imece-2002-32532 - duke universitypeople.duke.edu/~jthomas/papers/papers/hbwing.pdfble to the agard...

August 15, 2003 17:37

Proceedings ofASME International Mechanical Engineering Conference and Exposition

November 17-22, 2002, New Orleans, Louisiana, USA

IMECE-2002-32532

A HARMONIC BALANCE APPROACH FOR MODELING THREE-DIMENSIONALNONLINEAR UNSTEADY AERODYNAMICS AND AEROELASTICITY

Jeffrey P. Thomas, Earl H. Dowell, and Kenneth C. HallDepartment of Mechanical Engineering and Materials Science

Duke UniversityDurham, North Carolina 27708-0300

[email protected]

AbstractPresented is a frequency domain harmonic balance (HB)

technique for modeling nonlinear unsteady aerodynamics ofthree-dimensional transonic inviscid flows about wing configu-rations. The method can be used to model efficiently nonlinearunsteady aerodynamic forces due to finite amplitude motions of aprescribed unsteady oscillation frequency. When combined witha suitable structural model, aeroelastic (fluid-structure), anal-yses may be performed at a greatly reduced cost relative totime marching methods to determine the limit cycle oscillations(LCO) that may arise. As a demonstration of the method, non-linear unsteady aerodynamic response and limit cycle oscillationtrends are presented for the AGARD 445.6 wing configuration.Computational results based on the inviscid flow model indicatethat the AGARD 445.6 wing configuration exhibits only mildlynonlinear unsteady aerodynamic effects for relatively large am-plitude motions. Furthermore, and most likely a consequenceof the observed mild nonlinear aerodynamic behavior, the aeroe-lastic limit cycle oscillation amplitude is predicted to increaserapidly for reduced velocities beyond the flutter boundary. Thisis consistent with results from other time-domain calculations.Although not a configuration that exhibits strong LCO character-istics, the AGARD 445.6 wing nonetheless serves as an excellentexample for demonstrating the HB/LCO solution procedure.

NomenclatureAR = aspect ratio = wing span squared/wing area��

= semi-chord and chord respectively

��= vector of non-dimensional generalized forces�= total energy� � ��= scalar functions defining unsteady grid

motion in � , � , and � coordinates, respectively�= identity matrix�= � �� = mass of wing��= free-stream Mach number�= number of structural modes�= generalized mass matrix�! = number of harmonics used in harmonic balance

method�"�# = first harmonic unsteady pressure$ � = free-stream dynamic pressure% � % � = local and free-stream density, respectively�= vector of generalized aerodynamic forces& �(')�(* = flow velocity components in � , � , and � coordinate

directions, respectively+,�= free-stream velocity�'= volume of a truncated cone having stream-wise root

chord as lower base diameter, stream-wise tip chordas upper base diameter, and wing half span as height-

= reduced velocity or speed index-/.0+1�32 � 4�576 �8:9 = airfoil or wing root steady flow angle-of-attack;

= wing taper ratio; 7. � <2 �>=

4 = mass ratio, 4 . �� 2 % � �'?!@= � th structural mode shape

5 � �5 = frequency and reduced frequency�5 . 5 � 2A+B�576 = wing first torsional mode natural frequency

1 Copyright 2002 by ASME

�= matrix with structural frequency ratios squared

along main diagonal(e.g. � 5 # 2 576�� , ..., � 5�� 2 576�� ) @ ��

= � th modal structural coordinate and vectorof modal structural coordinates

Subscripts� � � = zeroth and first harmonic, respectively�= flutter onset (i.e. neutral stability) condition

1 IntroductionRecently, Hall et. al. [1] presented a new and and computa-

tionally efficient harmonic balance (HB) approach for modelingnonlinear periodic unsteady aerodynamics of two-dimensionalturbo-machinery configurations. The HB method can be usedto model nonlinear unsteady aerodynamic response due to fi-nite amplitude motions, and it is easily implemented withinthe framework of a conventional iterative CFD method. Sub-sequently, the present authors [2; 3] devised a computationalstrategy based on the harmonic balance technique for model-ing aeroelastic limit cycle oscillation (LCO) behavior of two-dimensional airfoil configurations.

In the following article, we now extend both the harmonicbalance technique for modeling nonlinear unsteady aerodynam-ics, and the harmonic balance based LCO solution procedure,to three-dimensional inviscid flows about wing configurations.The main challenge in going from two to three-dimensions isthat structural models in three-dimensions normally consist ofmany more degrees-of-freedom (DOF) other than the two “typi-cal” pitch and plunge DOF for two-dimensional airfoil sections.

