Improved Frequency Domain Flutter Analysis Using ... Frequency Domain Flutter Analysis Using Computational Fluid Dynamics by Ryan J. Beaubien B. Eng. (Aerospace) A thesis submitted

Im proved Frequency D om ain F lu tter A nalysis U sing C om putational F luid D ynam ics

by

Ryan J. Beaubien

B. Eng. (Aerospace)

A thesis subm itted to the Faculty o f Graduate Studies and Research

in pa rtia l fu lfillm ent of the requirements for the degree of

M aster’s o f Applied Science in Aerospace Engineering

Ottawa-Carleton Institu te for Mechanical and Aerospace Engineering

Department of Mechanical and Aerospace Engineering

Carleton University

Ottawa, Ontario, Canada

May 2006

© Copyright by Ryan J. Beaubien, 2006

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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A bstract

The most w ide ly used method for flu tte r certification is based on the linearized aero

dynamic po ten tia l theory, specifically, the Doublet-Lattice M ethod (D LM ). However,

th is method fails to accurately predict the aeroelastic behaviour in the transonic

regime.

The present work investigates two different approaches which are able to capture

the nonlinearities in the transonic regime: (1) perform ing a tim e marching sim ulation

using Euler and Navier-Stokes equations; and (2) correcting the D LM aerodynamic

data in a frequency domain analysis using Com putational F lu id Dynamics (CFD)

results. A f lu tte r analysis o f the A G A R D 445.6 wing was subsequently conducted

using the tim e marching and D LM correction approaches.

Various frequency domain correction techniques were considered w ith in the present

work resulting in the selection of an unsteady pressure matching method, based on a

downwash weighting approach. Unsteady pressures employed for th is technique were

obtained from Euler and Navier-Stokes CFD simulations undergoing a rig id body

p itch ing m otion.

ii i


Acknowledgements

I would like to thank Dr. Fred Nitzsche, my supervisor, and Dr. Daniel Feszty,

my co-supervisor, for the ir suggestions and constant support during th is research.

I am deeply indebted to: Ihor Gegar (Bombardier Aerospace) for his guidance on

structural and aerodynamic modelling; Dr. Roberto G il A lin es da Silva (Ins titu to

de Aeronautica e Espago) for his assistance w ith the A IC correction procedure; Dr.

Ken Badcock and A bdu l Rampurawala (University of Glasgow) for provid ing the

time marching code and assistance w ith CFD grid generation; and Johnny Cash for

producing music tha t motivates the soul.

This thesis is dedicated to my family.

iv


Contents

Abstract iii

Acknowledgem ents iv

List of Tables viii

List o f Figures ix

Nom enclature xi

1 Introduction 1

1.1 Description of F lu t te r ....................................................................................... 1

1.2 H istorical Review of F l u t t e r ......................................................................... 2

1.3 Com putational Analysis Techniques............................................................ 4

1.3.1 Frequency Domain M e th o d s .............................................................. 4

1.3.2 A IC Correction M e th o d s ..................................................................... 7

1.3.3 Tim e Marching M e th o d s ..................................................................... 8

1.4 Basis and Overview of Current W o r k ......................................................... 11

2 Frequency Dom ain Formulation 13

2.1 Unsteady Aerodynamic M o d e llin g ............................................................... 13

2.1.1 Overview of the Doublet-Lattice M e th o d ....................................... 14

2.2 Aeroelastic M o d e l............................................................................................. 18

2.2.1 Aeroelastic Equations of M o t io n ....................................................... 18

2.2.2 M odal A p p ro a c h .................................................................................. 20

v


2.2.3 Simple Harmonic M otion A p p ro a c h ............................................... 22

2.2.4 Interconnection o f the Structure w ith Aerodynam ics................. 22

2.3 F lu tte r Solution T e c h n iq u e s ......................................................................... 24

3 AIC Correction Formulation 26

3.1 Overview of A IC Correction M e th o d s ......................................................... 26

3.1.1 Force M atching M e th o d s .................................................................... 26

3.1.2 Pressure M atching M e th o d s ............................................................. 28

3.1.3 Dau-Garner Type M e thods ................................................................. 31

3.1.4 M odal Aerodynamic Influence Coefficients M a tr ix

M e th o d s ................................................................................................. 32

3.2 Derivation of Selected Correction M e th o d ................................................... 33

3.2.1 Theoretical D e r iv a t io n ....................................................................... 34

4 Tim e M arching Formulation 37

4.1 Numerical F lu id Methods .............................................................................. 37

4.1.1 Euler E q ua tion s ..................................................................................... 37

4.1.2 Navier-Stokes E q u a t io n s .................................................................... 39

4.1.3 Steady State S o lv e r .............................................................................. 42

4.1.4 Unsteady S o lve r..................................................................................... 44

4.1.5 Turbulence M o d e l.................................................................................. 46

4.2 S tructu ra l S o lv e r ............................................................................................... 48

5 Test Cases 51

5.1 V a lidation of the Com putational P rocedure................................................. 51

5.1.1 CFD F lu id M e s h e s .............................................................................. 52

5.1.2 Steady F lo w ............................................................................................ 53

5.1.3 Unsteady F l o w ..................................................................................... 54

5.2 F lu tte r Test C a s e .............................................................................................. 56

5.2.1 CFD F lu id M e s h e s .............................................................................. 59

5.2.2 D LM M e s h ............................................................................................ 63

v i


5.2.3 S tructu ra l M o d e l ................................................................................. 64

6 Results 68

6.1 Uncorrected Frequency Domain R e su lts ....................................................... 68

6.2 Time Marching R e su lts ..................................................................................... 70

6.2.1 G rid Density and F lu id Model Effect on F lu tte r Speed . . . . 70

6.2.2 F lu tte r B o u n d a rie s .............................................................................. 72

6.3 A IC Correction M ethod R e s u lts ..................................................................... 79

6.3.1 Present W ork R esu lts ........................................................................... 80

6.3.2 Referenced R e s u lts .............................................................................. 83

6.4 C o n c lu s io n s ......................................................................................................... 84

6.5 Future W o r k s ..................................................................................................... 85

References 87

v ii


List of Tables

5.1 Steady flow conditions for F-5 w ing................................................................ 54

5.2 Unsteady flow conditions for F-5 w ing........................................................... 54

5.3 Flow conditions for AG AR D 445.6 wing flu tte r analysis........................... 60

5.4 Comparison of modal frequencies for AG A R D 445.6 w ing ........................ 64

6.1 Comparison of flu tte r speed coefficients at Mach 0.96 for various grids. . 72

6.2 Reduced frequencies of the CFD simulations used in the A IC correction

m ethod................................................................................................................... 79

6.3 Com putational tim e required to calculate each flu tte r p o in t.................... 84

v iii


List of Figures

2.1 L ifting surface idealization in the D L M ............................................................... 15

2.2 DLM wing and element coordinates..................................................................... 16

5.1 F-5 wing flu id mesh................................................................................................... 52

5.2 Steady Cp over F-5 upper wing surface at M ^ = 0.95..................................... 53

5.3 Steady Cp and A Cp over F-5 wing at = 0.95............................................. 55

5.4 Steady state convergence for F-5 wing at M <*, = 0.95...................................... 55

5.5 Real and imaginary Cp over F-5 wing at = 0.95 for 2 and 3 cycles. . 57

5.6 Real and imaginary Cp over F-5 wing at = 0.95....................................... 58

5.7 Unsteady A Cp over F-5 wing at M ^ = 0.95....................................................... 58

5.8 Medium density AG AR D 445.6 CFD meshes..................................................... 61

5.9 Steady Cp over AG AR D 445.6 upper wing surface at = 0.96.............. 62

5.10 Steady state convergence for medium density AG AR D 445.6 wing meshes

at M qo = 0.96............................................................................................................. 63

5.11 AG AR D 445.6 structura l model............................................................................ 66

5.12 Calculated AG AR D 445.6 w ing mode shapes.................................................... 67

6.1 Uncorrected frequency domain flu tte r boundaries for A G A R D 445.6 wing. 69

6.2 Uncorrected frequency domain V - f and V-g plots at M 00 = 0.678................. 71

6.3 Time marching flu tte r boundaries......................................................................... 73

6.4 Time marching responses at M ^ = 0.678.......................................................... 74

6.5 Time marching responses at M ^ .= 0.901.......................................................... 75

6.6 Time marching responses at M ^ .= 0.960.......................................................... 76

6.7 Time marching responses at......= 0.990........................................................... 77

ix


6.8 Time marching responses at = 1.072........................................................... 78

6.9 Unsteady A Cp over AG A R D 445.6 wing at M 00 = 0.678.............................. 81

6.10 A IC corrected frequency domain V -f and V-g plots at M ,oo = 0.678. . . . 82

6.11 F lu tte r boundaries for AG AR D 445.6 w ing ........................................................ 83

x


N omenclat ur e

Acr Am plitude o f dynamic angle of attack

Body element length

A Cp L ifting pressure coefficient, A Cp = (piower -P u PPer)/qoo

A p L ifting pressure

w Dimensionless normal velocity, w = W /U Q0

K Kernel function

m M odal transform ation m atrix , composed of mode shape vectors

[A IC ] Aerodynamic influence coefficients m a trix

[D] Influence m a trix relating normalwash to lift in g pressure

IF} Substantial derivative operator

[G] Spline m a trix operator

m Aerodynamic transfer function m atrix

[.K ] Structura l stiffness m a trix

[M \ Structural mass m atrix

[Q] Aerodynamic loads influence coefficients m atrix


[5] Integration m a trix

[W T\ W eighting m a trix operator

{ L } Tota l aerodynamic force vector

{La} Induced aerodynamic force vector

{Le} External aerodynamic force vector

M Generalized coordinate vector

M Dimensionless normal velocity vector divided by A a

{w } Dimensionless normal velocity vector

{« } S tructu ra l displacement vector

p Reference viscosity

flT Eddy viscosity

V Poisson’s ra tio

n Angular velocity

UJ O scillation frequency associated w ith harmonic m otion

Re(), Im () Real and imaginary parts of a complex number

P Density

a Lateral coordinate in the plane of the lift in g surface

T = Uoot/C Non-dimensional time

Tij Stress tensor

$ Blending function

x ii


£, fj, 0 Coordinates of a po in t pressure doublet in element coordinates

£, T), ( Coordinates of a point pressure doublet

£1/4 One-quarter chord point o f lift in g surface element

b Reference length (root semi-chord)

cr W ing root chord

CDkuj Cross-diffusion term for the k — ui turbulence model

e Semi-width of lift in g surface element

E i Longitudinal elastic modulus

E 2 Lateral elastic modulus

F i SST coefficient blending function

E i SST closure forcing function

G Shear modulus

g S tructura l damping

H Enthalpy

h Displacement mode shape vector

i Imaginary number, i =

k Reduced frequency, k = u ib /U ^ or turbulent k inetic energy

(Chapter 5)

M qo Freestream Mach number

n l Nonlinear

x ii i


Qoo

Q x i Q y i Qz

s

T

t

U ,V ,W

U, V, w

U o o

UF

w

x , y , z

x , y , z

y

F, G ,H

FSGSH4

FV,G V,H V

Q

Turbulent P rand tl number

P rand tl number

Dynamic pressure, = \p U

Heat flows

Laplace variable, s = iuo

Static temperature

Time

Contravariant velocities

Cartesian velocities

Freestream velocity

F lu tte r speed coefficient

Normal velocity at surface

Coordinates o f a general field point given in terms of a coordi

nate system centred on and rotated in to the plane of a lift in g

surface element, i.e. element coordinate system

Cartesian coordinates o f a general field point

W all distance

F lux vectors

Inviscid viscous flux vectors

Diffusive viscous flux vectors

Heat flow vector

x iv


R Residual vector

W Solution vector

A IC Aerodynamic Influence Coefficients

B ILU Block Incomplete Lower Upper

CFD Computational F lu id Dynamics

CSD Com putational S tructura l Dynamics

C VT Constant Volume Tetrahedron

D LM Doublet-Lattice M ethod

LCO L im it Cycle Oscillation

M A IC M odal Aerodynamic Influence Coefficients

RANS Reynolds Averaged Navier-Stokes

Re Reynolds number

T F I Transfinite In terpo lation of Displacements

TSD Transonic Small Disturbance

XV


Chapter 1

Introduction

1.1 D escrip tion o f F lu tter

F lu tte r, the dynamic ins tab ility of an elastic body in an airstream, is commonly

encountered on bodies subjected to large la teral aerodynamic loads such as on a ircraft

wings, tails and control surfaces [1]. No aircraft is perfectly rig id due to weight

constraints and its fle x ib ility can interact w ith the airstream, sometimes creating

self-sustaining vibrations. For example, an aircraft w ing is re lative ly flexible and

can be observed to bend and tw is t under the influence o f a ir loads in flight. I f th is

motion occurs in a periodic manner under certain conditions, the dynamic loads may

begin to augment the elastic m otion o f the wing. As the am plitude o f the m otion

increases, the loads on the structure also increase, eventually causing the structure to

catastrophically fail. The speed at which th is phenomenon occurs is called the flu tte r

speed. Flights at speeds above and below the flu tte r speed yie ld unstable and stable

conditions, respectively.

1


2

Modern aircraft are subject to many kinds o f flu tte r phenomena. The classical type

of flu tte r is associated w ith potentia l flow and usually, but not necessarily, involves the

coupling of two or more degrees of freedom - wing bending and torsion being the most

common. The nonclassical type may involve separated flows, stalling conditions and

time-lag effects between the aerodynamic forces and the m otion o f the structure [1],

The simulation of the classical type o f flu tte r is the subject o f th is thesis.

1.2 H istorical R eview o f F lu tter

In 1916, the firs t recorded flu tte r incident occurred on a Handley Page 0 /4 0 0 tw in

engine biplane bomber. The bomber experienced violent self-excited antisym m etric

oscillations o f the fuselage and ta il. The problem was elim inated by increasing the

stiffness o f the fuselage and ta il structures.

In the period from 1918 to the 1930s, a systematic flu tte r study was conducted

for the van Berkel monoplane, a seaplane for reconnaissance. The flu tte r mechanism

was a coupling of the vertical bending of the wing w ith the m otion o f the ailerons

and was negated by moving the aileron to balance the wing mass [2],

In the 1930s, w ith the availab ility of higher performance engines, flu tte r began

to be recognized as a critica l safety concern. The solution o f increasing structura l

stiffness was not always feasible due to weight considerations. Experiments conducted

during this decade revealed tha t altering the mass d is tribu tion of the a ircraft was as

effective as m odifying the fligh t velocity at which flu tte r occurred as was increasing


3

the s tructura l stiffness [3].

