Computational Structural Mechanics & Fluid Dynamics. Advances and Trends

425 Pages·1988·18.594 MB·English

by  A.K. Noor and D.L. Dwoyer (Eds.)

COMPUTATIONAL STRUCTURAL MECHANICS
AND FLUID DYNAMICS
Advances and Trends
Papers presented at the Symposium on Advances and Trends in
Computational Structural Mechanics and Fluid Dynamics
Held 17-19 October 1988, Washington, D.C.
Editors
AHMED K. NOOR
Professor of Engineering and Applied Science, The George Washington University,
NASA Langley Research Center, Hampton, Virginia, U.S.A.
DOUGLAS L. DWOYER
Manager, Hypersonic Technology Office, NASA Langley Research Center, Hampton,
Virginia, U.S.A.
Sponsored by the George Washington University and NASA Langley Research
Center in cooperation with the National Science Foundation, the U.S. Association
for Computational Mechanics (USACM), the Air Force Office of Scientific
Research, and the American Society of Mechanical Engineers.
PERGAMON PRESS
OXFORD NEW YORK BEIJING · FRANKFURT
SAO PAULO SYDNEY TOKYO TORONTO
CAS 30:l/2-A
U.K. Pergamon Press pic, Headington Hill Hall,
Oxford 0X3 OBW, England
U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.
PEOPLE'S REPUBLIC Pergamon Press Room 4037, Qianmen Hotel, Beijing,
OF CHINA People's Republic of China
FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6,
OF GERMANY D-6242 Kronberg, Federal Republic of Germany
BRAZIL Pergamon Editora Ltda, Rua Eca de Queiros, 346,
CEP 04011, Paraiso, Säo Paulo, Brazil
AUSTRALIA Pergamon Press Australia Pty Ltd., P.O. Box 544,
Potts Point, N.S.W. 2011, Australia
JAPAN Pergamon Press, 5th Floor, Matsuoka Central Building,
1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan
CANADA Pergamon Press Canada Ltd., Suite No. 271,
253 College Street, Toronto, Ontario, Canada M5T 1R5
Copyright © 1988 Pergamon Press pic
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in any
form or by any means: electronic, electrostatic, magnetic tape,
mechanical, photocopying, recording or otherwise, without
permission in writing from the publishers.
ISBN 0 08 0371973
Published as a special issue of the journal Computers &
Structures, Vol. 30, Number 1/2 and supplied to subscribers
as part of their normal subscription. Also available to
non-subscribers.
Printed in Great Britain by A. Wheaton & Co. Ltd, Exeter
Tables of Contents
for
Pergamon Computer Journals
to be published as Softstrips
The Softstrip· (a registered trademark of Cauzin
Systems, Inc.) data strip printed on this page
includes this issue's entire Table of Contents in a
form which can be scanned directly into
commonly available computer data bases without
typing. The computer data base generated by
these data strips will be updated every issue and
it will allow you to construct a personal
computer-based data base to easily catalog and
later search for specific topics.
In 1986 Pergamon Press began using Softstrip
System technology. This new system is being
broadly adopted by book, journal, software, and
magazine publishers. It allows readers to
automatically scan and enter data (or computer
software) directly from the printed pages of
publications into all popular personal computers
— quickly, automatically, and error-free. Data
strips, like the one printed on the edge of this
page, will be appearing in a wide variety of
publications. They may all be scanned by using a
low-cost Softstrip Reader which can be obtained
from selected computer stores.
The Table of Contents information is structured
in a form which will directly enter into personal
computer software programs such as Living
Videotext ThinkTank or READY! Other data
bases may also be used, such as dBASE II, dBASE
III, Lotus 1-2-3, and Lotus Symphony.
For detailed information on the structure of the
data files, or for more information on the
Softstrip System, please write to:
Director of Publishing
Pergamon Press, Inc.
Fairview Park
Elmsford, NY 10523, USA
[phone]
Softstrip
CAS
Vol. 30
No. 1/2
1988
COMPUTERS &
STRUCTURES
2 3 4 5
I I I I I
1
ANNOUNCEMENT
Pergamon Press To Use
Softstrip· Technology
USING THE SOFTSTRIP PROCESS
FOR PUBLISHING SOFTWARE AND DATA
IN PERGAMON JOURNALS
Beginning in 1986, Pergamon Press began using Softstrip System technology in
selected publications. This new system and printing format allows readers to
automatically scan and enter data (or computer software) directly from the
printed pages of publications into all popular personal computers, quickly,
automatically, and error-free. Data strips will be appearing in a wide
variety of publications and they may all be scanned by using a low-cost
Softstrip reader which can be obtained from Pergamon Press, Inc., Maxwell
House, Fairview Park, Elmsford, NY 10523.