For two-dimensional flows, it has been observed that aNewton-Raphson root finding technique in conjunction with theharmonic balance method proves to be a very efficient procedurefor computing LCO solutions [2; 3]. However for the Newton-Raphson LCO solution procedure, one must compute gradients(or sensitivities) of the nonlinear unsteady aerodynamic loadingwith respect to each of the structural degrees-of-freedom. Fora two-dimensional airfoil section, this requires only two gradi-ent calculations. For three-dimensional configurations however,this step may have to be carried out for many more structuraldegrees-of-freedom. One focus of the research presented in thispaper has been to determine to what extent one may reduce thecomputational effort by using a limited number of the possiblestructural DOF. As will be shown subsequently, using only thestructural modes that dominate the flutter onset condition appearsto be sufficient to model LCO behavior accurately when using theHB/LCO solution procedure.

In the following, we demonstrate the harmonic balance tech-nique for modeling nonlinear unsteady aerodynamics, along withthe harmonic balance based LCO solution procedure, for the wellknown AGARD 445.6 transonic wing configuration [4; 5]. We

begin by presenting the aeroelastic system of equations applica-ble to the AGARD 445.6 wing configuration.

2 Aeroelastic Model Governing EquationsThe governing aeroelastic system of equations for the

AGARD 445.6 wing configuration (Fig. 1) may be written in thefrequency domain as:

�� 5 � 4 �� - � �� 5�� .��(1)

For the present analysis, we consider a linear structural modelexpressed in terms of structural natural modes together with anonlinear aerodynamic model. Structural damping is neglected. ��

is a constant dependent on the wing shape and overall massgiven by

� .�� AR � � � ; � � � � ; � ; � �� (2)

and� �

is the vector of generalized aerodynamic forces, the� th element of which is given by

�! @ . �$ � � �=#"$"&% �" #(' @*)(+, dA (3)

where the integral is evaluated over the surface of the wing.In addition to the generalized mass matrix

�, wing struc-

tural frequency ratio matrix�

, and mass ratio 4 , the govern-ing aeroelastic equations also include the reduced velocity

-,

reduced frequency�5 , and structural modal coordinates

�.

As was done for the aeroelastic LCO model of Ref. [3], onecan also include the static aeroelastic equations in the mathemat-ical model. However for the case of the AGARD 445.6 wingconfiguration, this is not necessary since the configuration con-sists of a wing based on a constant symmetric airfoil section atzero degree angle-of-attack. As such, no net steady, or static,aerodynamic load acts on the wing.

As will be discussed in the following section, in order tosolve the aeroelastic system (Eq. 1), one must be able to com-pute the generalized aerodynamic forces

� �for finite values of

the structural modal coordinates�

. This is where the harmonicbalance procedure becomes an invaluable tool.

3 Harmonic Balance Method3.1 Governing Equations

We consider the inviscid Euler equations (the Reynolds aver-aged Navier-Stokes equations can be treated in a similar manner),


which may be written in integral form as�� "�"�"�� dV� "�"&%� �� ) +, dA

.��(4)

where

is the vector of conservative fluid variables .�� %!% &3% ' % * � ��(5)

and �� .�� +� �� +� �� + (6)

where�

,�

, and�

are the � , � , and � direction component fluxvectors. i.e.

�3. !!!!" !!!!#% &% & � � "% & '% & *� � � " � &

$ !!!!%!!!!& � � . !!!!" !!!!#

% '% & '% ' � � "% ' *� � � " � '$ !!!!%!!!!& � � .

!!!!" !!!!#% *% & *% & '% ' � � "� � � " � *

$ !!!!%!!!!& �The unsteady motion of the control volume

� �� is given by� �� . �� + ' � �� +( � �� +) �(7)

and this accounts for the effect of wing surface and grid motion.

3.2 Fourier Series ExpansionWe consider the unsteadiness of the flow to be strictly peri-

odic in time with period * .,+ � 2 5 where 5 is the fundamentalunsteady frequency. Thus we can expand Eq. 4 in a Fourier se-ries. For example,

"�"�" �-� �� dV./. � � �10 2�34576�8 2 3

+. 5:9<; 57= (8)

so that ��>� " "�" �-� �� dV 0 � 5?2�34576�8 23@ +. 5 9 ; 57= � (9)

and similarly

"�" %� �� A�� ) +, dA.?B � � �10C2�34576-8 2�3

+B 5 9<; 57= � (10)

� is the number of harmonics used in the Fourier expansion.

3.3 Fourier CoefficientsSubstituting the Fourier expansions (Eqs. 9 and 10) into

Eq. 4, multiplying by 9 8 ; @ = , and integrating over one period,for each m, i.e.