During W orld W ar II, the importance o f flu tte r in aircraft design was increased

as designers incorporated new structura l materials and more powerful engines. In

1945, 146 flu tte r incidents occurred in Germany, resulting in 24 crashes [2]. A ll

of these incidents involved flu tte r of the control surfaces or the auxilia ry controls.

Only 54 flu tte r incidents were recorded during the 10 years following the war. This

decrease can be a ttr ib u te d to advances in com putational methods tha t complemented

improvement in experimental techniques for flu tte r models, ground resonance testing

and fligh t f lu tte r testing [2], [4].

Prom 1947 to 1956, several incidents of flu tte r were recorded which involved the

carriage of external stores, such as external fuel tanks or missiles, or pylon mounted

engines [2]. Th is problem is s till significant today as most a ircraft are capable of

several store configurations; certain combinations of external stores carried by the

American F-16, F-18 and F - l l l a ircraft produce a lim it cycle oscillation (LCO ). These

oscillations are characterized by sustained sinusoidal oscillations o f lim ited amplitude.

A lthough LCOs may not be destructive, they may be uncomfortable for the p ilo t, may

decrease the life o f the a ircraft and may negatively affect the precision o f the a ir-to -a ir

missile systems. In 1999, unexpected instabilities arose for the M itsubish i F -2A /B

fighter which delayed the program for over 9 months [5]. The problem was eventually

linked to the presence of stores. Stores were also responsible for a 30% reduction in


4

the fligh t envelope for the Royal Austra lian A ir Force F-18 [6].

O ther recent examples not involving LCOs include the Taiwan ID F fighter which

crashed due to flu tte r of the horizontal ta il during a high dynamic pressure fligh t

test in 1992, u ltim a te ly leading to the cancellation o f the project. Also in 1992, a

prototype of the American F-22 Raptor crashed in a f lu tte r related accident [7]. In

1997, an American F-117 Stealth fighter crashed due to a ileron/flaperon flu tte r on a

prim ary lift in g surface [8].

Today, few catastrophic flu tte r incidents occur due to the improvements in fligh t

flu tte r test technique, instrum entation and response data analysis [9]. However, f lu t

ter problems s till arise regularly as illus tra ted by the examples cited above. These

problems delay projects and incur substantial additional expense. F lu tte r testing is

s till considered hazardous due to the violent nature of the onset of flu tte r. Accurately

predicting the flu tte r boundary in the transonic regime using com putational methods

would result in a shorter and less expensive fligh t testing program as the pilots and

engineers would have greater confidence in the predictions.

1.3 C om putational A nalysis Techniques

1.3.1 F requency D om ain M eth o d s

The aeronautical industry m ainly employs commercial codes such as M S C /N A S TR A N

and ZAERO for flu tte r prediction. These codes perform a modal eigenvalue analysis

using a structura l dynamic solver and unsteady aerodynamics based on linearized


5

aerodynamic potentia l theories. The aerodynamic analysis, like the structura l anal

ysis, is based on a fin ite element approach. A brie f summary of the flu tte r solution

procedure follows; refer to Chapter 2 for the detailed derivation.

The aeroelastic response of a body in an airstream is a result of the m utua l

interaction of: (1) inertia l and elastic s tructura l forces; (2) aerodynamic forces in

duced by static or dynamic deformation o f the structure; and (3) external disturbance

forces [10]. The equations of m otion for the body are linearized and transformed in to

the frequency domain as the classical approach assumes tha t the structure undergoes

low amplitude elastic harmonic motion. The aerodynamic forces are computed which

are related to the structura l deformations using an aerodynamic influence coefficients

(A IC ) m atrix. The roots for the system of homogeneous equations o f m otion are

determined using various eigenvalue solution techniques. A root corresponding to

a decaying or stable condition is considered to be below the flu tte r speed. A root

corresponding to a divergent oscillation w ith positive damping is assumed to be be

yond the flu tte r speed. The root corresponding to neutral s tab ility gives the flu tte r

speed [3].

A ICs reduce development costs and are therefore beneficial in a production flu tte r

environment. Once the A IC matrices have been generated for a specific planform,

an un lim ited number of aeroelastic analyses can be performed in which only the

ine rtia l and stiffness properties are altered as the AICs are independent o f the a ircraft


6

v ib ra tion modes shapes [11].

Several subsonic and supersonic aerodynamic theories are able to accurately pre

d ict the flow in the Mach range of 0.0 < M m < 0.6 and M ,^ > 1 .1 . Common subsonic

theories used in production flu tte r analysis include the modified s trip theory, Doublet-

Lattice M ethod (D LM ) and ZO N A6. Common supersonic theories include the Mach

box method, Piston theory and ZONA51. S trip theory was the prim ary subsonic

unsteady aerodynamic theory used from the early 1940s to the m id 1960s.

In 1966, Reference [12] proposed m odifying strip theory to account for fin ite span

effects. The resultant method was referred to as modified strip theory. Coupled w ith

the solution procedure outlined above, modified strip theory formed the basis for

production flu tte r analyses in the late 1960s [13].

Today, industry p rim arily utilizes the D LM for calculating unsteady aerodynamic

loads. During the late 1960s, th is unsteady panel aerodynamic method was developed

by Albano and Rodden [14] and refined by the Douglas A irc ra ft Company [15]. The

D LM has been in use for over 30 years and is the standard by which new unsteady

aerodynamic codes are evaluated. I t is anticipated th a t the D L M w ill be used for

many years to come [13].

Since the governing equations over which the unsteady aerodynamic methods

were developed are based on a linearized unsteady potentia l flow hypothesis, the

application o f these methods are lim ited to purely subsonic or supersonic flows [16].


7

T h is is unfortunate as most passenger a ircraft cruise at transonic speeds [3]. These

methods are unable to capture the location and magnitude of local shock waves

and the associated shock wave-boundary layer interactions on w ing surfaces in the

transonic regime. The prediction o f these phenomena are crucial for assessing the

aeroelastic behaviour of a wing in transonic flow as they are the source for several

nonlinear aeroelastic effects, such as the transonic d ip and LCOs.

1 .3 .2 A IC C orrection M eth o d s

Several procedures to solve the transonic aeroelastic problem have been developed

over the past 40 years [17]. These procedures are commonly referred to as combined

procedures, as m ixed procedures or as semi-empirical corrections. Measured or com

puted data is re lated to the unsteady linear aerodynamic model so as to account for

nonlinear effects unpredicted by the linearized potential-based equations of the flu id

flow [16].

These corrections can be performed by the pre- or post-m ultip lication, add ition

or whole replacement of the A IC m atrix . This approach is adequate for engineering

applications as i t is less expensive than the direct use o f com putational flu id dynamic

techniques. The correction techniques, which have been applied to unsteady loading

calculations for aeroelastic analysis, are classified in four m ajor classes: (1) force

m atching methods [12], [18], [19], [20], [21]; (2) pressure matching methods [16], [22],

[23], [24], [25], [26], [27]; (3) Dau-Garner type methods [28], [29], [30], [31]; and (4)


modal aerodynamic influence coefficients (M A IC ) m a trix replacement [32], [33].

The first procedure matches reference nonlinear forces and moments which may be

obtained from experiments or Com putational F lu id Dynamics (C FD ) simulations. In

this case, nonlinearities such as pressure jum ps due to shock waves and viscous effects

are embedded in the reference quantities. The second procedure matches reference

nonlinear pressures, thereby m ainta in ing the same nonlinearities as the first method.

The th ird procedure employs steady nonlinear in form ation and semi-empirical rela

tions in order to compute the unsteadiness of the resulting nonlinear corrected pres

sures. The fourth procedure generates a M A IC m a trix which is referred to measured

or computed nonlinear pressures or loading due to given modal displacements o f the

lift in g surface. This new m a trix is substituted in the aeroelastic equations o f mo

tion where the generalized unsteady aerodynamic forces are related to the associated

modal displacements of the lift in g surface or downwash mode shapes. Commonly

employed methods for each class w ill be discussed in Chapter 3.

1.3 .3 T im e M arching M eth o d s

A feasible solution to the transonic aeroelastic problem is to complete a tim e marching

analysis where a CFD solver is coupled w ith a Computational S tructu ra l Dynamics

(CSD) solver. In these simulations, the structure is given an in it ia l velocity in one

o f the dominant modes and the subsequent tim e evolution of the m odal response is

calculated to see whether it grows or decays. The flow is usually simulated by solving


9

the Euler or Reynolds’ Averaged Navier-Stokes (RANS) equations. AERO -F and

AERO-S and CFL3D are examples o f C SD /C FD codes which have been successfully

used to simulate flu tte r on a complete a ircraft [34],

In flu id-structure in teraction problems, the flu id solution is usually computed

on an Eulerian coordinate system, whereas the s tructura l component is solved in a

Lagrangian system. There are three classes of tim e marching codes: (1) a loosely

coupled code; (2) a direct approach code; and (3) a simultaneously coupled code.

A loosely coupled code w ill solve the flu id and s tructura l systems u tiliz ing two

separate codes. Load and deformation in form ation is transferred between the sep

arate codes using an interfacing system. The modal approach is used in order to

reduce com putation costs. The advantage of this approach is th a t well established

flu id and structura l codes may be employed. The disadvantage is tha t errors can

be introduced during the transform ation of in form ation and sequencing between the

codes [3]. In the direct approach, the aerodynamic forces are calculated by the CFD

code and are mapped onto the structura l nodes. The CSD code calculates the struc

tu ra l response which is interpolated back onto the CFD grid. The CFD code then

again calculates the aerodynamic forces and this continues u n til a defined convergence

criteria is met. The direct approach is more accurate than the loosely coupled modal

approach. The disadvantage is tha t a high com putation cost is occurred in pu tting

and ou tpu tting the CFD and CSD responses. The fina l approach is to combine the


10

f lu id and structura l solvers in to a single code, referred to as a simultaneously coupled

code. The com putational tim e is reduced at the cost o f increasing the com plexity of

the code [3]. A brie f h istory of the development of tim e marching methods follows;

refer to Reference [3] for a detailed history.

In 1982, Reference [35] created a loosely coupled aeroelastic code, XTRANS3S.

Th is was one of the firs t nonlinear transonic f lu tte r analyses where the flu id was

modelled using the Transonic Small Disturbance (TSD) equation and the s tructura l

deformation was represented by the modes of the structure. Reference [36] fu rther

developed the TSD technique resulting in a new code, the Com putational Aeroe-

las tic ity Program-TSD (CAP-TSD). Reference [37] used CFL3D, a modified RANS

code, to calculate wing flu tte r using Navier-Stokes aerodynamics on the A G A R D

445.6 wing [3]. In 1993, Reference [38] introduced the method o f direct coupling of

a plate FE model w ith an Euler aerodynamic solver. In 1999, Reference [4] created

a loosely coupled aeroelastic code using a transform ation scheme based on the Con

stant Volume Tetrahedron (C V T) technique. F lu tte r calculations were performed on

the AG AR D 445.6 wing and the M DO wing. Reference [39] performed an aeroelastic

sim ulation o f an F -16 aircraft using a direct approach. In 2003, Reference [34] applied

the three field A rb itra ry Lagrangian-Euler form ulation of the Euler and Navier-Stokes

equations over an F-16 aircraft using a detailed s tructura l FE model.


11

1.4 Basis and O verview o f Current W ork

The aircraft industry requires a robust and efficient aerodynamic too l for use in

transonic flu tte r analyses. Presently, a subsonic linearized discrete lift in g surface

theory, the D LM , is employed but it cannot capture the flow nonlinearities in the

transonic regime. As discussed above, these nonlinearities can be captured using

two different approaches: (1) correcting the D LM aerodynamic data in a frequency

domain analysis using CFD or experimental results; or (2) perform ing a tim e marching

simulation using Euler or Navier-Stokes equations.

The objective of the present work is to compare flu tte r boundaries of a fin ite w ing

using both uncorrected and corrected D LM aerodynamics in a frequency domain

analysis w ith a tim e marching analysis using Euler and Navier-Stokes equations. A

single A IC correction procedure w ill be analyzed th a t matches reference unsteady

pressures obtained from CFD simulations undergoing a rig id body pitching m otion.

Uncorrected frequency domain results w ill be obtained using D LM aerodynam

ics in M S C /N A S TR A N and K E D LM P L. K E D LM P L, developed at the In s titu to de

Aeronautica e Espago in Brazil, is a subsonic D LM code which performs a K E -flu tte r

analysis using the A IC m a trix approach. The A IC correction procedure w ill be im

plemented in to this code as current versions of M S C /N A S TR A N w ill not allow direct

access to the A IC matrices.


12

The tim e marching simulations w ill be performed using the P M B (Parallel M u lti-

Block) code developed at the University of Glasgow. This im p lic it code features a

finite-volume Euler and RANS CFD solver w ith a proven capability of accurately

capturing transonic effects [4], [40].

Chapter 2 w ill present the theoretical background of the Doublet-Lattice method

and the derivation of the frequency domain flu tte r solution procedure. Chapter 3

w ill derive the A IC correction procedure. The tim e marching procedure using Euler

and Navier-Stokes equations w ill be detailed in Chapter 4. The test case, includ ing

structural and flu id models, w ill be defined in Chapter 5. Results and suggestions for

future works w ill be presented in Chapter 6.


Chapter 2

Frequency Domain Formulation

2.1 U n stead y A erodynam ic M odelling

The classic approach in solving the unsteady compressible flow problem is through

the use of lift in g surface theory. Basic lift in g surface theory assumes tha t flow is

inviscid, isentropic, subsonic, and contains no flow separation. The thickness of the

surface is neglected and the angle of attack is small so such tha t the small-disturbance

potentia l flow approach may be used to linearize the mixed boundary value problem.

The compressibility effect is taken in to account in the aerodynamic governing equation

using the Prandtl-G lauert transform ation [1], [41], [42],

Based on a fin ite element approach, the D LM achieved wide acceptance due to its

simplicity, accuracy and versatility. This section presents an overview of the D LM for

use in flu tte r analyses. A brie f review of the D LM follows; for a detailed derivation,

refer to References [11], [14] and [15].

13

Reproduced with permission o f the copyright owner. Further reproduction prohibited w ithout permission.