Pergamon Press offers the Tables of Contents of selected computer-related
journals on data strips. This allows subscribers to quickly and easily save
each issue's Table of Contents in a personal or larger data base. The data
base helps readers by simplifying article, author or topical information
searches.
Pergamon Press will also convert data and software provided by contributors
into data strip format. These data strips will be printed in our journals
with the contributor's article. By publishing data in the Softstrip format,
we will provide readers with greater utility and convenience. It will also
provide contributors with an enhanced publishing medium where publishing
materials relevant to use with personal computers is simplified.
CONTRIBUTOR'S INSTRUCTIONS
Submitting Articles for Publication
Contributors should follow standard manuscript submission procedures as
published in the journal. Upon acceptance by the Editor, you will be asked to
submit a diskette in accordance with Items 1-3 below.
Submitting Software/Data for Publication
Contributors are asked to submit their materials on carefully packaged
magnetic diskettes. The diskettes must be "unprotected." Each diskette must
be accompanied by a separate letter which clearly indicates the following
information:
[1] The type of computer and language from (and for) which the software and/or
data is meant to be used. NOTE: Initially, Pergamon Press will support
only three classes of machines; the IBM PC and compatibles, the Apple II
Series of computers, and the Apple Macintosh. (Other computers will be
added as demand for them is determined.)
[2] The name of the file(s) on the disk and the type of file(s). Files may be
programs in BASIC, Assembly Language, or Object Code. Alternatively,
files may be Lotus 1-2-3 worksheets or templates, text files, or simple
ASCII data files. Files for IBM PCs and compatibles should run under
PCDOS or MSDOS and those for Apple II Series computers should run under
DOS 3.3 or ProDOS.
[3] Brief instructions on the use of each program or data file explaining how
to run the program or how to access the data.
All software programs and data tables should be submitted only in conjunction
with the articles for which they are intended, both in hard copy form and
diskette. The publisher will determine how the strip will be presented.
Diskette cannot be returned.
Contributors are encouraged to use this new medium to supplement or enhance
their articles, especially where the articles contain software program
listings or data tables which readers may wish to use with computers.
Initially, to make it possible for Pergamon Journals to convert material into
data strips, we will support only certain computers and software languages.
For additional information on the Softstrip System in Pergamon publications,
please write to Director of Publishing, Pergamon Press, Inc., Maxwell House,
Fairview Park, Elmsford, NY 10523, USA.
Computers & Structures Vol. 30, No. 1/2, p. vii, 1988
Pergamon Press pic. Printed in Great Britain.
PREFACE
In the last two decades computational structural mechanics (CSM) and computational fluid dynamics
(CFD) have emerged as new disciplines combining structural mechanics and fluid dynamics with
approximation theory, numerical analysis, and computer science. The use of CSM and CFD has transformed
much of theoretical mechanics and abstract science into practical and essential tools for a multitude of
technological developments which affect many facets of our life.
Major advances in CSM and CFD continue to take place on a broad front. The new advances are
manifested by the development of sophisticated computational models to simulate mechanical, thermal and
electromagnetic behavior of fluids and structures; efficient discretization techniques; computational strategies
and numerical algorithms; as well as versatile and powerful software systems for solution of complex fluids
and structures problems. Examples of such problems abound and come from diverse engineering systems
including microelectronic devices, nuclear reactors, high-speed flight vehicles, and the space station.
Two major factors have contributed to the rapid pace of development of CSM and CFD in recent years.
The first factor is the significant advances in computational technology and the explosive growth in
computer hardware capability. This progress shows no sign of abating; all indications are that the changes
during the next decades will prove to be even greater, particularly with the introduction of novel forms of
computer architecture (e.g. multiprocessor systems, neural networks and optical computers). The second
major factor is the growing interaction among a number of disciplines including applied mechanics, control
technology, numerical analysis and software design.