" �9 �* 2�34576�8 2 3� � @ +. 5 � +B 5 � 9D; 5E= 9 8 ; @ = dt (11)

yields a system of equations for the Fourier coefficients. NamelyF +. � +B .*�(12)

where

F .HGIJ � � �! . . .� �

KMLN � +. . !!!" !!!#+. 8 23+. 8 2�3PO #...+. 2 3

$ !!!%!!!& � +B . !!!" !!!#+B 8 2�3+B 8 23PO #...+B 2 3

$ !!!%!!!& �3.4 Time Domain Variables

Next, via a Fourier transform matrix Q , one can relate theFourier coefficient variables

+.to time domain solution variables

stored at uniformly space sub-time levels within a given periodof motion. i.e., +. . Q�R. � +B . Q�RB (13)

where

R. . !!!" !!!#. � � 9 �. � � # �

.... � � � 2 3 �$ !!!%!!!& � RB .

!!!" !!!#B � � 9 �B � � # �

...B � � � 2 3 �$ !!!%!!!& � (14)

and � 5 . + � @� + � � � � 5 @ . � � � � � � � � +A�! � (15)

More specifically,

R. . !!!!" !!!!#

STSUS �-� �V�� 9 � dVSTSUS �-� XWY� � � # � dV...SXSXS �-� �ZT[ 3 � � � � 2 3 � dV

$ !!!!%!!!!& � (16)


and

�� 6 ��

�� V�� V � 8�� V � � �� "!# � V � dA�� W$�� XWY� 8�� XW � � �� &% �� "!# � XW � dA

.

.

.�� ZT[ 3 � �� Z [ 3 � 8�� Z [ 3 � � �� $'$( ) �� *!# � �Z [ 3 � dA

+ ��,��-. (17)

Thus F Q�R. � Q�RB . � �(18)

and

Q 8 # F Q�R. � Q 8 # Q�RB .��(19)

So now one can work in terms of the time domain variables,which is in general much easier to do. Note however that wetake full advantage of the assumed periodic motion and thus atransient time simulation and its associated computational cost isavoided. The resulting system of equations can then be writtenas

/ R. � RB .*�(20)

where

/ . Q 8 # F Q � (21)

3.5 Pseudo Time MarchingBy adding a pseudo time derivative term 0AR. 2 0 � to Eq. 20,

one can then develop an iterative technique for determining theharmonic balance solution R. . That is,

0 R.0 � �1/ R. � RB .*� �

(22)

whereby one then simply “marches” Eq. 22 in a fictitious time-like manner until a steady state (Eq. 20) is achieved. Solu-tion acceleration techniques such as local time-stepping, pre-conditioning, residual smoothing, and multi-grid can all be usedto accelerate the convergence of the harmonic balance solution.

So for example, in the case of a finite-volume based CFDmethod, Eq. 22 is solved for every computational “finite-volume”comprising the computational mesh. The overall thus consists ofpseudo time marching

�32 � + � �� dependent variables where�is the number of mesh points times the number of dependent

variables.

Modifying an existing steady CFD flow solver to implementthe harmonic balance technique is thus a relatively straight for-ward task as the main requirement is just a re-dimensioning ofthe primary arrays from

�elements to

�42 � + � � � � elements.Again since solution acceleration techniques such as local time-stepping, pre-conditioning, residual smoothing, and multi-gridcan be used, the computational cost of the method in determin-ing unsteady solutions thus scales by a factor of � + � � � � timesthe cost of a nominal steady flow solution.

Once a harmonic balance solution has been determined fora pre-specified wing motion and frequency, one can then obtainthe resulting generalized aerodynamic forces from Eq. 3.

4 Limit Cycle Oscillation Solution ProcedureBased on previous research efforts conducted for two-

dimensional airfoil configurations, we have found that a Newton-Raphson technique in conjunction with the harmonic balancenonlinear unsteady flow solver technique provides an efficientmethod to determine limit cycle oscillation response.

As demonstrated in Refs. [2] and [3], the harmonic balanceLCO solution method proceeds by choosing one of the struc-tural modal coordinates to be the independent variable. For thetwo-dimensional airfoil case, this was chosen to be the pitch co-ordinate

�8 . For three-dimensional flow about wings, we nowconsider

# , the modal coordinate of the first (typically bend-ing) structural mode shape, as the independent variable. We alsochose

# to be real valued.In formulating the HB/LCO solution technique, one then

proceeds by dividing Eq. 1 through by # , and re-expressing the

system of equations as:

B �65 � # � . �� 5 � 4 �$� �- � �� # � � � � � # � # � �5$� .*� �

(23)Considering both the real and imaginary parts of Eq. 23, the vec-tor 5 then represents the unknown LCO solution variables

5 .