14

2.1.1 O verview o f th e D o u b le t-L a ttice M eth o d

The linearized form ulation of the oscillatory subsonic lift in g surface theory relates the

normal velocity, W , at the surface

W = Uoo Re (weiuJt) (2.1)

to the pressure difference across the surface

A p = \U I Re ( A < V ” ‘ ) (2.2)

by a singular integral equation, transformed in to the reduced frequency domain,

where £ is the streamwise coordinate, a is the tangential spanwise coordinate (as

shown in Fig. 2.1), k = cub/U^ is the reduced frequency, ui is the frequency o f oscilla

tion, b is the root semi-chord, Ua0 is the freestream speed and A Cp is the dimensionless

lift in g pressure coefficient. The kernel, K , is the normalwash (or downwash) at a point

x, y, z induced by a pressure doublet of un it strength located at £, y, ( and is given in

Reference [15].

The lift in g surface is divided in to a grid o f trapezoidal elements over which the

lift in g pressure is assumed constant. Refer to Fig. 2.1. Element coordinates are used

to aid in perform ing the double integral of the kernel over each element. Let x, y, z and

A fj, 0 be coordinates of the receiving and sending points, respectively, in a coordinate

K (x - £, y - r], z - C, k, Moo) AC^ d£ da (2.3)


15

e i e

DOUBLET LINE

SENDING PANEL

DOWNWASH COLLOCATION POINT

RECEIVING PANEL

Figure 2.1: L ift in g surface idealization in the D LM .

system centred on and rotated in to the plane of the element. Refer to Fig. 2.2.

The double integral over the element is then performed in the £— and f j—directions.

The integration of K in the streamwise direction (£) is performed approximately by

concentrating the value o f the integrand at the one-quarter chord point, £1/4. Eq. (2.3)

becomes

A Gw [x8ir

K ( x - £1/4, V - i), 2, k, A L J dij (2.4)

where e is the semi-width of the element. The result is an unsteady horseshoe vortex

whose bound portion lies along the one-quarter chord line of the element. The change

in variable from a to fj indicates tha t element coordinates are to be used.

In Eq. (2.4), the downwash boundary condition w (x, y, z) is known and the lift in g


16

ELEMENT

Figure 2.2: D LM wing and element coordinates.

pressure, A C Pj, over each element is unknown. A set of linear algebraic equations

may be formed i f the downwash is satisfied for each element. Eq. (2.4) can be then

w ritten in m a trix form

The term D tj is the downwash factor at Xi,yi ,Zi due to element j . This term is

broken down in to a steady part and an unsteady increment and the solution must

satisfy two boundary conditions, the so-called K u tta condition and the flow tangency

condition. The K u tta condition states tha t the pressure difference at the tra iling edge

of a th in lift in g surface must be zero. The flow tangency condition states tha t for an

inviscid flow, the velocity vector must be tangent to the surface of the streamlined

body [42]. There is one control or receiving point for each element and the la tte r

{w } = [D] {AC „} (2.5)

where a typ ica l element of [D], D tj , is

(2.6)


17

boundary condition is satisfied at each of these points. The control po in t is centred

spanwise on the three-quarter chord line o f the element.

The inverse of the m atrix [D] m u ltip lied by the downwash vector, {w } , yields

the pressure d istribu tion. The resulting inverse m atrix is the A IC m a trix which is a

function of the reduced frequency and is related to the pressure d is tribu tion by

{ A Cp( ik ) } = [A IC ( ik ) ] (w (ifc )} (2-7)

The coefficients of the A IC m a trix may be interpreted as rates o f pressure variation

due to a given displacement am plitude inpu t associated w ith the boundary conditions.

The determ ination of the pressure coefficient vector in Eq. (2.7) is performed from the

known downwash which is related to the am plitude of the p itch and plunge m otion at

each element. The substantial derivative o f a given displacement mode is composed of

a derivative of the normal d irection displacement plus the associated velocity scaled by

the undisturbed speed. In a small disturbance sense, th is quan tity represents an angle

o f attack [16]. Therefore, from the boundary conditions for those small perturbations,

the relationship between the normalwash and a solid boundary displacement is w ritten

as

(w(iA :)} = dh(yXQ ^ ^ + ikh(x , y, 0) = [F( ik) \ {h (x , y, 0 )} (2.8)

The substantial derivative applied to a given modal displacement vector, {h } , is

denoted by the m atrix operator [F(zA:)]. The resulting aerodynamic loading vector,

{ L a( i k ) } may be expressed as the m ultip lica tion of the pressures by an integration


18

m atrix , [S'], which is constructed from the geometrical characteristics of the panel

elements [16]. The resulting final expression for the unsteady loading over the lift in g

surface is given by

{ L a( ik ) } = goo[S ] [A IC( ik ) } [F ( ik ) } { h } (2.9)

where q^ is the dynamic pressure and the m a trix [F( ik)} is given by

-3(.dx

+ ik(- (2.10)

2.2 A eroelastic M odel

2.2 .1 A eroelastic E q u ation s o f M otion

The equation of m otion for a structure in an airstream can be represented as an

equilibrium between the structura l and aerodynamic forces as

iM \ {« (* ) } + \K \ {« ( * ) } = { L ( u ( t ) , u ( t ) ) } (2.11)

where [M\ and [K] are the structura l mass and stiffness matrices, respectively, and are

usually obtained from a fin ite element model of the body. The structura l deformation

is denoted as {u } . The aerodynamic forces applied on the structure are denoted

as { L } . The terms [M\ { u ( t ) } and [K] ( u ( t ) } are referred to as the inertia l and

elastic structura l forces, respectively. The structura l damping m atrix is excluded

from Eq. (2.11) for illus tra tion purposes. S tructura l damping, typ ica lly small for an

aircraft, w ill appear later in the derivation as a rtific ia l damping. The aerodynamic


19

forces can be sp lit in to two components: (1) the aerodynamic forces induced by the

structural deformation, { L a(u ( t ) ,u ( t ) ) } ; and (2) the external forces, { L e(t)}. Typ ica l

examples of external forces are gust loads, atmospheric turbulence, store ejection

forces and p ilo t induced control surface aerodynamic forces [10]. These forces are

considered for dynamic system response investigations. The induced aerodynamic

loads may be generated by unsteady aerodynamic theories. Since these loads depend

on the structura l deformation, the relationship can be interpreted as an aerodynamic

feedback system [16]. Eq. (2.11) is rew ritten as

[M\ { u ( t ) } + [K] { u ( t ) } — { L a(u ( t ) , i i ( t ) ) } = { L e( t ) } (2.12)

The investigation of the dynamic s tab ility of Eq. (2.12) is known as the flu tte r

problem [16]. F lu tte r analysis usually involves the search of the structura l s tab ility

boundary of an aircraft structure in terms of its fligh t speed and a ltitude or the cor

responding dynamic pressure. This boundary is determined by examining the decay

or growth of the structura l response w ith respect to the fligh t speed. The struc

tu ra l dynamic model, which is the left hand side of Eq. (2.12), is generally linear

as amplitudes of deformations are small for aeroelastic phenomena [1]. When the

aerodynamic response may be assumed linear w ith respect to the structura l deforma

tion, Eq. (2.12) is a linear system. The flu tte r boundary is determined by solving the

complex eigenvalues of th is system.


20

The solution o f the eigenvalue problem requires the homogeneous aeroelastic sys

tem of equations

(2,13)

be w ritte n in to the Laplace domain. The transform ation o f the induced aerodynamic

forces to the Laplace domain is performed by a convolution integral

{ L a( u ( t ) , u ( t ) ) } = qoo Jo

(2.14)

where [H] represents the aerodynamic transfer function m atrix . The aerodynamic

loading in the Laplace domain is w ritten as

{Ta (^ (^ )) } Qoo Hsb

uZ. M s ) } (2.15)

w ith B ( & ) H { ^ f ^ ) and sb/Uoo being theas the Laplace domain counterpart of

non-dimensional Laplace variable. The resulting eigenvalue problem is obtained by

transform ing Eq. (2.13) to the Laplace domain as

s2 [M ] + [K \ - qoo

2 .2 .2 M od al A pproach

Hsb

UZ M s) } = o (2.16)

Since the fin ite element model of a ircraft structure norm ally contains a large amount

of degrees of freedom, the size of the mass and stiffness matrices can be substan

tia l. D irectly solving the eigenvalue problem of Eq. (2.16) would be com putationally


21

costly [10], [16]. Transforming the structura l dynamic system to modal space circum

vents this problem. The resulting eigenvalues are known as the natura l frequencies

and the eigenvectors are the associated mode shape vectors. Therefore, in a modal

representation, the structura l displacements are w ritten as a linear combination of

the generalized coordinates, {q ( t ) } , and the physical displacements of the structure,

{u ( f) } , as

{« (* ) } = [$] {? (*)} (2-17)

where [<f>] is the modal transform ation m a trix whose columns contain mode shape

vectors, each associated to a given natura l frequency o f the structure. Normally, no

more than 10 of the lowest natura l modes are sufficient for a flu tte r analysis o f a

wing. The rationale of the modal approach is based on the premise tha t the c ritica l

flu tte r modes are usually due to the coupling o f lower order structura l modes. Thus,

the structura l deformation of the flu tte r mode can be sufficiently represented by the

superposition o f lower order modes [10]. The resulting aeroelastic system of equations

w ritten in the modal space is

+ \ K \ — ?oo Q l — (g (s )} = 0 (2.18)

where [M ] = [(h]T [M] [<f>] and [ih ] = [$ ]T [K] [$] are the generalized mass and s tiff

ness matrices, respectively, and

e i —^ ' U.™ = t * r H' sb

[$ ] (2.19)


22

is the generalized aerodynamic force m atrix , w ith (? (s )} being the generalized coor

dinates vector in the Laplace domain.

2.2 .3 S im p le H arm onic M o tio n A pproach

The com putation of the aerodynamic transfer functions in the Laplace domain can be

extremely d ifficu lt. For this reason, unsteady aerodynamic methods are often form u

lated in the frequency domain assuming simple harmonic motion. The aerodynamic

transform function, in a simple harmonic reduced frequency domain, is called the A IC

m a trix [10].

The induced aerodynamic loading is obtained by a convolution integral which

transforms the aerodynamic loading from the tim e domain to the Laplace domain.

Since the A IC m a tr ix is only available in the simple harmonic reduced frequency do

main, the im aginary counterpart of aeroelastic system o f equations, given by Eq. (2.18),

is obtained w ith iuj replacing s leading to

’ -c u 2 [M ] + [ K \ - Qoo [Q{ik)\

w ith { u ( i u ) } = [<f>] {q( ic j) }.

2 .2 .4 In tercon n ection o f th e S tru ctu re w ith A erod yn am ics

The location and size of the aerodynamic panels may be independent of the struc

tu ra l fin ite elements in order to allow for discretization optim ization. A n in terpo lation

{q(ico)} = 0 (2.20)


23

procedure is required to pass the displacements of the s tructura l nodes to the aerody

namic control points and forces from the aerodynamic control points to the structura l

nodes. This procedure is based on spline function approximations [16]. A spline ma

trix , [G], is generated using the beam spline method, the in fin ite plate spline method,

the th in-p late spline method or the rig id-body attachment method, such tha t

{h(x , y, 0 )} = [G] { u (xs,ya, zs) } (2.21)

where { u ( x s,ys, zs) } is the fin ite element model nodal displacement vector and the

displacement vector at the aerodynamic control points is {h(x , y, 0 )}. Once the spline

m atrix has been generated, the force transferral from the aerodynamic control points

to structural nodes can be performed accordingly

{dh(x, y, 0 ) }T { L aaero (x, y, 0 )} = {5u (xs, ys, zs) } T { L satr (xs, ys, zs) } (2.22)

where {Sh(x, y, 0 ) }T and {Su (xs,ys, zs) } T are the v ir tu a l displacements. The aero

dynamic loading at the structura l nodes and at the aerodynamic control points are

denoted as { L sJ;r (xs, ys, zs) } and {T “ero (x, y, 0 )}, respectively. The relationship be

tween the loads is given by

{ L f (* .,» „* .) } = [G ? { L T ( x , y , 0)} (2.23)

Replacing Eq. (2.10) in Eq. (2.23), using Eq. (2.21), yields

{ L T (x s, ys, zs, i k ) } = qoo [G]T [S\ [A IC { ik ) \ [F( ik) ] [G] { u (x s, ys, zs) } (2.24)


24

The generalized aerodynamic forces m atrix , in the reduced frequency domain, is

given by

[Q(ik)] = [<f>JT [G]T [5] [A IC ( ik ) } [.F ( i k )] [G] [<f>J (2.25)

2.3 F lu tter Solution Techniques

The stab ility o f an aeroelastic system is evaluated by using f lu tte r solution techniques.

These methods are based on the solution of the eigenvalue problem w ith respect to a

given parameter variation. Three methods of analysis are available: (1) the American

(K) method; (2) a restricted but more efficient American (KE) method; and (3) the

B ritish (PK) method.

The K-m ethod of flu tte r analysis considers the aerodynamic loads as complex

masses. Therefore, the method becomes a v ib ra tion analysis using complex arithm etic

to determine the frequencies and a rtific ia l dampings required to sustain the assumed

harmonic motion. S tructura l damping is incorporated in to the analysis as a m atrix .

As the K-m ethod is a looping procedure, caution must be used i f a large number of

loops are specified; the analysis may take an excessive tim e to execute.

The KE-m ethod is ideal when hundreds of flu tte r analyses are required. The

KE-m ethod is sim ilar to the K-m ethod; by restricting the functionality, the KE-

method becomes more efficient than the K-method. Two m ajor restrictions are: (1)

no damping m a trix is allowed; and (2) no eigenvector recovery is made. The effects of

s tructura l damping are, however, included in the complex stiffness m atrix. Eq. (2.20)


25

becomes

- lo2 [M \ + (1 + i g ) [K \ - [Q{ik)\ { q ( iu ) } = 0 (2.26)

where g is the added a rtific ia l damping. For both the K - and KE-methods, the te rm

is a mathematical quantity tha t cannot be readily related to the physical system

damping. However, these methods are capable of provid ing the correct prediction of

the flu tte r boundary, which, in other words, is the flu tte r speed at zero damping.

The PK-m ethod not only determines stab ility boundaries bu t provides approxi

mate and realistic estimates of system damping at subcritica l speeds tha t can be used

to m onitor fligh t flu tte r tests. The PK-m ethod treats the aerodynamic matrices as

real frequency dependent springs and dampers. This method requires less eigenvalues

to solve, thereby resulting in a more efficient solver.

A lthough M S C /N A S TR A N is capable o f perform ing flu tte r analyses u tiliz ing the

K-, KE- and PK-methods, K E D LM P L can only u tilize the KE-m ethod. Thus, for

consistency between the frequency domain results, the KE-m ethod w ill be applied for

flu tte r computations in the present investigations.