Despite the significant advances made in each of the CSM and CFD disciplines, there has not been
enough interaction and cross fertilization between the two disciplines. Consequently, some of the techniques
used in one discipline were either rediscovered by, or remained unknown to, researchers working in the
other discipline. As a step to remedy this situation and to establish strong interaction among researchers
in CSM and CFD, a three-day symposium entitled Advances and Trends in Computational Structural
Mechanics and Fluid Dynamics was held in Washington, DC, 17-19 October 1988. The organizing committee
expected that by bringing together leading experts and active researchers in areas which could impact
future development of both the CSM and CFD disciplines, formal presentations and personal interaction
would increase communication and foster effective development of the computational mechanics technology.
The 41 papers contained in this volume document some of the major advances that have occurred in
both the CSM and CFD disciplines and help identify future directions of development in these fields. The
topic headings in the symposium are largely represented by the 10 section headings of this volume, namely:
Fluid Structure Interaction and Aeroelasticity; CFD Technology and Reacting Flows; Micromechanics,
Deformable Media and Damage Mechanics; Stability and Eigenproblems; Probabilistic Methods and
Chaotic Dynamics; Perturbation and Spectral Methods; Element Technology (Finite Volume, Finite
Elements and Boundary Elements); Adaptive Methods; Parallel Processing Machines and Applications; and
Visualization, Mesh Generation and Artificial Intelligence Interfaces.
The fields covered by this symposium are rapidly changing, and if new results are to have the maximum
impact and use, they must reach workers in the field as soon as possible. This consideration led to the
decision to publish the proceedings prior to the symposium. Special thanks go to Pergamon Press for their
cooperation in publishing this volume and to Dean Harold Liebowitz, School of Engineering and Applied
Science of The George Washington University for making arrangements for the publication.
The editors express their sincere thanks to the many individuals and organizations who contributed to
the planning of this symposium; in particular to the members of the technical program committee for their
contributions to the various phases of this symposium. Special thanks go to the authors of the papers, for
the effort they have put forth in the preparation of their manuscripts, and to the symposium secretary, Mrs
Mary Torian, for her constant help and support.
The assistance of the National Science Foundation, the U.S. Association of Computational Mechanics
(USACM), the Air Force Office of Scientific Research, and the American Society of Mechanical Engineers
are especially appreciated. It is our earnest hope that the publication of these proceedings will help broaden
awareness within the engineering community of the recent advances in computational structural mechanics
and fluid dynamics, and will serve the profession well.
The George Washington University Ahmed K. Noor
NASA Langley Research Center Douglas L. Dwoyer
Hampton, VA 23665
U.S.A.
Vll
Computers & Structures Vol. 30, No. 1/2, pp. 1-13, 1988 0045-7949/88 $3.00 + 0.00
Printed in Great Britain. © 1988 Pergamon Press pic.
INTERACTION OF FLUIDS AND STRUCTURES FOR
AIRCRAFT APPLICATIONS
GURU P. GURUSWAMYI
Applied Computational Fluids Branch, Ames Research Center, Moffett Field, California, U.SA.
Abstract—Strong interactions occur between the flow about an aircraft and its structural com-
ponents, which result in several important aeroelastic phenomena. These aeroelastic phenomena can
significantly influence the performance of aircraft. At present, closed-form solutions are available for
aeroelastic computations when flows are in either linear subsonic or supersonic range. However, for
complex nonlinear flows containing shock waves, vortices and flow separations, computational
methods are still under development. Several phenomena that can be dangerous and can limit the
performance of an aircraft are due to the interaction of these complex flows with flexible aircraft
components such as wings. For example, aircraft with highly swept wings experience vortex-induced
aeroelastic oscillations. The simulation of these complex aeroelastic phenomena requires coupling the
fluid and structural analysis. This paper provides a summary of the development of such coupled
methods and their applications to aeroelasticity. Results based on the transonic small perturbation
equations and the Euler equations are presented.