!!!!!!!!!" !!!!!!!!!#

-�5

Re � � � 2 #Im � � � 2 #...Re � � � 2 #Im � � � 2 #

$ !!!!!!!!!%!!!!!!!!!&�

(24)

which consists of the LCO reduced velocity-

(this variable iscustomarily the independent variable in time-domain LCO so-lution techniques), LCO reduced frequency

�5 , and the real andimaginary parts of the ratio of each of the LCO structural modal


coordinates, for modes two and higher, to the first structuralmodal coordinate amplitude

# . The real and imaginary partsof Eq. 23 as such represent a system of

+ �equations for the+ �

unknown LCO solution variables of 5 .As observed in Refs. [2] and [3], an efficient method for

solving Eq. 23 is to use a simple Newton-Raphson root find-ing technique. This is an iterative method for solving for theunknown LCO variables 5 whereby one “marches” the vectorequation

5 5 O # . 5 5 �� B � 5 5 �� 5 � 8 # B �65 5 � � (25)

until a suitable level of convergence is achieved.We have also observed that one can use simple forward

finite-differencing to compute the column vectors of� B �65 � 2 � 5 .

That is,

� � B �65 �� 5 � . GIIIIJ�� = � �� Re�� Z �� W � �� Im �� Z �� W , etc.

� � � �K LLLLN (26)

where for example� B �65 �� - 0 B �65 � -�� B � 5 � - � �

� B � 5 �� 5 0 B � 5 � �5 �� B � 5 � �5��

etc. for a small . Determining the column vectors of

� B 2 � 5 inthis manner thus requires numerous computations of

B � 5 � forvarious perturbations to 5 . This in turn means several computa-tions of the unsteady aerodynamic loading

� �for the different

perturbations of 5 , and this is where the harmonic balance solvercomes into play. This is also the most computationally expensiveaspect of the LCO solution methodology. However, the overallmethod does lend itself to a simple computational parallelizationstrategy as each of the gradient approximations can be calculatedon an individual computer processor.

4.1 The AGARD 445.6 Transonic Wing ConfigurationAs noted in the introduction, in order to demonstrate the

harmonic balance nonlinear aerodynamic and LCO solution pro-cedures, the the AGARD model 445.6 transonic wing config-uration [4; 5] is chosen as an example. This is a 45 degree

(a) Wing Surface and Symmetry Plane Grids

(b) Far-field Boundary Grids

Figure 1. Computational Grid Layout for AGARD 445.6 Wing Con-

figuration. 49(i) x 33(j) x 33(k) Grid Shown.

quarter chord swept wing based on the NACA 64A004 airfoilsection, which has an aspect ratio of 3.3 (for the full span),and a taper ratio of 2/3. Figure 1 illustrates the computationalmesh employed for this configuration with Fig. 1a showing aclose-up of the wing surface and symmetry boundary grids, andFig. 1b showing the outer boundary grid. The grid, which inthis instance is the “medium” resolution mesh, is based on an“O-O” topology that employs 49 (mesh

�coordinate) compu-

tational nodes about the wing in the stream-wise direction, 33(mesh

�coordinate) nodes normal to the wing, and 33 (mesh

coordinate) nodes along the semi-span. The total numberof fluid dynamic DOF for this CFD mesh is thus 266,805 (i.e.�

dependent flow variables2 � @�� 2 � @�� 2 @�� (33)).


(a) Structural Mode One (First Bending) - 60 hz

(b) Structural Mode Two (First Twisting) - 240 hz

(c) Structural Mode Three (Second Bending) - 304 Hz

Figure 2. AGARD 445.6 Wing “Weakened” Structural Configuration

Mode Shapes.

The outer boundary of the grid extends five semi-spans from themid-chord of the wing at the symmetry plane. The particularstructural configuration of the wing under consideration is re-ferred to as the “2.5 ft. weakened model 3” [4; 5].

Finally, Fig 2 shows the first three computed (via a finite ele-ment analysis) structural mode shapes and natural frequencies forthe AGARD 445.6 “weakened” wing configuration as presentedin [4]. The first mode shape is seen to consist of a first bendingtype of motion, the second mode shape then being a first twistingtype of motion, and finally the third mode being a second bend-ing motion. Two additional mode shapes, second twisting andthird bending, are also provided in [4]. These mode shapes haveall been mapped to the computational meshes used in this studyusing a sixth-order multi-dimensional least squares curve fittingtechnique.

4.2 Unsteady Grid Motion TreatmentAs noted in section 3.4, for the harmonic balance method,

the flow solution variables are stored at+A� � � sub-time levels

over a period of one cycle of motion. This means that the un-steady deformed shape of the wing is thus required at

+ � � �sub-time levels over the same period of motion. In the follow-ing analysis, we consider a linear structural model whereby weapproximate the finite-amplitude motions as a superposition of alimited number of the natural mode shapes of the structure. Thatis,

�� ; � � � � 5 � . � 9�� Re

�4@ 6 # @ ? @ 9 ; = �� (27)

where � 9�� is the nominal stationary grid and� �� is a blend-

ing function, which is equal to one at the wing surface boundary,and which decays to zero at the outer far-field grid boundary. Inthis instance, we use the following blending function� �� . �� 2 � �� max � (28)

where � �� is the distance, in the� th mesh coordinate, along

a grid line curve emanating from the � � � � th mesh point on thesurface of the wing.

