentson the NACA 0012 airfoil. Technical Report TP 1100, NASA, January 1978.

[69] K. W. McAlister, O. Lambert, and D. Petot. Application of the ONERA modelof dynamic stall. Technical Report TP 2399, NASA, November 1984.

[70] K. W. McAlister, S. L. Pucci, W. J. McCroskey, and L. W. Carr. An experi-mental study of dynamic stall on advanced airfoil sections. Volume 2: Pressureand force data. Technical Report TM 84245, NASA, September 1982.

[71] W. J. McCroskey, K. W. McAlister, L. W. Carr, and S. L. Pucci. An experi-mental study of dynamic stall on advanced airfoil sections. Volume 1: Summaryof the experiment. Technical Report TM 84245, NASA, July 1982.

[72] M. M. Munk. The aerodynamic forces on airship hulls. Technical Report 184,NACA, 1924.

[73] S. Munteanu, J. Rajadas, C. Nam, and A. Chattopadhyay. Reduced-order-model approach for aeroelastic analysis involving aerodynamic and structuralnonlinearities. AIAA Journal, 43(3):560–571, March 2005.

79

[74] R. H. Nichols and P. G. Buning. User’s Manual for OVERFLOW 2.1, 2.1tedition, August 2008.

[75] R. H. Nichols, R. W. Tramel, and P. G. Buning. Solver and turbulence modelungrades to OVERFLOW 2 for unsteady and high-speed applications. In 25th

Applied Aerodynamics Conference, San Francisco, CA, June 2006. AIAA Paper2006-2824.

[76] T. E. Noll, J. M. Brown, M. E. Perez-Davis, S. D. Ishmail, G. C. Tiffany, andM. Gaier. Investigation of the Helios Prototype Aircraft Mishap: Volume 1.Technical report, NASA, January 2004.

[77] R. Palacios and C. E. S. Cesnik. Static nonlinear aeroelasticity of flexibleslender wings in compressible flow. In 46th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics and Materials Conference, Austin, TX, April2005. AIAA Paper 2005-1945.

[78] M. J. Patil and D. H. Hodges. Flight dynamics of highly flexible flying wings.Journal of Aircraft, 43(6):1790–1799, November–December 2006.

[79] M. J. Patil, D. H. Hodges, and C. E. S. Cesnik. Limit cycle oscillations inhigh-aspect-ratio wings. Journal of Fluids and Structures, 15:107–132, 2001.

[80] M. J. Patil, D. H. Hodges, and C. E. S. Cesnik. Nonlinear aeroelasticity andflight dynamics of high-altitude long-endurance aircraft. Journal of Aircraft,38(1):88–94, January–February 2001.

[81] D. A. Peters. Toward a unified lift model for use in rotor blade stabilityanalyses. Journal of the American Helicopter Society, 30:32–42, 1985.

[82] D. A. Peters and W. Cao. Finite state induced flow models, Part 1: Two-dimensional thin airfoil. Journal of Aircraft, 32(2):313–322, March–April 1995.

[83] D. A. Peters and M. J. Johnson. Finite-state airloads for deformable airfoilson fixed and rotating wings. In P. P. Friedmann and C. I. Chang, editors,Aeroelasticity and Fluid/Structure Interaction, volume 44, pages 1–28, NewYork, 1994. American Society of Mechanical Engineers.

[84] D. Petot. Progress in the semi-empirical prediction of the aerodynamic forcesdue to large amplitude oscillations of an airfoil in attached or separated flow.In 9th European Rotorcraft Forum, Stresa, Italy, September 1983.

80

[85] D. Petot. Modelisation du decrochage dynamique du profil NACA 0012. LaRecherche Aerospatiale, 6:439–442, November–December 1984.

[86] D. Petot. Differential equation modeling of dynamic stall. La RechercheAerospatiale, 5:59–72, 1989.

[87] D. Petot and R. Dat. Unsteady aerodynamics loads on an oscillationg air-foil with unsteady stall. In Workshop on Dynamics and Aeroelastic StabilityModeling of Rotorcraft Systems, Florida Atlantic University, Boca Raton, FL,November 1987.