INTRODUCTION developed for computing time-accurate aeroelastic
Strong interactions of structures and fluids are responses of typical sections by using the modal
common in many engineering environments. Such equations of motion [4]. This was successfully in-
interactions can give rise to physically important corporated in the two-dimensional, unsteady, tran-
phenomena such as those which occur for aircraft sonic code LTRAN2 (the present improved ver-
due to aeroelasticity. Aeroelasticity deals with the sion is called ATRAN2)[5]. The method was
science that studies the mutual interaction between demonstrated to compute the transonic flutter
aerodynamic forces and elastic forces for aircraft. boundaries of typical sections. This procedure was
Aeroelasticity significantly influences the perfor- later extended for wings and was incorporated in
mance of aircraft. Correct understanding of aero- the ATRAN3S[6] code, the Ames version of Air
elastic characteristics is important for safe and Force/NASA XTRAN3S[7,8], a code for tran-
efficient performance of aircraft. sonic aeroelastic analysis of aircraft. ATRAN3S is
To date, exact methods are available for making the most advanced code for aeroelastic analyses
aeroelastic computations when flows are in either based on the transonic small perturbation (TSP)
the linear subsonic or supersonic range. However, equations. Currently, ATRAN3S is being used for
for complex flows containing shock waves, vor- generic research in unsteady aerodynamics and
tices, and flow separations, computational methods aeroelasticity of almost full aircraft config-
are still under development. Several phenomena urations^]. Though codes based on the potential
that can be dangerous and limit the performance of flow theory give some practically useful results,
an aircraft occur due to the interaction of these they cannot be used for cases such as separated
complex flows with flexible aircraft components flows. Now, given the availability of new, efficient,
such as wings. For example, aircraft with highly numerical techniques and faster computers[10] the
swept wings experience vortex-induced aeroelastic time has come to consider Euler/Navier-Stokes
oscillations [1]. Several undesirable aeroelastic (ENS) equations for aeroelastic applications.
phenomena occur in the transonic range which are Codes based on the ENS equations have already
due to the presence and movement of shock waves. been applied for practically interesting problems
Limited wind tunnel and flight tests have shown a involving steady flows. Generic codes such as
critical aeroelastic phenomena such as a low tran- ARC3D[11], NASA Ames Research Center's
sonic flutter speed due to shock wave mo- three-dimensional ENS code, have been used for
tions [2, 3]. For the hypersonic vehicles, panel several scientific investigations. A state-of-the-art
flutter may play an important role in its design. survey of ENS codes is given in [12]. Research
In order to accurately compute the interactions codes such as ARC3D have produced practically
between fluids and structures, it is necessary to useful codes such as the TNS code, NASA Ames
solve the fluid and structural equations of motion Transonic Navier-Stokes code based on zonal
simultaneously. Such a procedure was first grids. TNS has successfully computed complex,
separated, steady flows about wings and wing-body
configurations[13]. Most of the work to date, by
tPrincipal Analyst, Sterling Federal Systems. using the ENS equations, are limited to either
1
2 GURU P. GURUSWAMY
NS-T - THIN LAYER NAVIER-STOKES
FP- FULL POTENTIAL
TSP - TRANSONIC SMALL PERTURBATION
LIFTING AIRFOILS
(EULER) WING-BODY SWEPT WINGS WING-BODY WING-BODY
(TSP) (FP) (EULER) (NS-T)
έ- -v
STEADY T
SWEPT WING AIRFOILS FULL AIRCRAFT WING-BODY SWEPT WING
(TSP) (FP) (TSP) (FP) (NS-T)
AIRFOILS AIRFOILS SWEPT WINGS WING-BODY
(TSP) (FP) (FP) (TSP)
r^r
UNSTEADY i— Λ"
SWEPT WINGS AIRFOILS WING
(TSP) (EULER) (EULER RESEARCH)
WINGS WINGS
AIRFOILS AIRFOILS
(FP) (EULER
(TSP) (EULER)
^Λ -RE+SE ARCH)
AEROELASTIC
SWEPT WINGS WING-BODY
(TSP) (TSP)
1970 1975 1980 1985 1990
Fig. 1. Milestones in the development of computational aerodynamics for aircraft.
steady flow or, at the most, unsteady flows about GOVERNING AERODYNAMICS EQUATIONS AND
rigid bodies. Coupling of the ENS equation with APPROXIMATIONS
the structural equation of motion has just Computations in this paper were done by using
begun [14], the Euler equations. However, all techniques
The present paper summarizes the development presented in this paper can be easily extended to
of advanced computational fluid dynamics tech- compute results using the Navier-Stokes equations.
niques and their application to aeroelasticity.
The Euler equations of motion
The strong conservation law form of the Euler
equations are used for shock-capturing purposes.