This procedure for modeling the finite amplitude wing sur-face and grid motion is very simple and efficient. However, itcan eventually lead to negative volumes when the amplitude ofthe motion is too great. This is in part due to the “O-O” gridtopology, which has the tendency to have a rather highly skewedmesh near the wing tip.

5 Nonlinear Unsteady AerodynamicsWe next compute the nonlinear aerodynamic response for

a��

=0.960 background steady flow about the AGARD 445.6wing configuration due to various finite amplitudes of unsteadywing motion. We choose a reduced frequency of

�5 . � � � sincethis is near the reduced frequency at which the wing flutters.


0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

1.8

1.9

2.0

2.1

2.2

Mea

n F

low

Pre

ssur

e, p−

0/q ∞

33x25x25 Grid49x33x33 Grid65x49x49 Grid

(a) Mean (Zeroth Harmonic) Surface Pressure

0.0 0.2 0.4 0.6 0.8 1.0


−120−100−80−60−40−20

020406080

100120

Rea

l Uns

tead

y P

ress

ure,

Re(

p− 1)

/q∞


(b) Real Unsteady (First Harmonic) Surface Pressure

0.0 0.2 0.4 0.6 0.8 1.0


−10

−5

0

5

10

Imag

Uns

tead

y P

ress

ure,

Im(p−

1)/q

∞


(c) Imaginary Unsteady (First Harmonic) Surface Pressure

Figure 3. Mesh Convergence Characteristics for the AGARD 445.6

Wing Surface Pressure at 70% of Wing Semi-span: �� (deg),� � ��

, �� , and ��

.

5.1 Mesh Convergence

First, we consider the quality of the mesh. To get a sense ofrequired mesh resolution, we consider a small, in effect linear,amplitude motion of the first modal coordinate. i.e.

# . � � �� for a reduced frequency

�5 . � � � . Of course grid resolution is

Structural Mode Shape � max �� max ��! "�$#&% �� max �('*)1 - First Bending 0.0120 0.2416 0.3309

2 - First Twisting 0.0025 0.0500 0.0685

2 - Second Bending 0.0030 0.0547 0.0749

Table 1. AGARD 445.6 Wing Configuration Maximum Modal Coordi-

nate Amplitudes - 49(i) x 33(j) x 33(k) Grid.

also an important issue for larger amplitude motions. Howeverfor larger motion amplitudes, the number of harmonics one usesin the HB method, in addition to the grid resolution, plays a rolein overall solution accuracy. As such, a comprehensive study ofmodel resolution becomes quite a bit more involved. For thisreason, we have chosen to examine a small amplitude motion.Future grid convergence studies will consider larger amplitudemotions and higher harmonics.

Figure 3 shows the computed steady and unsteady flow sur-face pressure distributions at 70% span for three different gridresolutions. As can be seen, there is not much difference in thesolutions amongst the three different grid resolutions. As such,we have chosen to employ 49x33x33 grid for all subsequent cal-culations. As can be seen, since the AGARD 445.6 wing usesonly a 4% thick symmetric airfoil section, the pressure distribu-tion is also symmetric about the upper and lower surfaces, andbecause the section is so thin, no definitive shock is readily ap-parent at even this high transonic Mach number.

5.2 Finite Amplitude Motion LimitsAs mentioned in section 4.2, negative computational cell

volumes can occur once the amplitude of the wing motion be-comes too large. Table 1 shows the maximum value for eachof the first three structural modal coordinates, when taken sepa-rately, at which point negative volume problems start to occur forthe 49x33x33 mesh. Also listed is the corresponding maximumunsteady vertical displacement of the wing surface

�� # max duringan interval of motion.

5.3 Harmonic ConvergenceNext, Figs. 4 and 5 show computed real and imaginary

parts, respectively, of the first three generalized force coefficients(� # , � � , and

� ,+) normalized by the each modal coordinate

( # , � , and

+) as a function of the magnitude of each modal

coordinate ( # , � , and

+). Shown are results when using one,

two, and three harmonics in the harmonic balance solution pro-cedure. As can be seen, the amplitude of the modal coordinatehas an effect on the generalized force, and particularly so for thereal part. Also evident is how one should use more harmonicswhen considering ever increasing modal coordinate amplitudes.However, in this example, the use of only two harmonics can be


0 2 4 6 8 10 12

First Bending Modal Coordinate, ξ1 x 103

−270

−265

−260

−255

−250

−245

−240

Rea

l Gen

eral

ized

For

ce, R

e(C

Q1)/

ξ 1

1 Harmonic2 Harmonics3 Harmonics

LInear Analysis

(a) Structural Mode One

0.0 0.5 1.0 1.5 2.0 2.5

First Torsional Modal Coordinate, ξ2 x 103

900

1000

1100

1200

1300

Rea

l Gen

eral

ized

For

ce, R

e(C

Q2)/

ξ 2


LInear Analysis

(b) Structural Mode Two

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Second Bending Modal Coordinate, ξ3 x 103