Chapter 3

AIC Correction Formulation

3.1 O verview o f AIC C orrection M eth od s

Several procedures have been developed which relate measured or computed data to

the unsteady linear aerodynamic model in order to account for nonlinear effects. The

correction techniques are classified in four m ajor classes: (1) force matching methods;

(2) pressure m atching methods; (3) Dau-Garner type methods; and (4) M A IC m a trix

replacement. Comm only employed methods for each class w ill be brie fly discussed.

Detailed in fo rm ation is available in Reference [16].

3.1.1 F orce M atch in g M eth o d s

The modified s trip theory was the firs t correction technique; l i f t and moment coeffi

cient derivatives obtained from Theodorsen’s method were replaced by the derivatives

determined from measurements. The published results were in good agreement w ith

experimental results for the case of high aspect ra tio wings only [12], [16].

26


27

Reference [18] developed a method to correct the subsonic discrete lift in g sur

face theory based on a reference experimental or nonlinear computed loading. Th is

method included a procedure to obtain a pre- or post-m ultip licative operator to cor

rect the A IC m a trix by the use of weighting factors. The purpose o f such operators

was to match a single reference steady spanwise loading o f the lift in g surface. Each

spanwise station corresponded to a set o f panels aligned in the streamwise direction,

thereby resulting in less known quantities than the necessary weighting factors. In

order to circumvent th is problem, the weighting factors were obtained using the least

squares method and thus, there is no guarantee tha t the modified pressure d is tr i

bution resulting from the correct A IC m a trix w ill be identical to the experimental

or nonlinear computed pressures. In the presence of shock waves, th is pressure dis

to rtion can severely m odify the aeroelastic behaviour in the transonic regime as the

nature o f the transonic d ip depends on the position and strength o f the shock wave.

Nevertheless, the modified pressure d is tribu tion w ill guarantee correct sectional l i f t

and moments [16], [18].

A method to correct a linear stripwise A IC m atrix was presented by Reference [20]

which related s trip displacements to the ir loads. This procedure was based on a post

m ultip lica tion of the stripwise A IC m a trix by weighting factors which were obtained

from the inversion of the A IC m atrix m u ltip lied by the reference loading condition.

The reference loading condition was obtained from the solution o f the transonic small


28

disturbance equation in a steady flow condition. The results showed agreement w ith

experimental flu tte r speeds. Small differences between the results were a ttribu ted to

the inaccuracy of the predicted strength and location of the shock wave using the

numerical solution of the transonic small disturbance equation.

3.1 .2 P ressu re M atch in g M eth o d s

Reference [22] presented a procedure based on the pre-m ultip lica tion of the A IC

m atrix by a diagonal weighting operator which was obtained from the ra tio be

tween the experimental and theoretical steady pressures. Sim ilarly, Reference [23]

pre-m ultip lied the A IC m a trix by a diagonal weighting operator based on the ra tio

between measured and theoretical unsteady pressures related to a given downwash

mode shape. Reference [24] calculated a diagonal weighting operator obtained from

the ra tio between the nonlinear and the linear quasi-steady pressure slopes. The

correction was performed in two forms, as a m ultip lica tive weighting operator or

as an additive weighting operator. The results showed incorrect trends for computed

unsteady pressure results as well as nonconservative flu tte r speeds [16]. The computa

tion of the correction factors presented in References [22], [23] and [24] also produced

badly conditioned corrected A IC matrices for zero or very low pressure values.

Reference [25] presented an additive correction procedure based on the increment

of the linear A IC m atrix by a frequency dependent nonlinear term. The nonlinear


29

term was obtained from a nonlinear unsteady CFD com putation o f the lift in g sur

face undergoing an impulse-type displacement. The pressures were computed from

a discrete Fourier transform of the aerodynamic response in the tim e domain. The

resulting frequency dependent unsteady pressures were then used to obtain the non

linear term of the A IC m a trix at each reduced frequency. This method was more

expensive than previously discussed approaches as i t required a refined tim e march

ing CFD solution. The results showed agreement w ith experiments for standard

aeroelastic w ing configuration flu tte r computations only [16], [25].

A n extension o f the correction method of Reference [18] was presented by Ref

erences [26] and [27] in which pressures were used instead of loads as reference con

ditions. This procedure employed m ultip le reference pressure conditions associated

w ith a reference mode shape. The number of the given reference steady pressures was

smaller than the order of the A IC m atrix . Therefore, the resulting pressure m atrix ,

which contained the given pressures due to prescribed mode shapes, was a rectangu

lar m atrix. The com putation of the weighing operator is performed by the inversion

of the m atrix of given pressures, requiring a m in im ization technique. M in im ization,

however, was avoided by computing of a set of linear steady pressures based on a

linearly independent a rtific ia l mode shape. Since these complementary pressure con

ditions were known, they were included in the given reference pressure m atrix . A

modified square pressure m a trix was then constructed in which each of its columns


30

were related to a given modal displacement [16]. This pressure m a trix was inverted

and m ultip lied by a linear pressure m atrix . The resulting weighting operator was a

fu lly populated m atrix . As D LM interference effects were not strongly modified, th is

procedure avoided severe pressure distortions. Subsonic flu tte r calculations were pre

sented showing agreement w ith the experimental results; however, fu rther extension

in to the transonic regime yielded poor results [26].

Reference [16] presented a pressure correction based on downwash weighting which

preserved the mean steady flow nonlinear characteristics. The control point displace

ment vector was modified in order to satisfy a given reference pressure d is tribu tion .

The advantage of this method was tha t the computed weighting factors depended

on a linear system of equations instead of a simple pressure ratio. The solution of

th is system d id not yie ld incorrect weighting factors even for reference conditions

containing nu ll displacements or pressures. The second advantage was tha t the A IC

matrices resulting from the conventional modelling, based on discrete element solu

tions of the linearized potentia l flow equations (i.e. the D LM ), related pressures to

displacements at control points. Therefore, i t was feasible to employ th is pressure

matching technique for the correction o f the discrete element models. Reference [16]

presented results using nonlinear reference conditions computed from steady state

and unsteady CFD solutions. The results showed agreement w ith experimental f lu t

ter speeds for unsteady CFD solutions as a weighting operator was computed for all


31

reduced frequencies in the aeroelastic analysis. Pressure matching procedures based

on steady reference conditions failed in obtaining unsteady pressures at higher reduced

frequencies due to the absence of unsteady transonic flow in form ation in the refer

ence conditions. The magnitude of the pressures were well approximated; however,

the phases com putation presented wrong trends.

3.1 .3 D au-G arner T yp e M eth o d s

The objective o f Dau-Garner correction methods was to compute approximate non

linear unsteady pressures in the frequency domain for use in transonic flu tte r calcu

lations. The pressure com putation was performed w ith the aid o f steady state semi-

empirical relations. The theoretical background for th is class o f correction methods

was introduced by Reference [28] and the extension of the theory was performed by

Reference [29], resulting in the Dau-Garner method.

The basic hypothesis of the Dau-Garner method was an em pirical relation tha t

assumed an equal ra tio between unsteady and steady velocity potentia l gradients

whether the flow was transonic or purely subsonic. The unsteady velocity potentia l

gradient was expressed as a function of the given steady state velocity potentia l gra

dient. The ra tio between the linear unsteady and steady velocity potentia l gradients

was computed from a linear theory such as the D LM [16].

An additional condition was the relation between unsteady pressures and the

unsteady velocity potential, which was derived from the Euler momentum equations.


32

The steady velocity potentia l gradient was w ritten as a function of a steady state

pressure d is tribu tion , as given by the pressure to velocity potentia l re lation derived

from the Euler momentum equations. Therefore, the unsteady velocity potentia l

d is tribu tion was w ritten as a function of steady pressure d is tribu tion times the ra tio

between the unsteady and steady velocity potentia l gradients. The in troduction of

the nonlinearities associated w ith transonic flow was performed by the substitu tion

of the steady state linear pressures by the nonlinear pressures. This method m odified

both the real and imaginary parts o f the pressures w ith respect to a nonlinear steady

state pressure d is tribu tion [16], [29], [28], [30].

The published results showed inconsistencies w ith the computed phase compo

nents. In most cases, the computed phases differed by approximately 50 degrees from

the experimental results.

3 .1 .4 M od al A erodyn am ic Influence C oefficients M atrix M eth o d s

Body motions may be w ritten in terms of modal generalized coordinates instead of

physical displacements coordinates. The m a trix operator which relates the resulting

pressures modes to displacement mode shapes is called the modal aerodynamic in flu

ence coefficient m a trix (M A IC ). The M A IC m a trix computed from linear theory is

replaced by an externally computed m a trix which accounts for nonlinear aerodynamic

behaviour [16], [33].


33

The only M A IC method published is the Transonic Equivalent S trip (TES) method

of Reference [32], This procedure required two consecutive correction steps, a strip -

wise correction followed by a spanwise correction. In the first step, a chordwise mean

flow correction is performed based on steady pressures obtained from CFD or exper

im enta l sources. A n inverse a irfo il design procedure is employed to generate a new

profile of an equivalent a irfo il for each o f the spanwise stations o f the lift in g surface.

The com putation o f the unsteady pressures follows and was based on a tim e domain

impulse response fin ite difference solution of the transonic small disturbance equa

tion . The unsteady pressures in the frequency domain are computed from the Fourier

transform of the tim e response of these pressures. The second step consisted o f a

spanwise correction of the two-dimensional unsteady pressures in order to account for

three-dimensional effects [16], [32], [33].

The published flu tte r results for cases involving isolated lift in g surfaces were in

good agreement w ith the experimental data. The TES method is, however, unable

to efficiently handle complex typ ica l section profiles such as a combination o f wing-

pylon-nacelles or external stores w ith the wing profile [16].

3.2 D erivation o f Selected C orrection M ethod

The selected correction procedure for the present work was the unsteady pressure

m atching method of Reference [16]. This method, based on a downwash weighting

approach, has the follow ing advantages: (1) the nonlinear flow characteristics in the


34

transonic regime are preserved; (2) reference conditions can be based on unsteady

CFD or experimental data; (3) reference conditions containing nu ll displacements or

pressures do not yield incorrect weighting factors; and (4) ease o f im plementation in to

the DLM.

The following section w ill detail the A IC correction procedure. Refer to Refer

ence [16] for the detailed derivation of the steady and unsteady correction procedure.

3.2.1 T h eoretica l D erivation

The basic equation over which the procedure is developed is the algebraic pressure to

downwash relationship tha t is derived from the application of discrete element kernel

function methods to model the linearized potentia l flow equation [16]. This re lation

is rew ritten in a simplified form as a function only o f the reduced frequency as

{ A C p{ ik ) } = [ A I C ( i k ) ] { w ( i k ) } (3.1)

The reference conditions are based on frequency-dependent pressures computed

from the tim e domain aerodynamic response. Pressure tim e histories result from

lift in g surface displacements due to a prescribed motion. These displacements are

arb itrary and may be an impulse-type or an oscillating harmonic m otion [16]. For

the present work, the unsteady pressure differences com putation was performed based

on a tim e domain CFD solution of the Euler and Navier-Stokes equations, using the

fin ite volume approach presented in Chapter 4.


35

The chosen displacement may be any unsteady motion associated w ith a displace

ment mode shape of the lift in g surface. In the present work, the m otion selected was

a rig id body harmonic p itching ro ta tion around the root m idchord axis of the lift in g

surface at a reduced frequency, k. This m otion leads to an unsteady downwash, re

garded as an unsteady perturbation in angle of attack w ith the am plitude equal to

the amplitude of the p itching motion, A a [16].

Nonlinear pressure coefficient differences, denoted by the superscript nl, are com

puted from the solutions of the Navier-Stokes or Euler equations as

{ACp"‘ } = {<7'} - {C«} (3.2)

where the superscripts I and u denote the lower and upper surfaces o f the lift in g

surface, respectively.

As the linear aerodynamic model is defined in the frequency domain, the reference

tim e domain pressure differences must be transformed in to the frequency domain. The

following Fourier transformations were employed in order to obtain the firs t harmonic

frequency domain components of the pressure differences

b. (‘Ti+n/kRe (A C f { , k ) ) = ^ / A C„( t ) s in (fc r)d r (3-3)

U n T i + n / k

Im (A C ? ( i k ) ) = — / A C „(t ) cos(fcr)dr (3-4)

where r is the non-dimensional time, as defined in the non-dimensionalization of

the Euler and Navier-Stokes equations; refer to Chapter 5 for the validation o f the


36

computational procedure for unsteady flow computations.

The complex first harmonic components of the pressure coefficient differences are

obtained from the real and im aginary part of the firs t harmonic components, given

by Eqs. (3.3) and (3.4) as

{ A C ;1} = (R e { A C ; l ( i k ) } + Im { A C%l ( i k ) } ^ (3.5)

Eq. (3.1) is rew ritten as a function o f a frequency dependant weighing operator,

W T, given by

{ A C ^ } = [ A IC { i k ) \ [W T ( i k ) ] (w (iA :)} (3.6)

where {w (i/c )} is the downwash vector divided by the am plitude o f the p itching

motion, A n . The diagonal weighing m a trix coefficients are obtained from the ra tio

between a modified unsteady downwash, computed by

{w n l(ifc)} = [ A I C { i k ) Y l { { \ C ; 1} (3.7)

and the known downwash associated w ith the prescribed lift in g surface motion, lead

ing to [16]

( W T ( i k ) it) = Y c k ) \ ‘ (3 '8)

Therefore, the nonlinear unsteady aerodynamic loading is performed by incorpo

rating the correcting weighting operator in to Eq. (2.10), leading to an approximate

aerodynamic loading given by [16]

{ L f ( i k ) } = qoo [5] [A IC ( ik ) \ [W T ( i k ) \ (w(zA;)} (3.9)


Chapter 4

Time Marching Formulation

4.1 N um erical Fluid M ethods

The tim e marching simulations w ill be performed using the PM B (Parallel M u lt i-

Block) code developed at the University of Glasgow. This im p lic it code features a

finite-volume Euler and RANS CFD solver w ith a proven capability o f accurately

capturing transonic effects. A brief derivation of the Euler and RANS solvers follows.

For detailed derivations, refer to References [4], [40], [43] and [44].

4.1 .1 Euler E quations

The three-dimensional Euler equations can be w ritten in non-dimensional conservative

form as

a w aF o g a H , .~m + ~ d i + ~ d i + ~d7 ~ ° (4 1 )

37


38

where the vector W = (p, pu , pv, pw , p E )T denotes the vector of conservative variables.