HISTORICAL PERSPECTIVE
The equations in Cartesian coordinates in non-
A brief chronology of the development of com- dimensional form can be written as
putational fluid dynamics (CFD) for aircraft ap-
plications is shown in Fig. 1 (based on Fig. 2 of dO dE dF dG
[10]). Figure 1 shows the lag in the development of — + — + — +— = 0Λ, (1)
dt dx dy dZ
aeroelastic methods using CFD. The main reason
for this lag is the lack of efficient unsteady methods where
in CFD for computationally intensive aeroelastic
calculations. The typical computational time for P
aeroelastic studies is about two orders more than pu
that required for steady state studies. This increase Q = pv
in computational time is due to the need of un- pw
steady computations and additional complexities in L e
physics associated with aeroelasticity. The history
of the CFD applications to aeroelasticity is sum- Γ PW " - pv i pW Ί
marized in Fig. 2. This figure shows the fact that a IP"2 + P puv puw
lot needs to be done in the area of the use of exact E = puv , F = pv2 + p , G = pvw
flow equations for aeroelastic applications. As seen puw pvw pw2 + p
in Fig. 2, most advanced aeroelastic applications l(e + p)u. J,e + p)v. -{e + p)w\
use the TSP equations. A major part of this paper (2)
discusses the use of the TSP theory to investi-
gate several practically interesting aeroelastic The Cartesian velocity components M, V and w are
phenomena. The current development of aeroelas- nondimensionahzed by a«, (the free-stream speed of
tic computational methods based on the ENS sound), density p is nondimensionahzed by p; the
x
equations is also presented. total energy per unit volume e is nondimen-
Interaction of fluids and structures for aircraft applications 3
BASED ON UNSTEADY TIME ACCURATE METHODS
TSP FP EULER NAVIER
STOKES
1978 ? 1986 ?
1982 1984 1988 ?
1986 ? ? ?
1988 ? ? ?
Fig. 2. History of CFD applications to aeroelasticity.
sionalized by p^ai; and the time t is nondimen- reported by Pulliam and Chaussee [16], both based
sionalized by c/a where c is the root chord. Pres- on implicit approximate factorization, are used.
sure can be found from the ideal gas law as Both algorithms were implemented in a new code,
ENSAERO, a general-purpose aeroelastic code
p = (y-l)[e- 0.5 p(u2 + v2 + w2)] (3) based on the ENS equations and the modal struc-
tural equations of motion with time-accurate
and throughout γ is the ratio of the specific heats. aeroelastic configuration adaptive grids. Results
To enhance numerical accuracy and efficiency presented in this paper are from ENSAERO ver-
and to handle boundary conditions more easily, the sion 1, which uses the diagonal algorithm to solve
governing equations are transformed from the the Euler equations.
Cartesian coordinates to general curvilinear coor- The diagonal algorithm used in this paper is a
dinates by using simplified version of the Beam-Warming scheme.
In the diagonal algorithm, the flux Jacobians are
r= t diagonalized so that the computational operation
count is reduced by 50%. The diagonal scheme is
ξ=ξ(χ,γ,ζ,ή first-order-accurate in time and yields time-ac-
(4) curate shock calculations in a non-conservative
7} = η(χ, y, z, i) mode. More details of this scheme can be found in
[16].
£=£(x,y,z,f).
Transonic small perturbation equations
Several numerical schemes have been developed A decade ago when CFD was becoming popular
to solve the transformed form of eqn (1). In this for aeronautical applications, the use of the com-
work, the algorithm developed by Beam and plete ENS equation was not practical because of
Warming [15] and the diagonal algorithm extension the lack of efficient methods and computer
4 GURU P. GURUSWAMY
resources. As a result, several simplified equations where [φ] is the modal matrix and {q} is the
were derived from the Euler form of eqn (1). For generalized displacement vector. The final matrix
unsteady transonic calculations, among the most form of the aeroelastic equations of motion is
useful of the simplified equations has been the
m{q} + [G]{q} + [K]{q} = {Fl, (7)
transonic small perturbation (TSP) equations based
on the potential flow theory[17]. The TSP theory
where [M], [G] and [K] are modal mass, damp-
has resulted in production codes such as
ing and stiffness matrices, respectively. {F} is
ATRAN3S, which has been successfully used for
the aerodynamic force vector defined as
advanced aeroelastic applications [18].
(i)p^[0]T[A]{AC} and [A] is the diagonal area
The TSP equation used in ATRAN3S is p
matrix of the aerodynamic control points.