−890

−880

−870

−860

−850

−840

Rea

l Gen

eral

ized

For

ce, R

e(C

Q3)/

ξ 3


LInear Analysis

(c) Structural Mode Three

Figure 4. Real part of Unsteady Generalized Force as a Function

of Mode Shape Amplitude: � � � �� (deg),

� �� , and

�� .

seen to produce a very high level of harmonic convergence foreven the largest modal amplitudes. It has been our experiencethat two harmonics in many cases are all that are necessary forachieving a good level of accuracy as long as the amplitudes arenot too large, say for example, a maximum of five degrees inpitch amplitude for the case of an airfoil section.

0 2 4 6 8 10 12

First Bending Modal Coordinate, ξ1 x 103

−45

−44

−43

−42

−41

Imag

Gen

eral

ized

For

ce, I

m(C

Q1)/

ξ 1


LInear Analysis

(a) Structural Mode One

0.0 0.5 1.0 1.5 2.0 2.5

First Torsional Modal Coordinate, ξ2 x 103

−620

−600

−580

−560

−540

−520

−500

−480

Imag

Gen

eral

ized

For

ce, I

m(C

Q2)/

ξ 2


LInear Analysis

(b) Structural Mode Two

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Second Bending Modal Coordinate, ξ3 x 103

−40

−30

−20

−10

0

10

Imag

Gen

eral

ized

For

ce, I

m(C

Q3)/

ξ 3


LInear Analysis

(c) Structural Mode Three

Figure 5. Imaginary part of Unsteady Generalized Force as a Func-

tion of Mode Shape Amplitude: �,� � ��(deg),

� � �� ,

and �� .

Next, Fig. 6 shows surface pressure distributions for the ze-roth harmonic (Fig. 6a) (i.e. mean flow), along with the real(Fig. 6b) and imaginary (Fig. 6c) parts of the first harmonic (un-steady flow), again normalized by the modal coordinate ampli-tude

# at a location approximately 70% of wing semi-span. Inthis instance, we consider four different modal coordinate ampli-


0.0 0.2 0.4 0.6 0.8 1.0


1.7

1.8

1.9

2.0

2.1

2.2

Mea

n F

low

Pre

ssur

e, p−

0 /q

∞

ξ1=0.1x10−3

ξ1=2x10−3

ξ1=5x10−3

ξ1=10x10−3


0.0 0.2 0.4 0.6 0.8 1.0


−100

−80

−60

−40

−20

0

20

40

60

80

100

Rea

l Uns

tead

y P

ress

ure,

Re(

p− 1)

/q∞ξ 1

ξ1=0.1x10−3

ξ1=2x10−3

ξ1=5x10−3

ξ1=10x10−3


0.0 0.2 0.4 0.6 0.8 1.0


−10

−5

0

5

10

Imag

Uns

tead

y P

ress

ure,

Im(p−

1)/q

∞ξ 1

ξ1=0.1x10−3

ξ1=2x10−3

ξ1=5x10−3

ξ1=10x10−3


Figure 6. Mean and Unsteady Flow Surface Pressure Distribution

Dependence on Mode Shape Amplitude: �� (deg),

� �� , ��

, and�� .

tudes ( # =0.0001, 0.0020, 0.0050, 0.0100) for the case of three

harmonics used in the harmonic balance technique. Readily ap-parent are nonlinear effects due to the ever increasing modal co-ordinate amplitudes. In fact, this is surprisingly so for the zeroth

0.0 0.2 0.4 0.6 0.8 1.0


1.7

1.8

1.9

2.0

2.1

2.2

Mea

n F

low

Pre

ssur

e, p−

0 /q

∞



0.0 0.2 0.4 0.6 0.8 1.0


−100

−80

−60

−40

−20

0

20

40

60

80

100

Rea

l Uns

tead

y P

ress

ure,

Re(

p− 1)

/q∞ξ 1



0.0 0.2 0.4 0.6 0.8 1.0


−15

−10

−5

0

5

10

15

Imag

Uns

tead

y P

ress

ure,

Im(p−

1)/q

∞ξ 1



Figure 7. Mean and Unsteady Flow Surface Pressure Distribution

Dependence on Number of Harmonic Used in HB Solver: �,� ��(deg),

� � �� , ��

, and �� .

harmonic of the solution (Fig. 6a).

Finally, Fig. 7 shows normalized zeroth and first harmonicsurface pressure distributions for the largest modal coordinate


0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.2

0.3

0.4

0.5

0.6

0.7

Flu

tter

Vel

ocity

, Vf=

U∞/µ

1/2 ω

αb Yates − ExperimentalPOD/ROM Computational Model

Figure 8. Mach Number Flutter Speed Dip Trend for the AGARD

445.6 Wing Configuration

amplitude ( # =0.0120) while varying the number of harmonics

used in the harmonic balance technique. Two harmonics can beseen to produce good harmonic convergence.