The flux vectors, F , G and H , are

PC \

puU + p

F = pvU

pwU

y U (pE + p) + x tp

( p v N

puV

G = pvV + p

pwV

^ V ( p E + p ) + ytp

( p W ^

puW

H = pvW

p w W + p

^ W (p E + p ) + ZtP y

where p, u, v, w, p and E denote the density, the three Cartesian components o f the

velocity, the pressure and the specific to ta l energy, respectively. The terms U , V and

W are the contravariant velocities defined by

U = u — x t

V = v - y t

W = w - z t (4.3)

where x t , yt and zt are the grid speeds in the Cartesian directions.


39

4.1.2 N a v ier -S to k es E q u ation s

The three-dimensional Reynolds Averaged Navier-Stokes (RANS) equations can be

w ritten in non-dimensional conservative form as

<9W dFi - Fu dGl - G*' dW -+ ~ ~ + — ----------= 0d t dx dy dz

(4.4)

The conservative form of the Navier-Stokes equations are obtained through a

derivation based on a spatia lly fixed control volume. The flux vectors F, G and

H are composed o f inviscid terms as described w ith the superscript i and diffusive

viscous terms as described by the superscript v [45]. The inviscid flux terms are

/ pU ^

puU + £xp

F* = pVU + £yP

pwU + £zp

p U H

pV

puV + r) xp

G l = pvV + rjyp

pw V + pzp

p V H

pW

puW + qxp

H l = pvW + qyp (4.5)

pw W + qzp

p W H

where H denotes enthalpy. The terms U, V and W are defined in the local curvilinear

V


£, r), c, coordinate system as

The viscous terms are

U = i xu + £yv + £zw

V = 7]xu + rjyV + r)zw

W = qxu + qvv + qzw

= __1

Re

TV

T,xy

y UTXX T VTXy T WTXZ T Qx J(

G v =1

Re

\T,xy

T,yy

T ,yz

y UTXy T TTyy “I" WTyZ “f" ([y J

0/

a v = —i

Re

Tx

T,yz

t .

U T ZX + V T zy + W T ZZ + q z


41

where Re denotes the Reynolds number. The stress tensors are defined as

T xx = + P i )

yy — (// + Hr)

Tzz = - (M + pT )

r Xy = - ( f i + p r )

r Xz = ~ + A r)

V = - 0 + /at)

du 2 f du dv dw2 1 1-------

dx 3 \ d x dy dzdv 2 / du dv dw

^d y 3 y d x ^ dy~^~ dzdw 2 f du dv d w \

2— - - I — + — + — I

2 , + 5 pk

dz 3 \ d x % dz y du dv dy dx du dw dz dx dv dw dz dy

2 , +r k

(4.8)

where k, [It and P tt are the tu rbu len t k inetic energy, the eddy viscosity and the

turbulent P rand tl number, respectively. The heat flux vectors are defined as

1Qx —

% =

Qz =

( t - m ii

( t - m :i

2oo

P r P vt

M MrP r P r i

V> HrP r P r i

d Tdxd T

dyd T

dz(4.9)

(7 - m i

where T denotes the static temperature.

The Navier-Stokes equations perm it the complete and accurate solution o f a phys

ical flow. In order to allow th is solution to occur in a tim e ly manner, turbulence

modelling is introduced as a means o f predicting turbulent transition, flow separation

and flow reattachment. The RANS equation decomposes the flow variables in to a

tim e averaged component and a turbu lent fluctua tion describing the instantaneous

variation from the mean value [43].


42

The turbulence properties o f the flow are modelled by the turbu lent viscosity,

turbulent kinetic energy and an additional set of closure equations. The closure

equations, known as turbulence models, present a m athematical resolution for the

effects o f turbulence. Refer to References [43] and [45] for detailed derivations.

4.1 .3 S tead y S ta te Solver

The Euler and Navier-Stokes equations are discretized onto curvilinear m ulti-b lock

body conforming grids using a cell-centred fin ite volume method. The pa rtia l d iffer

ential equations of Eqs. (4.1) and (4.4) are converted in to a set o f ord inary differentia l

equations which can be w ritten as

4 = - R u , * (W ) (4.10)

where W and R are the vectors o f the cell conserved variables and the cell residuals,

respectively. The volume of the cell is denoted by Vy,fc. The convective terms are

discretized using Osher’s upwind method. M USCL variable extrapolation is used

to provide second-order accuracy w ith the Van Albada lim ite r to prevent spurious

oscillations around shock waves [44], Two layers of ghost cells are generated outside

of the com putational domain provid ing boundary conditions. For the far-field, the

ghost cells are set equal to the freestream conditions. For viscous flows, the no-slip

boundary condition is imposed at the wall. For inviscid flows, the normal component

of velocity is set equal to zero [45].


43

A numerical in tegration allows the solution of the set o f differentia l equations

created by the fin ite volume method. The steady state solution is determined from

an im p lic it tim e-m arching scheme according to

W n+} - W " ■ j. 1A t t4-11)

where the superscripts n and n + 1 refer to tim e n A t and (n + l ) A f , respectively. The

numerical in tegra tion expressed above represents a system of non-linear equations.

In order to s im p lify the solution procedure, the residual term is linearized

(w “ ) + (-4-12)

I W ( W " + 1) = R ijj, (W ” ) + + O (A t2)

~ R U,fc(W ) + a w . . , ^ A t

d R ij,

9 W i,j,k

where A W y ^ = W a n d the integration becomes linear according to

= -R?Jifc(W ") (4.13)Vi,j,k j ^ j,/cA t d W i jykj

The above linear system is solved through a Generalized Conjugate Gradient

(GCG) method. The GCG method provides an approximate solution of a linear

system by m in im iz ing an error function in a finite-dimensional space of potentia l so

lu tion vectors. The preconditioning strategy for the linear system is based on a block

incomplete lower-upper (B ILU ) factorization. This was selected because the sparsity

pattern of the B IL U method is the same as tha t of the Jacobian m atrix. The B ILU


44

factorization is decoupled between blocks and thus, does not restrict the efficiency of

parallei com putation [45].

Since im p lic it schemes require special treatm ent p rio r to the fu ll fledged resolution

of a com putational domain, the flow is in it ia lly solved explicitly. This allows for some

smoothing of the flow field before the solver can be switched to the less robust bu t

much faster im p lic it scheme. The resulting Jacobian m a trix has more than one non

zero entries per row [43].

The solution for the steady state turbu lent case is identical to the steady state

mean flow case described above. The eddy viscosity is determined from the tu rb u

lent kinetic energy, and other pertinent values, and then used to calculate the mean

flow values, which are in tu rn used to update the turbulent terms. A n approximate

Jacobian m atrix is used as the source term by disregarding the production terms and

accounting only for the dissipation terms [45].

4 .1 .4 U n stea d y Solver

The unsteady solver enables time-accurate simulations of unsteady flows on meshes.

The case described here is for turbulent flow and accounts for mesh deformation.

The lam inar and inviscid cases are calculable through the e lim ination of terms and

constant mesh size is solvable w ithou t adjustment [45].

The basis of the unsteady solver is the m u ltig rid false-time form ulation developed

by Jameson. The Jameson form ulation for the updated steady state problem is given


+ (w '“ *, = 0 (4.15)

where the superscripts km, kt , It and l t represent the tim e level o f the spatia lly de-

scritized variables. In th is form ulation, the only m odification to the lam inar case is

the addition o f the eddy viscosity from the previous tim e step. Since the turbulence

model is tied only to the eddy viscosity, i t allows a two equation turbulence model to

be implemented w ith o u t m odification to the mean flow solver and is the method of

choice for the com putational code [45].

In order to solve Jameson’s equations for the updated steady state problem, a

false or pseudo tim e, r , is introduced, yielding

where the superscript n continues to describe the real tim e step and the superscript

m is introduced in order to describe the pseudo tim e step. A n im p lic it tim e stepping

method for pseudo tim e is employed in PM B, w ith km = lm = l t = n — l , m — 1 and

kt = n — 1, m. The solution o f the equations is decoupled by freezing values and the

real tim e stepping proceeds w ithou t sequential error thereby provid ing second order

accuracy in tim e [45].

n-|-l,m+lT rn+\,m+lW i , j , k V i J , k +

A t- k m y k m _ ^ i 1 h v r 'x h


46

4.1 .5 T urbulence M od el

Closure equations in the form of a turbulence model must be implemented in order

to have a closed solution to the RANS equations. The Shear Stress Transport (SST)

turbulence model was selected as i t is applicable to cases involving complex surface

interactions [45].

The SST model of Reference [46] blends the k — e model (typ ica lly accepted for

turbulence modelling outside of the boundary layer) and the k —oo model (accepted for

turbulence modelling w ith in the boundary layer). A two equation turbulence model

can be introduced through the eddy viscosity term [45]. The eddy viscosity term

becomes

W = n Pn v h Vv ai = ° '31 (417)max [1; OF2/ (aicu)j

which describes tha t in a turbulent boundary layer the maximum value o f eddy vis

cosity is lim ited by forcing the shear stress to be bound by the turbu lent kinetic

energy uj times a constant a\. Additionally, the absolute vo rtic ity is m u ltip lied by a

forcing function F2 which is defined as a function of the wall distance y according to

\ f k bOO/i' ' 2Fo = tanh max (4.18)

0.09 u y ’’ py2uj

The two equations of the SST model are defined in terms of the blending function

F i, which established the appropriate m ix of the e and u model equations [45]. The


47

transport equation is given by

1p— + p V V k - — V [(/X + a*pT) V k] = Pk - P*pku (4.19)

and the specific dissipation rate by

o 1p ^ 7 + dy V a ; - [(/* + Va;] = Pu - ppto2 + 2 ( 1 - F x) ^ V / c V u ; (4.20)

C/t IlG L0

The blending function F i is set to un ity for a no-slip boundary and goes to zero

towards the outer edge of the boundary layer. The function is defined as

Ft = tanh mm max. 4p<7a;2A: \

09aV py2ujJ ’ C D kuly2 JV k 500/x

(4.21)

where C D kuj is the cross-diffusion term for the k — u) turbulence model and is defined

as

C D kui = max2pcrw2

UJVftVu;; 10 -2 0 (4.22)

The appropriate constants must also be determined from the k — e and k — oj

models. Three coefficients remain constant for a ll cases:

o i = 0.31, p* = 0.09, k = 0.41 (4.23)

The remainder o f the coefficients, /?, 7 , ak and a:jJ are found by blending the con

stants from the other two turbulence models. A llow ing p to represent a characteristic

coefficient w ith p i representing the value o f the coefficients in the k — u> model and

P2 for the k — e model, the pattern o f coefficient blending according to F\ is [45]

P — F ip i + (1 — Fi)p2 (4.24)


48

The coefficients of the k — lo model are

(jfcx = 0.85, <7*1 = 0.85, A = 0.075, 7 l = A / / T - au lK2/ s / f ^ = 0.553 (4.25)

and for the k — e model are

<7fc2 = 1.0, <7*2 = 0.856, p2 = 0.0828, 72 = [32/f3 * - a ^ K 2/ ^ = 0.440 (4.26)

4.2 Structural Solver

As the geometries of interest deform during the prescribed m otion, the aerodynamic

mesh must deform rather than rig id ly translate and rotate. Th is is achieved for both

flow models using Transfinite In terpo lation of Displacements (T F I) w ith in the blocks

containing the wing [44]. The wing surface deflections are interpolated to the volume

grid points x t]k as

&%ijk ^j^a^ik (4.27)

where cfj are values of a blending function which varies between one at the wing

surface ( j = 1) and zero at the block face opposite. The surface deflections x a^k are

obtained from the transform ation of the deflections on the s tructura l grid. The grid

speeds can be obtained by differentiating Eq. (4.27).

F in ite element methods allow for the static and dynamic response of a structure

to be determined. Mass [M ] and stiffness [K ] matrices are used to determine the

equation of m otion of an elastic structure subjected to an external force { f s}

\M \ { « } + {K \ { « } = { / , } (4.28)


49

where {u } denotes the vector o f s tructura l displacements. Note, the linear s tructura l

system is determined prio r to perform ing flu tte r calculations. In the present work,

the commercial package M S C /N A S TR A N was employed to determine the [M \ and

[.K } matrices.

The structura l displacements of the structure are w ritten as a linear combination

of the generalized coordinates, { q } : and the physical displacements of the structure

as

{« } = f f l {« } (4.29)

where [<f>] is the modal transform ation m atrix . The eigenvalues for th is system are

solved and scaled so tha t

[$ ]T [M \ [$] = 1 (4.30)

Projecting the fin ite element equations onto the mode shapes results in

A f l l + u>2 {? } = f f l T { /» } (4.31)

This equation can be solved by a two stage Runge-Kutta method, which requires a

knowledge of / " and / ” +1. To avoid in troducing sequencing errors by approxim ating

the term / ” +1, the Runge-Kutta solution is iterated in pseudo tim e along w ith the

flu id solution. A t convergence of the pseudo tim e iterations, the structura l and flu id

solutions are properly sequenced in tim e [44],

Achieving effective in form ation transfer between the s tructura l and flu id grids is

complex; the coupled solver needs to accurately determine the displacement of the


50

flu id grid points based on the deflections o f the structura l grid points. The la tte r

are caused by the flu id forces on the structura l grid points which must be accurately

determined from the known forces on the flu id surface grid points. The two grids

are not, in general, coincident and where CFD is used to derive the aerodynamic

forces it is unlike ly tha t the grids w ill have common interfaces. The Constant Volume

Tetrahedron (C V T) scheme was proposed to provide a consistent and accurate trans

form ation procedure for coupling flu id and structure under these circumstances [44].

Refer to Reference [4] for the detailed C V T derivation.


Chapter 5

Test Cases

5.1 V alidation o f the C om putational P rocedure

The fin ite difference solutions of the Euler and Navier-Stokes equations were applied to

the simulation o f the F-5 wing. This wing was selected as significant three-dimensional

effects occur due to the low aspect ratio. The F-5 wing is a fighter type wing w ith a

high leading edge sweep and a thickness of 5% of chord. Refer to References [47] and

[48] for the description of the wing geometry, test conditions and the experimental

results. Note in Chapter 5.2, the AG A R D 445.6 wing was selected as the test case

for flu tte r; however, experimental pressure data was not measured during the flu tte r

tests. Therefore, the F-5 wing was employed to validate the fin ite difference solutions

of the Euler and Navier-Stokes equations and the AG A R D 445.6 wing was employed

to validate the tim e and frequency domain flu tte r solutions.