[88] D. Poirel and S. J. Price. Random binary (coalescence) flutter of a two-dimensional linear airfoil. Journal of Fluids and Structures, 18:23–42, 2003.

[89] L. Prandtl. Uber die ausgebildete Turbulenz. Zeitschrift fur Angewandte Math-ematik und Mechanik, 5:136–139, 1925.

[90] L. Prandtl. Uber ein neues Formelsystem fur die ausgebildete Turbulenz.Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematisch-Physikalische Klasse, pages 6–19, 1945.

[91] W. H. Press, S. A. Tenkolsky, W. T. Vetterling, and B. P. Flannery. NumericalRecipes: The Art of Scientific Computing. Cambridge University Press, NewYork, 3 edition, 2007.

[92] O. Reynolds. On the dynamical theory of incompressible viscous fluids and thedetermination of the criterion. Philosophical Transactions of the Royal Societyof London, Series A, 186:123, 1895.

[93] J. P. Rogers. Applications of an analytic stall model to time-history and eigen-value analysis of rotor blades. Journal of the American Helicopter Society,29(1):25–33, 1984.

[94] C. Runge. Uber die numerische Auflosung von Differentialgleichungen. Math-ematische Annalen, 46:167–178, 1895.

[95] G. Schewe, H. Mai, and G. Dietz. Nonlinear effects in transonic flutter withemphasis on manifestations of limit cycle oscillations. Journal of Fluids andStructures, 18:3–22, 2003.

[96] E. J. Shaughnessy, I. M. Katz, and J. P. Schaffer. Introduction to Fluid Me-chanics. Oxford University Press, New York, 2005.

81

[97] C. M. Shearer and C. E. S. Cesnik. Nonlinear flight dynamics of very flexibleaircraft. Journal of Aircraft, 44(5):1528–1545, September–October 2007.

[98] P. R. Spalart and S. R. Allmaras. A one-equation turbulence model for aero-dynamic flows. In 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno,NV, January 1992.

[99] G. R. Srinivasan, J. A. Ekaterinaris, and W. J. McCroskey. Evaluation ofturbulence models for unsteady flows of an oscillating airfoil. Computers andFluids, 24(7):833–861, 1995.

[100] G. G. Stokes. On the theories of internal friction of fluids in motion. Transac-tions of the Cambridge Philosophical Society, 8:287–305, 1845.

[101] D. M. Tang, M. D. Conner, and E. H. Dowell. Reduced-order aerodynamicsmodel and its application to a nonlinear aeroelastic system. Journal of Aircraft,35(2):332–338, March–April 1998.

[102] D. M. Tang and E. H. Dowell. Flutter and stall response of a helicopter bladewith structural nonlinearity. Journal of Aircraft, 29(5):953–960, September–October 1992.

[103] D. M. Tang and E. H. Dowell. Experimental and theoretical study for nonlinearaeroelastic behavior of a flexible rotor blade. AIAA Aircraft, 31(6):1133–1142,June 1993.

[104] D. M. Tang and E. H. Dowell. Flutter and stall response of a helicopter bladewith structural nonlinearity. Journal of Aircraft, 32(5):1062–1071, September–October 1995.

[105] D. M. Tang and E. H. Dowell. Comments on the ONERA stall aerodynamicmodel and its impact on aeroelastic stability. Journal of Fluids and Structures,10:353–366, 1996.

[106] D. M. Tang and E. H. Dowell. Experimental and theoretical study on aeroe-lastic response of high-aspect-ratio wings. AIAA Journal, 39(8):1430–1441,August 2001.

[107] D. M. Tang and E. H. Dowell. Experimental and theoretical study of gust re-sponse for high-aspect-ratio wing. AIAA Journal, 40(3):419–429, March 2002.

[108] D. M. Tang and E. H. Dowell. Limit-cycle hysteresis for a high-aspect-ratiowing model. Journal of Aircraft, 39(5):885–888, September–October 2002.

82

[109] D. M. Tang and E. H. Dowell. Effects of geometric structural nonlinearity onflutter and limit cycle oscillations of high-aspect-ratio wings. Journal of Fluidsand Structures, 19:291–306, 2004.