6 Nonlinear Unsteady Aeroelasticity6.1 Flutter Onset Trend for the AGARD 445.6 Wing

ConfigurationFigure 8 shows the computed Mach number flutter onset

reduced velocity trend for the AGARD 445.6 wing configura-tion. Computational model results are shown together with ex-perimental results. The six Mach numbers are the same as thoseof the experimental investigation of Ref. [5]. The reduction ofthe flutter onset velocity in the transonic region is readily appar-ent. The flutter onset conditions are determined using the properorthogonal decomposition (POD) reduced order model (ROM)time-linearized unsteady aerodynamic approach of Ref. [7]. ThePOD/ROM approach has been demonstrated to provide a goodapproximation of the flutter onset conditions, which can thenbe used as initial conditions for the iterative HB/LCO solutionprocedure. The agreement between theory and experiment isgenerally good except at the supersonic free-stream Mach num-bers. Other inviscid computational studies by investigators suchas Lee-Rausch and Batina [8] and Gordnier and Melville [9] havedemonstrated similar differences between theory and experimentfor the supersonic Mach numbers.

Table 2 shows computed values for the reduced velocity-

,reduced frequency

�5 , and the shape of the unsteady motion of thewing at the flutter onset condition, which is presented in terms ofthe ratio of the amplitude of each of structural modal coordinatesnormalized to the amplitude of first structural modal coordinate.As can be seen, the first bending structural motion dominates the

0.2 0.3 0.4 0.5 0.6 0.7

Reduced Velocity, U∞/µ1/2ωαb

0.000

0.002

0.004

0.006

0.008

0.010

0.012

LCO

Am

plitu

de o

f Firs

t Str

uctu

ral M

ode,

ξ1

M∞=0.960

M∞=1.141M∞=1.072

M∞=0.678

M∞=0.499

M∞=0.901

Figure 9. Computed AGARD 445.6 Wing Configuration LCO Behav-

ior Trends.

flutter onset unsteady motion for the six Mach numbers consid-ered, particularly in the transonic region. In fact, beyond the firstthree structural modes, higher modes only contribute a fraction ofa percent to the overall unsteady motion at the flutter onset con-dition. As will be shown, one can typically neglect these highernumbered modes in the HB/LCO solution procedure.

6.2 LCO Behavior Trends for the AGARD 445.6 WingConfiguration

Based on the LCO solution procedure presented in the Sec-tion 4, Fig. 9 shows computed LCO behavior trends for theAGARD 445.6 wing configuration for the six different Machnumbers. Unfortunately, Ref. [5] does not provide any specificexperimental LCO data. Plotted in Fig. 9 is the LCO amplitudeof the first structural modal coordinate,

# , versus the reducedvelocity

-for each of the six different Mach numbers. The data

points for what appear to be zero LCO amplitude actually corre-spond to a very small LCO amplitude of

# . � � �� .In this instance, only the first three structural modes are

used for the HB/LCO solution process. The convergence of theHB/LCO solution procedure is also very rapid. Only one or twoiterations of Eq. 25 are required. The LCO amplitudes in thetransonic region go to larger values than those in the subsonicand high supersonic region since the first bending mode shapevery much dominates in the transonic region. As such, negativevolume problems with the unsteady grid are less of an issue inthe transonic region.

Note how the LCO behavior trend curves are nearly verti-cal. This is indicative of very linear aeroelastic LCO responsebehavior, which is not all that surprising for this particularly thinwing that has a maximum thickness ratio of 4%. A recent time-


Mach Number, �� # ��

� �� # ��

�� # �� # � ��

Re � �� % � � � � �� # � �� # �� # � ��

Im � �� % � � � �� # � �� # �� #Re � �� % � � � �� # ��

Im � �� % � � � �� # � # �� # ��

Re � �� % � � � � ��

Im � �� % � � � �� # ��

Re � ��! % � � � �� # ��

Im � ��! % � � � � �� # � ��

Table 2. AGARD 445.6 Wing Configuration Mach Number Flutter Onset Conditions.