51


(a) Root Section (b) Zoom of Root Section

(c) Upper Wing Surface

Figure 5.1: F-5 wing flu id mesh.

5.1.1 C F D F lu id M esh es

Structured, m ultiblocked C-H medium and coarse grids were generated by a propri

etary in-house code for use in both the Euler and Navier-Stokes simulations. Refer

to Fig. 5.1. The medium grid contained 613,972 nodes w ith 7,730 points on the wing

surface. The coarse grid contained 85,274 nodes w ith 2,374 points on the w ing surface.

A grid convergence study was conducted for a steady case at zero angle o f attack,


53

35.2% Span 97.7% Span

-0.4 -0.4

- 0.2 - 0 . 2

Q.o0.2 0.2□ '

0.4 o N -S Coarse N -S Medium□ Euler Coarse Euler Medium

0.4,

0.6 0.6

0.8 0.80.5x/c

1 0 0.5 1

Figure 5.2: Steady Cp over F-5 upper wing surface at A400 = 0.95.

Mach number o f 0.95 and a Reynolds number of 11 x 106 based on mean aerodynamic

chord. The results for the Euler and Navier-Stokes solutions are shown in Fig. 5.2 for

two wing stations. Both the medium and coarse grids show good agreement for the

Euler and Navier-Stokes solutions. Therefore, the coarse F-5 w ing grid was employed

to validate the com putational procedure.

5.1 .2 S tead y Flow

Steady state simulations of the F-5 wing were performed for validation of the Euler

and Navier-Stokes numerical methods. The undisturbed flow conditions are shown

in Table 5.1. Transonic effects are high for these flow conditions due to the presence

o f a strong shock wave. Note, the Reynolds number is scaled by the wing root chord

cr = 0.6396 m.

The steady pressures and experimental measurements are presented in Fig. 5.3


54

Table 5.1: Steady flow conditions for F-5 wing.

Description ValueMach Number 0.95

Reynolds Number 11 x 106Angle of A ttack 0.0°

for three spanwise stations. In order to evaluate the numerical method for use in the

A IC correction procedure, the steady lift in g pressures, A Cp, are also shown. Good

agreement is obtained for both upper and lower surfaces. The Euler solution required

65% less com putational tim e to complete in comparision to the Navier-Stokes solution,

yet produced v ir tu a lly identical results. Refer to Fig. 5.4 for the convergence history.

5.1 .3 U n stea d y F low

The tim e domain unsteady flow simulations were performed from a steady state con

verged solution. The unsteady flow simulation in itia tes a rig id body harmonic p itch ing

ro ta tion about the root m idchord axis of the wing at a reduced frequency, k. The

simulation parameters are shown in Table 5.2. Refer to Chapter 3.2.1 for the Fourier

transformations tha t were applied to the pressure tim e histories.

Table 5.2: Unsteady flow conditions for F-5 wing.

D escription ValueMach Number 0.95

Reynolds Number 11 x 106Reduced Frequency 0.264

P itch ing M otion a (t) = 0.0° + 0.222° sin(2Atf)Number of T im e Steps per Cycle 20


55

35.2% Span 72.1% Span 97.7% Span

-0 .5 -0 .5 -0 .5

Q.o

0.5 0.5 0.5

0 0.5 1 0 0.5 1 0 0.5 1x/c

0.5 0.5 0.5

Q.o<

-0 .5 -0 .5 ■0.5 N -S Eulero Exp.

0 0.5 1 0 0.5 1 0 0.5 1x/c

Figure 5.3: Steady Cp and A Cp over F-5 wing at M 00 = 0.95.

-0.5

-1.5<0 - 2 TJ8-2 5

DCo>o_i

-3.5-4

— — Euler-4.5Navier-Stokes

-50 50 100 150 200 250 300 350CPU (work units)

Figure 5.4: Steady state convergence for F-5 wing at = 0.95.


56

The simulation was run for two and three complete cycles in order to avoid unde

sired transient effects [16]. Refer to Fig. 5.5. Two cycles are adequate to avoid these

effects for both the Euler and the Navier-Stokes solutions. Therefore, for the present

investigation, unsteady pressures are acquired from the second cycle.

Computed unsteady pressures and experimental measurements are presented in

Fig. 5.6 for three spanwise stations. The unsteady lift in g pressures are presented

in Fig. 5.7. Good agreement of the unsteady pressures was observed. The strength

and location of the prim ary shock wave, however, varied slightly; the Euler solution

overestimated the strength whereas the Navier-Stokes solution underestimated the

strength.

5.2 F lu tter Test Case

The AG AR D 445.6 wing was selected as the test case since flu tte r measurements

are available for a wide range of Mach numbers. Results have also been published

in various papers on com putational aeroelasticity, including References [4], [49] and

[50]. The wing has a quarter chord sweep of 45°, an aspect ra tio o f 1.65, a taper ra tio

of 0.66 and a constant N A C A 65A004 symmetric profile. W ind tunnel testing was

conducted in the NASA Langley Transonic Dynamics Tunnel and the results were

published in 1963. Various wing models were tested; the case most commonly used

in computational aeroelasticity papers is the weakened wing model at zero angle of

attack in air. This model was constructed of lam inated mahogany and had holes


57

35.2% Span 97.7% Span20

Q. Q.oCD

DC

OCD0C

-10 -100 0.5 1 0 0.5 1

x/c35.2% Span

x/c97.7% Span

Q. CLoE

OE

- 5 -52 Cycles3 Cycles

-10 -100 0.5 1 0 0.5 1

x/c

35.2% Span

(a) Euler

x/c

97.7% Span20 20

Q_ Q.o<Dcc

-100 0.5 1 0 0.5 1

x/c35.2% Span

x/c97.7% Span

Q. Q.oE

OE

- 5 - 52 Cycles3 Cycles

-10 -100 0.5 1 0 0.5 1

x/c x/c

(b) Navier-Stokes

Figure 5.5: Real and imaginary Cp over F-5 wing at — 0.95 for 2 and 3 cycles.


58

35.2% Span 72.1% Span 97.7% Span

-10 -10 -100.5 1 0.5 0.5x/c

3-0. n 0 O .

-5 -5 -5 N -S Eulero Exp.

-10 -10 -100.5 1 0.5 1 0.5x/c

Figure 5.6: Real and im aginary Cp over F-5 wing at = 0.95.

35.2% Span 72.1 % Span 97.7% Span20

-5 0 0.5 1

Q.O<CDcr

-5 0.50 1-5 0 0.5 1

x/c

-5

-10 0 0.5 1

-5

-100 0.5 1

Q.o<E

-5 N-S Euler

-100.5x/c

Figure 5.7: Unsteady A Cp over F-5 wing at M 00 = 0.95.


59

drilled through the w ing to reduce its stiffness. F lu tte r speed coefficients, Up, for

Mach numbers in the range of 0.338 to 1.141 were reported. Refer to Reference [51]

for the description of the wing geometry, test conditions and the experimental results.

The flu tte r speed coefficient is expressed as

Up = h Uo°i/2 (5-1)6swq/R/2

where U00 is the freestream velocity at flu tte r, bs is the wing semispan, uia = 39.44

Hz is the frequency o f the firs t torsional mode (shown in Chapter 5.2.3) and p =

W (PooK) where m = 1.693 kg is the mass o f the wing, V = 0.130 m 2 is related to

the volume of the wing and p00 is the freestream density at flu tte r.

F lu tte r speed coefficients w ill be calculated for: (1) tim e marching simulations

using the Euler and Navier-Stokes equations; (2) uncorrected frequency domain sim

ulations using M S C /N A S TR A N and K E D LM P L; and (3) A IC corrected frequency

domain simulations using K E D LM P L and PM B CFD data. The flow conditions for

the various Mach numbers are shown in Table 5.3. Results w ill be shown in Chapter 6.

The following sections w ill describe and validate the AG A R D 445.6 wing aerodynamic

and structura l models.

5.2.1 C FD F lu id M esh es

An 0 - 0 grid and a C-H grid were generated by a proprie tary in-house code for use

in the Euler and Navier-Stokes calculations, respectively. M edium and coarse grids


60

Table 5.3: F low conditions for AG AR D 445.6 w ing flu tte r analysis.

M a c h N u m b e r R e y n o ld s N u m b e r D e n s ity [kg /m 30.678 1.410 x 106 0.20820.901 0.911 x 106 0.09950.960 0.627 x 106 0.06340.990 0.534 x 106 0.05931.072 0.442 x 106 0.0552

were created for both the 0 - 0 and the C-H grids. The medium 0 - 0 grid had 190,000

nodes w ith 4,453 points on the wing surface. The coarse 0 - 0 grid had 27,000 nodes

w ith 1,131 points on the wing surface. The medium C-H grid had 324,000 nodes w ith

2,862 points on the wing surface. The coarse C-H grid had 45,000 nodes w ith 979

points on the wing surface. Refer to Fig. 5.8.

A grid convergence study was conducted for a steady case at zero angle of attack,

Mach number of 0.96 and a Reynolds number of 4.51 x 105 based on mean aerodynamic

chord. The results for the Euler and Navier-Stokes solutions are shown in Fig. 5.9 for

four wing stations. Convergence history for the medium grids is shown in Fig. 5.10.

Both the medium and coarse grids show good agreement for the respective topologies.

The Euler and Navier-Stokes solutions show m inor disagreement near the tra iling edge

of the wing due to the difference between the grid topologies.

Therefore, the coarse 0 - 0 and C-H grids were employed in the present investiga

tions.


61

(a) Euler Root Section (b) Navier-Stokes Root Section

/ / / / / / J

(c) Euler and Navier-Stokes Upper Wing Surfaces

Figure 5.8: Medium density AG A R D 445.6 CFD meshes.


62

Span = 0.0%- 0.2

0.2Q.O0.41

0.60

0.80 0.6 0.8 10.2 0.4x/c

Span = 90.0%- 0.2

0.60 0.6 0.80.2 0.4 1x/c

Figure 5.9: Steady Cp over AG AR D

Span = 25.0%- 0.2

0.4U

0.60.2 0.4 0.6 0.8 10

x/c

Span = 95.0%- 0.2

0°" 0.2

O RANS Coarse RANS Medium

□ Euler Coarse Euler Medium

0.60.2 10 0.4 0.6 0.8

x/c

.6 upper wing surface at M 00 = 0.96.


63

-2

CDD■DCOCD

DCCDO

-4

-5— Euler— Navier-Stokes

0 50 100 150 200 250 300CPU (work units)

Figure 5.10: Steady state convergence for medium density AG A R D 445.6 wing meshes at M qq = 0.96.

5.2 .2 D L M M esh

A linear aerodynamic model was created for use in the D LM simulations employing

M S C /N A S TR A N and K E D LM P L. The wing was modelled by panels and the lift in g

surfaces were assumed to lie parallel to the flow. Note, the D LM does not account for

thickness effects o f the liftin g surfaces and assumes the undisturbed flow is uniform or

varying harmonically. The model consisted o f 21 points in the spanwise direction and

21 points in the chordwise direction. The spanwise and chordwise point d istributions

were un ifo rm ly set in order to m aintain near un ity box ratios as recommended by

Reference [52],


64

5.2 .3 S tructural M od el

The linear structura l model was created in M S C /N A S TR A N using the model param

eters in the aeroelastic optim ization study by Reference [53]. This model was selected

for comparative purposes as i t was also used by References [4], [49] and [50] for tim e

marching studies. The wing was modelled w ith plate elements as a single layer or

thotrop ic material. The model consisted o f 231 nodes and 200 elements. Refer to

Fig. 5.11. The thickness d is tribu tion was governed by the a irfo il shape. The m ateria l

properties used were E i = 3.1511 GPa, E 2 = 0.4162 GPa, v = 0.31, G = 0.4392 GPa

and p = 381.98 k g /m 3 where E i and E 2 are the m oduli of e lastic ity in the long itud i

nal and lateral directions, v is Poisson’s ratio , G is the shear modulus in each plane

and p is the wing density. Table 5.4 compares the measured and calculated firs t four

fundamental modes and Fig. 5.12 shows the calculated mode shapes.

Table 5.4: Comparison of modal frequencies for AG A R D 445.6 wing.

M ode 1 [Hz] M ode 2 [Hz] M ode 3 [Hz] M ode 4 [Hz]Experiment [51] 9.60 38.10 50.70 98.50Reference [4] 9.67 36.87 50.26 90.00Reference [53] 9.63 37.12 50.50 89.94Reference [54] 9.69 37.84 51.00 92.36Calculated 9.46 39.44 49.71 94.39

References [4] and [54] employed the density and model defin ition of Reference [53].

Reference [4] reported a match in the model mass, suggesting tha t the thickness o f the

plate elements were adjusted in order to force match the model mass. Reference [54],


65

however, does not report the mass.

Using the density and model defin ition of Reference [53] resulted in a wing mass

of 1.693 kg, 9% lighter than the Reference [53] model (mass equal to the experimental

model). The modal frequencies and shapes showed good comparison between a ll three

models and thus, the density was not adjusted to match the w ing mass.

The linear structura l model was used for both the Euler and Navier-Stokes tim e

marching as well as the frequency domain simulations. The add ition of s tructura l

damping is not included for the tim e marching or K E D LM P L solutions. Thus, for

comparative purposes, the s tructura l damping for the M S C /N A S TR A N model was

set to zero.


66

\\ [ \\ \\ \\ \\ \\ \\ \\ \

X \

X \X

N\ N

Figure 5.11: AG A R D 445.6 s tructura l model.


67

(a) Mode 1 (1-B)

(b) Mode 2 (1-T)

(c) Mode 3 (2-B)

(d) Mode 4 (2-T)

Figure 5.12: Calculated AG A R D 445.6 wing mode shapes.


Chapter 6

Results

6.1 U ncorrected Frequency D om ain R esu lts

Based on D LM aerodynamic theory and the KE-method, uncorrected frequency do

m ain results were obtained using M S C /N A S TR A N and K E D LM P L . Refer to Chap

ter 2 for the frequency domain form ulation and to Chapter 5 for the description of

the test case.

F lu tte r boundaries for the AG AR D 445.6 weakened w ing are shown in Fig. 6.1

at Moo = 0.678 to = 0.990. The M S C /N A S TR A N and K E D LM P L solutions

produced significantly higher flu tte r boundaries in comparison to the experimen

ta l results. These programs also failed to predict the transonic dip located after

Moo = 0.90. Instead, M S C /N A S TR A N and K E D LM P L produced erroneous linearly

decreasing flu tte r boundaries.