Αφ„ + Βφ = [Εφ + Ρφ2 + G02L + [φ + Ηφφ\ The aeroelastic equation of motion [eqn (7)] is
χι χ χ y ν χ ν solved by a numerical integration technique based
on the linear acceleration method[4].
+ [<fcL, (5)
where AEROELASTIC RESULTS FROM THE TSP
THEORY
A = Ml; B = 2Mi; E = (1 - Ml); This section describes aeroelastic results from
the TSP equations coupled with the modal struc-
F = -Ö)(7+l)Aß; 0 = -Θ(γ-3)Λβ; tural equations of motion. All results presented are
from ATRAN3S, the NASA Ames version of Air
H = -(y-\)Ml Force/NASA, XTRAN3S. Following this code
several other codes have been developed. Results
The timely success in using eqn (5) for aero- shown in this section are representative of all such
elastic applications was because of its simplicity in developments.
grid requirements and boundary conditions. In this
Transonic aeroelastic calculations of rectangular
paper, results obtained using both the Euler equa-
wings
tion [eqn (1)] and the TSP equation [eqn (5)] will
be presented. The successful development of the two-dimen-
sional code LTRAN2, which employs an alternat-
ing-direction-implicit (ADI), finite-difference
AEROELASTIC EQUATIONS OF MOTION scheme, led to the development of three-dimen-
The governing aeroelastic equations of motion sional, unsteady, transonic, aerodynamic codes.
of a flexible wing are obtained by using the Ray- LTRAN3, the earlier low-frequency version of
leigh-Ritz method (Chap. 9 of [19]). In this ATRAN3S[6] was developed for time-accurate
method, the resulting aeroelastic displacements at calculations. The time-accuracy of this code was
any time are expressed as a function of a finite set validated against unsteady experimental data[21].
of assumed modes. The contribution of each Figure 4 taken from [21] shows the magnitude and
assumed mode to the total motion is derived by the the phase angle of the unsteady pressures for a
Lagrange's equation. Furthermore, it is assumed rectangular wing oscillating in its first bending
that the deformation of the continuous wing struc- mode. Time-accurate computations have ac-
ture can be represented by deflections at a set of curately captured the effects of unsteady motion of
discrete points. This assumption facilitates the use the shock wave. The rise in the phase angle behind
of discrete structural data, such as the modal the shock wave, which is one of the salient features
matrix, the modal stiffness matrix, and the modal of the unsteady transonic flow, has been predicted
mass matrix. These are generated by a finite-ele- accurately by LTRAN3. This code was success-
ment analysis or by experimental influence fully applied to compute the flutter boundaries of
coefficient measurements. In this study, the finite- rectangular wings by using coupled and uncoupled
element method is employed to obtain the modal methods. Figure 4 shows the good comparison of
data. Figure 3 shows the first five modes of a unsteady pressures and flutter boundary computed
rectangular wing computed by modeling the wing from LTRAN3 with the experiment and NAS-
with a 16 degrees-of-freedom rectangular finite TRAN, respectively.
element[20]. The scheme used in LTRAN3 is not adequate
It is assumed that the deformed shape of the for fighter wings since it uses the classical shearing
wing can be represented by a set of discrete dis- transformation technique [6]. As a result, a new
placements at selected nodes. From the modal code, ATRAN3S[6], was developed for general-
analysis the displacement vector {d} can be purpose applications to both transport and fighter
expressed as wings by using a modified shearing trans-
formation [6]. Additional capabilities such as
{d} = W{ql (6) supersonic free streams were added to this code.
Interaction of fluids and structures for aircraft applications 5
Fig. 3. Mode shapes and frequencies of a rectangular wing using the finite element method.
Figure 5(a) shows the stable, near neutrally plots of dynamic pressure vs Mach number from
stable, and unstable responses at M=l.l for a both computations and the experiment are shown
rectangular wing of aspect ratio 5 with a 6% thick in Fig. 5(b). The 'transonic dip' in the flutter curve
parabolic airfoil computed by solving eqn (7). which extends to the zone of supersonic free
The flutter speeds were computed by numerically streams can be seen in the figure. These results
interpolating the dynamic pressures to match a illustrate the importance of CFD codes to compute
response that corresponds to zero damping. The aeroelasticity in the transonic regime.