[110] D. M. Tang, D. Kholodar, and E. H. Dowell. Experimental and theoreti-cal study on aeroelastic response of high-aspect-ratio wings. AIAA Journal,38(9):1543–1557, September 2000.

[111] J. C. Tannehill, D. A. Anderson, and R. H. Pletcher. Computational FluidMechanics and Heat Transfer. Taylor & Francis, Levittown, PA, 2 edition,1997.

[112] H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT Press,Cambridge, MA, 1983.

[113] T. Theodorsen. General theory of aerodynamic instability and the mechanismof flutter. Technical Report TR 496, NACA, 1935.

[114] J. P. Thomas, C. H. Custer, E. H. Dowell, and K. C. Hall. Unsteady flow com-putation using a harmonic balance approach implemented about the OVER-FLOW 2 flow solver. In 19th AIAA Computational Fluid Dynamics Conference,San Antonio, TX, June 2009. AIAA Paper 2009-4270.

[115] J. P. Thomas, E. H. Dowell, and K. C. Hall. Nonlinear inviscid aerodynamiceffects on transonic divergence, flutter, and limit-cycle oscillations. AIAA Jour-nal, 40(4):638–646, April 2002.

[116] J. P. Thomas, E. H. Dowell, and K. C. Hall. Modeling viscous transoniclimit-cycle oscillation behavior using a harmonic balance approach. Journal ofAircraft, 41(6):1266–1274, November–December 2004.

[117] P. D. Thomas and C. K. Lombard. The geometric conservation law - a linkbetween finite-difference and finite-volume methods of flow computation onmoving grids. In 11th AIAA Fluid and Plasma Dynamics Conference, Seattle,WA, July 1978. AIAA Paper 1978-1208.

[118] P. D. Thomas and C. K. Lombard. Geometric conservation law and its applica-tion to flow computations on moving grids. AIAA Journal, 17(10):1030–1037,1979.

[119] E. F. Toro, M. Spruce, and W. Speares. Restoration of the contact surface inthe HLL-Riemann solver. Shock Waves, 4(1):25–34, July 1994.

83

[120] C. T. Tran and D. Petot. Semi-empirical model for the dynamic stall of airfoilsin view of the application to the calculation of responses of a helicopter bladein forward flight. Vertica, 5(1):35–53, 1981.

[121] L. N. Virgin, E. H. Dowell, and M. D. Conner. On the evolution of determin-istic non-periodic behavior of an airfoil. International Journal of Non-LinearMechanics, 34:499–514, 1999.

[122] D. C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., LaCanada, CA, 1993.

[123] J. Zhang and J. Xiang. Nonlinear aeroelastic response of high-aspect-ratioflexible wings. Chinese Journal of Aeronautics, 22:355–363, 2009.

84

Biography

Justin W. Jaworski was born on May 28, 1981 in Ft. Wayne, IN, the firstborn child of

Melissa and Warren Jaworski. He graduated from H.B. Plant High School in Tampa,

FL in 1999 and matriculated to Duke University the fall of the same year. As an

undergraduate, Justin received a number of named merit-based awards including the

Peggie C. Cleveland Scholarship, T.R. Mullen, Jr. Scholarship, and the Pratt Un-

dergraduate Fellowship. In 2003, Justin graduated with a Bachelor of Engineering

degree magna cum laude with distinction and was awarded the Mechanical Engineer-

ing and Materials Science Faculty Award for the highest scholastic average among

mechanical engineering graduates.

From 2003 to 2004, Justin worked as a research assistant on the Silent Air-

craft Initiative within the Aero/Astro Department at the Massachusetts Institute of

Technology. In August 2004, he returned to Duke to pursue graduate research with

Professor Earl Dowell, and completed a Master’s degree in May 2006. As a graduate

student, Justin has published three journal papers and has received external grants

from the North Carolina Space Grant Consortium and Sigma Xi to support his re-

search activities. He was awarded the Dean’s Award for Excellence in Mentoring

by the Graduate School and has served as a graduate student representative on the

Duke University Board of Trustees Facilities and Environment Committee.

Justin is a member of AIAA and ASME, and the honor societies of Sigma Xi, Pi

Tau Sigma, and Tau Beta Pi. He is also an avid classical singer and pianist.

85