K E D LM P L produced a boundary sim ilar to M S C /N A S TR A N , the code employed

by industry for flu tte r certification. The variance in flu tte r speed between K E D LM P L

68


69

and M S C /N A S TR A N was unexpected as both programs utilized the same aerody

namic and structura l models as well as the same im plementation o f the D LM . Further

investigation was not possible as the source code of M S C /N A S TR A N was unavailable.

0.46

0.44

0.42

0.4

■g 0.38

0.36

0.34

— • — Experiment — ▼— MSC/NASTRAN ■ KEDLMPL

0.32

0.7 0.75 0.8 0.85 0.9 0.95 1Mach Number

Figure 6.1: Uncorrected frequency domain flu tte r boundaries for AG A R D 445.6 wing.

The mechanism of flu tte r for all tested Mach numbers was between modes 1 (first

bending) and 2 (first torsion). This mechanism can be observed in the velocity-

damping (V-g) and velocity-frequency (V -f) plots. Refer to Fig. 6.2 for the V-g and

V - f plots at M qo = 0.678 .

The uncorrected frequency domain simulations were conducted on a machine w ith

a single 2.8 GHz processor. The com putational tim e required to calculate each f lu t

te r data point for M SC /N A S TR A N and K E D LM P L was 5 minutes and 4 minutes,


70

respectively.

6.2 T im e M arching R esu lts

Time marching simulations based on the Euler and Navier-Stokes equations were

conducted using PM B. Refer to Chapter 4 for the tim e marching form ulation.

In these simulations, the structure was given a small in it ia l dimensionless velocity

in the first mode. The subsequent tim e evolution of the m odal response was then

calculated in order to determine whether the perturbation had grew or had decayed.

I f the perturbation decayed, a new sim ulation was performed using an increased

freestream density. This process was repeated u n til flu tte r occurred, i.e., when the

perturbation grew.

6.2.1 G rid D en sity and F lu id M od el E ffect on F lu tter S p eed

The comparison of the tim e marching solutions at M x = 0.96 (near the bottom of

the transonic dip) to various published grid densities and flu id models is shown in

Table 6.1. The results of Reference [50] show a 6% downward trend in the flu tte r

speed between the medium and coarse grids. The current results show an upward

trend (6% for Navier-Stokes, 4% for Euler) sim ilar to the results o f Reference [49].

The tim e marching simulations were conducted on a Beowulf cluster consisting of

four machines w ith 3.2 GHz processors. The com putational tim e required to calculate

each flu tte r data point using the Euler equations w ith the coarse 0 - 0 grid was 4 hours.


71

100

NX>»ocCD3crCD

100 120 140 160 180 200 220 240 260 280 300Velocity [m/s]

0.5

c'a .EasO -■— Model (1-B)

-m — Mode 2 (1-T) -V — Mode 3 (2-B) ■A— Mode 4 (2-T)

-0.5

100 120 140 160 180 200 220 240 260 280 300Velocity [m/s]

(a) MSC/NASTRAN

100

NX>, ouCDcCD

CT 40CD

Lt_

100 120 140 160 180 200 220 240 260 280 300Velocity [m/s]

0.5

O)-■q .EcoQ -■— Mode 1 (1-B)

- • — Mode 2 (1-T) — Mode 3 (2-B)

-A — Mode 4 (2-T)

-0.5

100 120 140 160 180 200 220 240 260 280 300Velocity [m/s]

(b) KEDLMPL

Figure 6.2: Uncorrected frequency domain V - f and V-g plots at M 0Q = 0.678.


72

The com putational tim e required using the Navier-Stokes equations w ith coarse C-H

grid was 6 hours.

The coarse grids showed good agreement w ith the medium grids and required

approximately three times less com putational time. Therefore, the coarse grids were

employed to determine the AG A R D 445.6 wing flu tte r boundary.

Table 6.1: Comparison of flu tte r speed coefficients at Mach 0.96 for various grids.

R eference G r id V o lu m e F lu id M o d e l u FCurrent Medium Navier-Stokes 0.308Current Coarse Navier-Stokes 0.327Current Medium Euler 0.317Current Coarse Euler 0.330

Reference [49] Fine Euler 0.175Reference [49] Medium Euler 0.192Reference [49] Coarse Euler 0.227Reference [50] Medium Navier-Stokes 0.314Reference [50] Medium Navier-Stokes 0.304Reference [50] Medium Navier-Stokes 0.285

6.2 .2 F lu tter B oundaries

F lu tte r boundaries for tim e marching simulations including previously published re

sults by Reference [4] are shown in Fig. 6.3. The modal responses o f the AG AR D

445.6 wing before and after flu tte r using the Euler and Navier-Stokes equations are

shown in Figs. 6.6 to 6.8.

The coarse grid tim e domain Euler and Navier-Stokes simulations produced sim

ila r flu tte r speeds and boundary shapes between M ^ = 0.678 and = 0.96 in


73

0.45

04c0‘o3=0OO"S 0.35 ®Q.CO

LL0.3

— — Experiment ▼ Tim e Marching, Euler■ Tim e Marching, N avier-StokesA Reference [4]0 .25

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1Mach Num ber

Figure 6.3: T im e marching flu tte r boundaries.

comparison to the experimental results. The tim e domain boundaries, however, are

approximately 6% lower. This difference may be a ttribu ted to the lack o f s tructura l

damping in the tim e domain solutions; previous results have shown using a value

of structural damping of 0.5% w ill shift the boundary towards the experimental re

sults [49].

The Euler and Navier-Stokes simulations predicted identical values for the bo ttom

of the transonic dip. The dip is more pronounced in the Navier-Stokes sim ulation than

the Euler simulation and less pronounced than Reference [4],


74

CDE<DOJ®Q.

■aCDN1®ocCDCD

1

0.5

0

-0.5

-1

x 10 Mode 1 x 1 o Mode 2

p = 0.20 kg/mp = 0.21 kg/m

0

x 10

50x

Mode 3

100

-20 50 100

-20 50 100

x 1 o Mode 4

-50 50 100

(a) Navier-Stokes

x 1 o Mode 1 x 1 o Mode 2cCDECDOroQ.COQ

CDcCDCD

p = 0.19 kg/mp = 0.20 kg/m

x 1 o Mode 3

150

-20 50 100 150

-20 50 100 150

10 Mode 4

-2

0 50 100 150

(b) Euler

Figure 6.4: Time marching responses at M 00 = 0.678.


75

CDOffiQ.

COSCCD

CD

1

0.5

0

-0.5

-1

x 10 Mode 1 x 10 Mode 2

p = 0.08 kg/m0.09 kg/m

x 10

50i

Mode 3

100

0.5

-0.5

0 50 100

-20 50 100

x 10 Mode 42

1

0

1

•20 50 100

(a) Navier-Stokes

x 10 Mode 1

0.5

— — p = 0.07 kg/m' p = 0.08 kg/m'

2 -0.5

0 100 15050

x 10

xMode 3

-5

-100 50 100 150

10 Mode 2

-20 50 100 150

x 10 5 Mode 4

0.5

-0.5

0 50 100 150

(b) Euler

Figure 6.5: Time marching responses at M,x = 0.901.


76

oECDO«Q .

T)CDN

roCDcCDa

1

0.5

0

-0.5

-1

x 1 o Mode 1 x 1 o Mode 2

p = 0.05 kg/mp = 0.06 kg/m

0

x 10

50x

Mode 3

100

-50 50 100

-20 50 100

x 10 Mode 4

0.5

-0.5

0 50 100

(a) Navier-Stokes

x 1 o Mode 1

p = 0.04 kg/mp = 0.05 kg/m

Mode 3

150

x 10

-10o 50 100 150

x 1 o Mode 2

-20 50 100 150

x 10 Mode 4

0.5

-0.5

0 50 100 150

(b) Euler

Figure 6.6: T im e marching responses at = 0.960.


x 10 Mode 1 x 1 o Mode 2

Q.

— — p = 0.04 kg/m' p = 0.05 kg/m'

2 -0.5

0 50 100

x 10 Mode 3

-50 50 100

-20 50 100

x 1 o Mode 4

0.5

-0.5

0 50 100

(a) Navier-Stokes

x 1 o 3 Mode 1 x 1 o Mode 2

-a— ~ p = 0.04 kg/m' p = 0.05 kg/m'

0 50 100 150

x 10 Mode 3

-100 50 100 150

-20 50 100 150

x 1 o Mode 4

0.5

-0.5

0 50 100 150

(b) Euler

Figure 6.7: Time marching responses at M ^ = 0.990.


78

x 10 Mode 1 x 10 Mode 2

CDECDO_C0Q.

T3CD.N2aicCD

CD

1

0.5

0

— — p = 0.08 kg/m' p = 0.09 kg/m'

-0.5

0 50 100

x 10

xMode 3

0.5

-0.5

0 50 100

-20 50 100

x 10 Mode 4

-20 50 100

(a) Navier-Stokes

cCDECDOffiQ.CODT3CDN2CDCCD

CD

1

0.5

0

-0.5

-1

x 1 o Mode 1 x 1o Mode 2

p = 0.04 kg/mp = 0.05 kg/m

0

x 10

50 100x

Mode 3

-5

-100 50 100 150

-20 50 100 150

x 1 o Mode 4

0.5

-0.5

0 50 100 150

(b) Euler

Figure 6.8: Time marching responses at M,x = 1.072.


79

6.3 AIC C orrection M ethod R esu lts

The unsteady pressure matching correction method, based on a downwash weighting

approach, was conducted using K E D LM P L. Unsteady lift in g pressures employed for

this technique were obtained from Euler and Navier-Stokes CFD simulations under

going a rig id body pitching m otion. These results were employed to calculate the

weighting vector, W T . Refer to Chapter 3 for the derivation of the unsteady pressure

matching correction method.

Reduced frequencies o f the rig id p itching m otion were calculated for each Mach

number based on experimental flu tte r speeds and frequencies. Refer to Table 6.2 for

the experimental values and the resulting reduced frequencies, k. The am plitude o f

the pitching m otion was set to 1° based on the aeroelastic analysis of Reference [16]

for the AG AR D 445.6 wing.

The rig id p itch simulations were conducted on a single 3.2 GHz processor. The

computational tim e required using the Euler and Navier-Stokes equations for two

rig id p itch cycles was approximately 30 minutes and 45 minutes, respectively.

Table 6.2: Reduced frequencies of the CFD simulations used in the A IC correction method.

M a c h N u m b e r Loo m /s ] 00F [Hz] k0.678 231.37 17.98 0.2440.901 296.69 16.09 0.1700.960 309.01 13.89 0.141


80

6.3.1 P resen t W ork R esu lts

Unsteady lift in g pressures obtained from the Euler and Navier-Stokes rig id p itch

simulations were interpolated to mid-span and three-quarter chord for each o f the

400 panels contained in the linear aerodynamic model.

Refer to Fig. 6.9 for the unsteady lift in g pressures for three spanwise stations.

Good agreement between the Euler and Navier-Stokes simulations was observed. The

real portion o f the unsteady lift in g pressures were consistent w ith the results presented

in Reference [16]; however, vast differences were noted for the imaginary portions. The

resulting phase angle along the entire wing span was approxim ately —45°, thereby

suggesting an error in the Fourier transform ation. Experim ental pressure measure

ments are unavailable for the AG A R D 445.6 wing, thus, a detailed investigation was

not conducted to determine the cause o f the errors in the unsteady lift in g pressures.

The unsteady lift in g pressures are employed to calculate the weighting vector,

W T for use in the A IC correction procedure. The error in the im aginary portion

of these pressures resulted in a h ighly scaled weighting vector, W T . The calculated

weighting vector was approximately 10 — 30i whereas the expected weighting vector

is near unity, 1.0 ± 1.0b

The A IC correction procedure was conducted in K E D LM P L employing the erro

neous unsteady lift in g pressures obtained from the Navier-Stokes simulations. Refer

to Fig. 6.10 for the V-g and V - f plots at M 00 = 0.678. No coupling between the modes


81

32.5% Span 52.5% Span 92.5% Span3

a- 2

1

00 0.5 1

3

2

1

00 0.5 1

x/c

Q.o < -2 E

-3

-2

- 3

- 4 - 40 0.5 1 0 0.5 1

x/c

32.5% Span

(a) Navier-Stokes

52.5% Span3

A 2

1

00 0.5 1

x/c1

3

2

1

n0.5 1

CL

o < -2 E

-3

-2

- 3

- 4 - 40.5x/c

0.5

3

2

1

00 0.5 1

-2

- 3

-40 0.5 1

92.5% Span

1

3

2

1

00.5

-2

- 3

- 40 0.5

(b) Euler

Figure 6.9: Unsteady A Cp over AG A R D 445.6 wing at M ,^ = 0.678.


82

200

X 150

S 100

u_

0 100 200 300 400 500Velocity [m/s]

0.5

-0.5

Mode 1 (1-B) Mode 2 (1-T) Mode 3 (2-B) Mode 4 (2-T)

0 100 200 300 400 500Velocity [m/s]

Figure 6.10: A IC corrected frequency domain V - f and V-g plots at M ,^ = 0.678.

was observed in the V - f plot. The resulting flu tte r speed coefficient was calculated as

0.31, approximately 30% lower than the experimental, uncorrected frequency domain

and tim e marching results.

As previously discussed, the correction procedure of the present work was devel

oped and successfully implemented in Reference [16]. Therefore, the following subsec

tion w ill employ the results from Reference [16] in order to illus tra te the effectiveness

of the correction procedure.


83

0.45

0.4

"§ 0.35CDQ.

0.3

Experiment■ ■ Uncorrected

A AIC Correction, Ref. [16]■ Time Marching, N avier-S tokes0.25

0.65 0.7 0.80.75 0.85 0.9 0.95 1Mach Number

Figure 6.11: F lu tte r boundaries for AG AR D 445.6 wing.

6.3 .2 R eferen ced R esu lts

In Reference [16], a flu tte r analysis was conducted for the A G A R D 445.6 wing using

the A IC correction procedure for M ^ = 0.678, M,x = 0.901 and = 0.960. The

flow was modelled using the Navier-Stokes equations. Sim ilar to the present works,

the structura l model was based on Reference [53].

F lu tte r boundaries for the wing are shown in Fig. 6.11. The A IC correction pro

cedure showed good agreement w ith the experimental and tim e marching solutions.