0.25 0.30 0.35 0.40 0.45 0.50 0.55

Reduced Velocity, U∞/µ1/2ωαb

0.000

0.002

0.004

0.006

0.008

0.010

0.012

LCO

Am

plitu

de o

f Firs

t Str

uctu

ral M

ode,

ξ1

2 Structural Modes3 Structural Modes4 Structural Modes

M∞=0.960 M∞=0.499

Figure 10. Computed AGARD 445.6 Wing Configuration LCO Be-

havior Trends When Using Different Numbers of Structural Modes in

HB/LCO Solutions Procedure.

domain, and viscous flow, computational study of this same con-figuration for

� �=0.960 and

� �=1.141 conducted by Gord-

nier and Melville [9] also determined that no LCO conditionsappear to exist. Note that

# . � � � � + corresponds to an un-steady wing amplitude of approximately one third of the wingroot chord (Section 5.2).

Finally, Fig. 10 shows computed LCO behavior trends for��=0.499 and

� �=0.960 when using two, three, and four

structural mode shapes in the HB/LCO solution procedure. As

can be seen, only the first few structural mode shapes are nec-essary to produce converged LCO solutions. Again, this is mostlikely due to the fact that the first bending mode tends to domi-nate the aeroelastic flutter motion for the AGARD 445.6 wing.

It is interesting to note that the strongest nonlinear LCObehavior occurs for

� �=1.072 and

� �=1.141 as shown in

Fig. 9. For these Mach numbers, the nonlinear effect is to de-crease the reduced velocity at which LCO may occur below theflutter onset condition. This may be at least a partial explanationfor the observed differences between theory and experiment inFig. 8. There the calculated (linear) reduced velocity at the onsetof flutter is compared to the reduced velocity at which flutter wasobserved experimentally. As shown in Fig. 8, the flutter/LCOobserved experimentally does indeed occur at a reduced velocitybelow that predicted theoretically for the onset of flutter.

7 ConclusionsA harmonic balance method for modeling nonlinear periodic

unsteady three-dimensional inviscid transonic flows about wingconfigurations is presented. Demonstrated is the ability of themethod to model nonlinear aerodynamic effects due to large am-plitude motions for a well known aeroelastic configuration. Theuse of only two harmonics in the procedure is shown to be quitesufficient for producing harmonic convergence. A limit cycleoscillation solution methodology is also presented and demon-strated for the same benchmark transonic configuration. A weakLCO is observed, which is consistent with results from time-domain calculations by Gordnier and Melville [9]. Nevertheless,it is suggested that nonlinear aerodynamic effects may at least


partially explain some of the previously observed differences be-tween experiment and linear aeroelastic theory.

AcknowledgmentsThis work was supported by AFOSR grant, ”Limit Cycle

Oscillations and Nonlinear Aeroelastic Wing Response: Re-duced Order Aerodynamics” and STTR grant, ”Nonlinear Re-duced Order Modeling of Limit Cycle Oscillations of AircraftWings/Stores”. The latter grant is joint work with Zona Technol-ogy. Dr. Dan Segalman and now Dr. Dean Mook are the AFOSRProgram Managers.

References[1] Hall, K. C., Thomas, J. P., Clark, W. S., “Computation ofUnsteady Nonlinear Flows in Cascades Using a Harmonic Bal-ance Technique”, AIAA Journal, Vol. 40, No. 5, 2002, pp. 879–886.[2] Thomas, J. P., Dowell, E. H., Hall, K. C., “Nonlinear Invis-cid Aerodynamic Effects on Transonic Divergence, Flutter andLimit Cycle Oscillations”, AIAA Journal, Vol. 40, No. 4, 2002,pp. 638–646.[3] Thomas, J. P., Dowell, E. H., Hall, K. C., “ModelingViscous Transonic Limit Cycle Oscillation Behavior Using aHarmonic Balance Approach”, AIAA Paper 2001-1414, 43rdAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynam-ics and Materials (SDM) Conference, Denver, CO, April 2002.[4] Yates, E. C., Jr., “AGARD Standard Aeroelastic Config-urations for Dynamic Response I - Wing 445.6”, NASA TM100492, August 1987; also Proceedings of the 61st Meeting ofthe Structures and Materials Panel, Germany, AGARD-R-765,1985, pp. 1–73.[5] Yates, E. C., Jr., Land, N. S., and Foughner, J. T. , Jr., “Mea-sured and Calculated Subsonic and Transonic Flutter Character-istics of a 45 Deg Sweptback Wing Planform in Air and in Freon-12 in the Langley Transonic Dynamics Tunnel”, NASA TN D-1616, March 1963.[7] Thomas, J. P., Dowell, E. H., Hall, K. C., “Three-Dimensional Aeroelasticity Using Proper Orthogonal Decompo-sition Based Reduced Order Models”, AIAA Paper 2001-1526,42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy-namics and Materials (SDM) Conference, Seattle, WA, April2001.[8] Lee-Rausch, E. M. and Batina, J. T., “Wing Flutter Bound-ary Prediction Using Unsteady Euler Aerodynamic Method”,Journal of Aircraft, Vol. 32, No. 2, 1995, pp. 416–422.[9] Gordnier, R. E., Melville, R. B., “Transonic Flutter Simula-tions Using an Implicit Aeroelastic Solver”, Journal of Aircraft,Vol. 37, No. 5, 2000, pp. 872–879.


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