84

6.4 C onclusions

The aircraft indus try requires a robust and efficient aerodynamic too l for use in

transonic f lu tte r analyses. Presently, a subsonic linearized discrete lift in g surface

theory, the D LM , is employed but i t cannot capture the flow nonlinearities in the

transonic regime as illustrated in Fig. 6.2.

These nonlinearities can be captured using two different approaches: (1) per

form ing a tim e marching sim ulation using Euler and Navier-Stokes equations; or (2)

correcting the D L M aerodynamic data in a frequency domain analysis using CFD

results. Refer to Table 6.3 for the comparison of required com putational time. Note

tha t the tim e marching simulations were conducted on a Beowulf cluster consisting

of four machines w ith 3.2 GHz processors. The uncorrected and corrected frequency

domain simulations were conducted on a single 2.8 GHz processor.

Table 6.3: Com putational tim e required to calculate each flu tte r point.

M e th o d T im e [hr]K E D LM P L (Uncorrected) 0.06

M S C /N A S TR A N 0.08Time Marching, Euler 4.00Time Marching, N-S 6.00

K E D LM P L (A IC Corrected, Euler) 0.58K E D LM P L (A IC Corrected, N-S) 0.83

As illustra ted in Fig. 6.3, the Euler and Navier-Stokes tim e marching simulations


85

adequately predicted the transonic flu tte r boundary. However, the large computa

tiona l tim e requirement for th is method negates its use in industry. Also, s tructura l

design changes occur during most design cycles. The uncorrected and corrected fre

quency domain approaches produce aerodynamic influence coefficient (A IC ) matrices

which are independent of the structura l model. These matrices can be reapplied to

new structura l models thus reducing the com putational requirements. In contrast,

for new structura l models, the tim e marching solutions would require a completely

new simulation.

As illustrated in Fig. 6.11, the flu tte r boundary obtained using the A IC correction

frequency domain method demonstrated good agreement w ith the tim e marching

simulations. The largest error for the AG AR D 445.6 wing flu tte r boundary was 2.8%

at Moo = 0.901. Therefore, th is approach is ideally suited for industria l applications

as th is approach: (1) captures the flow nonlinearities in the transonic regime; (2)

requires only one-tenth of the com putational tim e of the tim e marching simulations;

and (3) produces A IC matrices which are applicable to new s tru c tu ra l models.

6.5 Future W orks

The A IC correction method should be further developed once the error in the Fourier

transform ation has been elim inated from the post-processing o f the unsteady lift in g

pressures. Possible areas to investigate are: (1) correct the A IC m atrix using un

steady pressures obtained from modal motions instead of a r ig id pitching m otion;


86

(2) incorporate higher harmonics of the unsteady pressures in to the analysis; and (3)

extend the method for use on a complete aircraft.

A m ajor lim itin g factor for fu tu re extensions is the lack o f experimental data

available for validation. A new dataset (M A V R IC -I) o f a business je t wing-fuselage

flu tte r model has been generated at NASA Langley Research Center. Unfortunately,

however, i t has not been released for general use. Ideally, new experimental flu tte r

datasets should contain: (1) aerodynamic unsteady pressure measurements; (2) a

detailed structura l model; and (3) flu tte r results in the transonic regime.


References

[1] R. L. Bisplinghoff, H. Ashley, and R. L. Halfman, Aeroelasticity. Cambridge,

M A: Addison-Wesley Publishing Company Inc., 1955.

[2] I. E. Garrick and W . H. Reed, “H istorical development of a ircraft f lu tte r,” Jour

nal of A ircra ft, vol. 18, pp. 898-912, 1981.

[3] A. M. Rampurawala, Assessment o f In te r-G rid Transformation fo r Complete

A ircra ft Aeroelastic Analysis. M aster’s thesis, University of Glasgow, 2002.

[4] G. S. L. Goura, Time Marching Analysis o f F lu tte r Using Computational F lu id

Dynamics. PhD thesis, University of Glasgow, 2001.

[5] A. Jeziorski, “F-2 wing cracks delay development completion,” Flight In te rna

tional, June 16, 1999.

[6] G. Norris, “Asraam flight tests reveal lower-speed F /A -18 flu tte r,” Flight In te r

national, October 10, 2000.

[7] M. A. Dornheim, “Report pinpoints factors leading to YF-22 crash,” A via tion

Week & Space Technology, pp. 53-54, November 9, 1992.

[8] M. A. Dornheim, “ Elevon v ib ra tion leads to F-117 crash,” Avia tion Week &

Space Technology, p. 30, September 22, 1997.

[9] M. W. Kehoe, “A historical overview of fligh t flu tte r testing,” NASA-TM-4720,

1995.


[10] ZONA Technologies Inc., ZAERO Software System - Theoretical M anual - Ver

sion 7.0. 2003.

[11] J. P. Giesing, T. P. Kalman, and W . P. Rodden, “Subsonic unsteady aerody

namics for general configurations, direct application o f the non-planar Doublet-

Lattice M ethod,” A ir Force F ligh t Dynamics Lab, A ir Force Systems Command,

W right-Patterson A ir Force Base, Report AFFD L-TR -71-5 Part 1, 1972.

[12] E. C. Yates Jr., “Modified-strip-analysis method for predicting wing flu tte r at

subsonic to hypersonic speeds,” Journal o f A ircraft, vol. 3, no. 1, pp. 25-29,

1966.

[13] R. N. Yurkovich, “Status of unsteady aerodynamic prediction for flu tte r o f high-

performance a ircra ft,” Journal o f A ircraft, vol. 40, no. 5, pp. 832-842, 2003.

[14] E. Albano and W . P. Rodden, “A doublet-lattice method for calculating l i f t

d istributions on oscillating surfaces in subsonic flows,” A IA A Journal, vol. 7,

no. 2, pp. 279-285, 1969.

[15] T. P. Kalman, W . P. Rodden, and J. P. Giesing, “Aerodynamic influence coef

ficients by the Doublet Lattice M ethod for interfering nonplanar lift in g surfaces

oscillating in a subsonic flow,” Douglas Report DAC-67977, 1969.

[16] R. G. A. da Silva, A study on correction methods fo r aeroelastic analysis in

transonic flow. PhD thesis, Sao Jose dos Campos, 2004.

[17] R. Palacios, H. Climent, A. Karlsson, and B. W inzell, “Assessment of strageies for

correcting linear unsteady aerodynamics using CFD or test results,” Proc. of the

C E A S /A IA A International Forum on Aeroelasticity and S tructura l Dynamics,

Madrid, Spain, pp. 195-210, 2001.

[18] J. P. Giesing, T . P. Kalman, and W . P. Rodden, “Correction factor techniques

for improving aerodynamic prediction methods,” NASA-CR-144967, 1976.


89

[19] R. J. Zwaan, “Verification of calculation methods for unsteady airloads in the

prediction o f transonic flu tte r,” Journal o f A ircraft, vol. 22, no. 10, pp. 833-839,

1985.

[20] D. M. P it t and C. E. Goodman, “ F lu tte r calculations using Doublet Lattice

aerodynamics modified by the F u ll Potential equations,” Proc. of the 28th

A IA A /A S M E /A S C E /A H S Structures, S tructura l Dynamics, and M aterials Con

ference, Monterey, California, Paper AIAA-87-0882-CP, 1987.

[21] J. Brink-Spalink and J. M. Bruns, “ Correction o f unsteady aerodynamic

influence coefficients using experimental or CFD data,” Proc. of the 41st

A IA A /A S M E /A S C E /A H S /A S C Structures, S tructura l Dynamics and M ate ri

als Conference and Exh ib it, A tlanta , Georgia, Paper AIAA-2000-1498, 2000.

[22] W . P. Rodden and J. D. Revell, “The status of unsteady aerodynamic influence

coefficients,” Fairchild Fund Paper FF-33 , 1962.

[23] H. Bergh and R. J. Zwaan, “A method for estimating unsteady pressure d is tribu

tions for a rb itra ry v ib ra tion modes from theory and from measured d istribu tions

for one single mode,” N LR -TR F.250, National Aerospace Laboratory, Nether

lands, 1966.

[24] W . Luber and H. Schmid, “ F lu tte r investigations in the transonic flow regime

for a fighter type aircraft,” AG AR D Report No. 703, 1982.

[25] M. L. Baker, “ CFD based corrections for linear aerodynamic methods,” AG A R D

Report No. 822, 1997.

[26] I. Jadic, D. Hartley, and J. G iri, “A n enhanced correction factor technique for

aerodynamic influence coefficient methods,” Proc. o f MacNeal-Schwendler Cor

poration ’s Aerospace User’s Conference, 1999.


90

[27] I. Jadic, D. Hartley, and J. G iri, “ Im proving the linear aerodynamic approxima

tion in linear aeroelasticity,” Proc. o f the 41st A IA A /A S M E /A S C E /A H S /A S C

Structures, S tructura l Dynamics and Materials Conference and E xh ib it, A tlanta ,

Georgia, Paper AIAA-2000-1450, 2000.

[28] H. C. Garner, “A practical framework for evaluation o f oscillating aerodynamic

loadings on wings in supercritical flows,” A E Technical Memorandum Structures

900, United Kingdom, 1977.

[29] K. Dau, “A semi-empirical method for calculating pressures on oscillating wings

in unsteady transonic flow,” Report No. DA-EF24-B08/92, Deutche Airbus,

1992.

[30] G. SenGupta, “Evaluation of the Dau-Garner method for predicting unsteady

pressures in transonic flow,” Proc. of the International Symposium on Aeronau

tica l Science & Technology, Jakarta, Indonesia, Paper ISASTI-96-1.3.4, 1996.

[31] K. Yonemoto, “A practical method predicting transonic w ing flu tte r phenom

ena,” Proc. of the 14th International Council o f the Aeronautical Sciences

Congress, Toulouse, France, pp. 724-732, 1984.

[32] D. D. L iu, Y. F. Kao, and K. Y. Fung, “An efficient method for computing

unsteady transonic aerodynamics of swept wings w ith control surfaces,” Journal

o f A ircraft, vol. 25, no. 1, pp. 25-31, 1988.

[33] P. C. Chen, D. Sarhaddi, and D. D. L iu, “Transonic aerodynamic influence co

efficient approach for aeroelastic and M DO applications,” Journal o f A ircraft,

vol. 37, no. 1, pp. 85-94, 2000.

[34] P. Guezaine, C. Farhat, and G. Brown, “Application of three field nonlinear

flu id-structure form ulation to the prediction of aeroelastic parameters of an F-16

fighter,” Computers and Fluids, vol. 32, pp. 3-29, 2003.


91

[35] C. J. Borland and D. P. Rizetta, “ Nonlinear transonic flu tte r analysis,” A IA A

Journal, vol. 20, no. 11, pp. 1606-1615, 1982.

[36] J. T. Batina, H. J. Cunningham, and R. M. Bannett, “ Modern wing flu tte r

analysis by com putational flu id dynamics methods,” Journal o f A ircraft, vol. 25,

no. 10, pp. 962-968, 1988.

[37] E. M. Lee-Rausch and J. T. Bantina, “ Calculation o f A G A R D 445.6 wing flu tte r

using Navier-Stokes aerodynamics,” Proc. of the A IA A Applied Aerodynamics

Conference, Monterey, California, Paper AIAA-93-3476-CP, 1993.

[38] G. P. Guruswamy and C. Byun, “ D irect coupling o f Euler flow equations w ith

plate fin ite element structures,” A IA A Journal, vol. 33, no. 2, pp. 375-377, 1994.

[39] R. M elville, “ Nonlinear sim ulation of F-16 aeroelastic instab ility ,” Proc. o f the

39th A IA A Aerospace Sciences Meeting and E xh ib it, Reno, Nevada, Paper

AIAA-2001-0570, 2001.

[40] K. J. Badcock, M. A. Woodgate, F. C antariti, and B. E. Richards, “Solution

of the unsteady euler equations in three dimensions using a fu lly unfactored

method,” Aerospace Engineering Report 9909, University o f Glasgow, 1999.

[41] W . P. Rodden, “A comparison of methods used in in terfering lift in g surface

theory,” AG AR D Report No. 643, 1976.

[42] E. Sulaeman, Effect o f Compressive Force on Aeroelastic S tability o f a Strut-

Braced Wing. PhD thesis, V irg in ia Polytechnic Ins titu te and State University,

2001.

[43] K. J. Badcock, B. E. Richards, and M. A. Woodgate, “ Elements of com putational

flu id dynamics on block structured grids using im p lic it solvers,” Progress in

Aerospace Sciences, vol. 36, pp. 351-392, 2000.


92

[44] M. A. Woodgate, K. J. Badcock, A. M . Rampurawala, and B. Richards, “Aeroe

lastic calculations for the hawk a ircraft using the euler equations,” Aerospace

Engineering Report 0313, University o f Glasgow, 2003.

[45] G. L. Davis, Trailing Edge Flap Control o f Dynamic Stall on Helicopter Rotor

Blades. M aster’s thesis, Carleton University, 2005.

[46] C. R. Group, “M odelling o f turbulence in eros-uk,” Doc. No. TN02-012, Univer

s ity o f Glasgow, 2002.

[47] H. T ijdem an and J. W . G. van Nunen, “Transonic w ind-tunnel tests on an os

c illa ting wing w ith external store, part 1: General description,” N LR T R 78105

U Part 1, 1978.

[48] J. W . G. van Nunen and H. Tijdem an, “Results o f transonic w ind tunnel mea

surements on an oscillating wing w ith external store (data report),” N LR T R

78030 U, 1978.

[49] K. J. Badcock, M. A. Woodgate, and B. Richards, “D irect aeroelastic b ifu r

cation analysis of a symmetric wing based on the euler equations,” Aerospace

Engineering Report 0315, University of Glasgow, 2003.

[50] R. E. Gordnier and R. B. Melville, “Transonic flu tte r simulations using an im

p lic it aeroelastic solver,” Journal o f A ircra ft, vol. 37, no. 5, pp. 872-879, 2000.

[51] E. C. Yates Jr., AG AR D Standard Aeroelastic Configurations fo r Dynam ic Re

sponse I-W ing J45.6. AG A R D Report 765, 1988.

[52] W . P. Rodden and E. H. Johnson, M S C /N A S T R A N Aeroelastic Analysis User’s

Guide. Los Angeles, CA: The MacNeal-Schwendler Corporation, Version 68,

1994.


93

[53] R. M. Kolonay, Unsteady Aeroelastic O ptim ization in the Transonic Regime. PhD

thesis, Purdue University, 1996.

[54] K. Kavukcuolu, Wing F lu tte r Analysis w ith an Uncoupled Method. M aster’s

thesis, M iddle East Technical University, 2003.


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