Fundamentals of Modern Unsteady Aerodynamics

402 Pages·2015·10.51 MB·English

by  Ülgen Gülçat

Ülgen Gülçat
Fundamentals of
Modern Unsteady
Aerodynamics
Second Edition
Fundamentals of Modern Unsteady Aerodynamics
Ü ü ç
lgen G l at
Fundamentals of Modern
Unsteady Aerodynamics
Second Edition
123
ÜlgenGülçat
Istanbul
Turkey
ISBN978-981-10-0016-4 ISBN978-981-10-0018-8 (eBook)
DOI 10.1007/978-981-10-0018-8
LibraryofCongressControlNumber:2015952985
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Preface to the Second Edition
The first edition of this book appeared in the fall of 2010 both as a hard copy and
e-book. Since then there has appeared, in the literature, numerous unsteady
aerodynamics-related material, which deserves to be presented in a graduate text-
book. Most of the new material is relevant to Chap. 8: Modern Topics. Here, a
calculation method for propulsive force, lift generation and induced drag of a
pitching-plunging thin finite wing is provided with a numerical example as an
additional material. The unsteady 3-D boundary layer solution technique is intro-
ducedforpredictionoftheviscousdragtoseeifthepropulsiveforceovercomesthe
drag. In addition, the ground effect on the air vehicles performing near ground is
formulatedtoseehowtheliftandthepropulsiveforcesarealteredforthehighand
low aspect ratio wings.
Thestate-spacerepresentationofaerodynamicswasintroducedbrieflyinthefirst
edition. In the present edition, more detailed discussion of the method is provided
vianumericalsolutionsforairfoilsandfinitewingsofvariousaspectratiosevenin
the presence of ground.
Additional material, including bio-inspired and biological flows, related to the
unsteady flows is also provided at the end of Chap. 9 to emphasize the present
developments and future prospects.
SomemorematerialisaddedtotheAppendixsothatnoderivationofequations
is left incomplete but not overdone in the text.
Needless tosay,in thefirst editionthere werea few typographical errors which
have been detected and corrected for the second edition.
Dr. Christoph Baumann read the new material and took the necessary steps for
the second edition, and K. M. Govardhana prepared the metadata of the book.
Mehmet Tan provided the figure for the cover page. My wife Zeliha, once more,
stood by me in all these times with great patience. Finally, I would like to express
my gratitude and appreciation to all who made publication of this book possible.
Istanbul Ülgen Gülçat
October 2015
v
Preface
Flying animate objects were present in the earth’s atmosphere about hundreds of
millionyearsbeforetheappearanceofhuman-kindonearth.Onlyatthebeginning
ofthetwentiethcentury,wasproperanalysisoftheliftingforcemadetoprovidethe
possibility of powered and manned flight. Prior to that, one of the pioneers of
mechanics, Sir Isaac Newton had used ‘his impact theory’ in an attempt to for-
mulate the lifting force created on a body immersed in a free stream. In the late
seventeenth century, his theory was a failure due to calculation of insufficient lift
generation which made him come to the conclusion that ‘flying is a property of
heavenly bodies’. In a similar manner, almost after two centuries, William
Thomson(LordKelvin),whosecontributionstothermoandgasdynamicsarewell
known, proved that ‘only objects lighter than air’ can fly!
Perhaps it was the adverse influence of these two pioneers of mechanics on
Western Europe, where contributions to the discipline of hydrodynamics is
unquestionable,thatdelayedthetrueanalysisofliftgeneration.Theproperanalysis
of lifting force, on the other hand, was independently made at the onset of the
twentieth century by the theoretical aerodynamicists Martin Kutta and Nicolai
JoukowskiofCentralandEasternEuroperespectively.Inaboutthesameyears,the
Wright brothers, whose efforts on powered flight were ridiculed by the authorities
of their time, were able to fly a short distance. Thereafter, in a time interval of a
littlemorethanacentury,whichisaconsiderablyshortspancomparedtothedawn
of civilization, we see not only tens of thousands of aircraft flying in the earth’s
atmosphereatagivenmomentbutwealsowitnessunmannedormannedmissions
to the moon, missions to almost every planet in our solar system and to deeper
spacetolettheexistenceoflifeonearthbeknownbytheotherpossibleintelligent
life forms.
The foundation of the century-old discipline of aeronautics and astronautics
undoubtedlyliesintheprogressmadeinaerodynamics.Theimprovementmadeon
the aerodynamics of wings, based on satisfying the Kutta condition at the trailing
edgetogiveacirculationnecessaryforliftgeneration,wassorapidthatinlessthan
aquarter centuryitledtothebreakingofthesoundbarrier andtothediscovery of
vii
viii Preface
sustainablesupersonicflight,whichwasunprecedentedinnatureandoncethought
to be not possible! In many engineering applications involving motion we
encountereitherforcedorvelocity-inducedoscillatorymotionathighspeeds.Ifthe
changes in the excitations are rapid, the response of the system lags considerably.
Similarly, the response of the aerodynamic systems cannot be determined using
steadyaerodynamicsforrapidlychangingexcitations.Theunsteadyaerodynamics,
on the other hand, has sufficient tools to give accuratelythe phase lag between the
rapid motion change and the response of the aerodynamic system. As we observe
the performances of perfect aerodynamic structures of nature, we understand the
effect of unsteady phenomena to such an extent that lift can be generated with
apparent mass even without a free stream. In some cases, when the classical
unsteady aerodynamics does not suffice, we go beyond the conventional concepts,
with observing nature again, to utilize the extra lift created by the suction force of
strongvorticies shedfromthesharpleadingedgeoflowaspectratiowingsathigh
anglesofattack.Weimplementthisfactindesigninghighlymaneuverableaircrafts
athighanglesofattackandlowfreestreamvelocities.Ifwegotoanglesofattack
higher than this, we observe aerodynamically induced but undesirable unsteady
phenomena called wing rock. In addition, quite recently the progress made in
unsteady aerodynamics integrated with electronics enable us to design and operate
Micro Air Vehicles (MAVs) based on flapping wing technology having radio
controlled devices.
This book, which gives the progress made in unsteady aerodynamics in about
less than a century, is written to be used as a graduate textbook in Aerospace
Engineering. Another important aim of this work is to provide project engineers
with the foundations as well as knowledge needed about the most recent devel-
opments involving unsteady aerodynamics. This need emerges from the fact that
thedesignandanalysistoolsusedbyresearchengineers aretreatedasblackboxes
providing results with inadequate information about the theory and practice. In
addition, the models of complex aerodynamic flows and their solution method-
ologies are provided with examples, and enhanced with problems and questions
askedattheendofeachchapter.Unlikethisfulltext,therecentdevelopmentsmade
in unsteady aerodynamics together with the fundamentals have not appeared as a
textbookexceptinsomechaptersofbooksonaeroelasticityorhelicopterdynamics!
The classical parts of this book are mainly based on ‘not so terribly advanced’
lecture notes of Alvin G. Pierce and basics of vortex aerodynamics knowledge
provided by James C. Wu while I was a PhD student at Georgia Tech. What was
then difficult to conceive and visualize because of the involvement of special
functions, now, thanks to the software allowing symbolic operations and versatile
numerical techniques, is quite simple to solve and analyze even on our PCs.
Although the problems become more challenging and demanding by time, the
development of novel technologies and methods render them possible to solve
provided that the fundamentals are well taught and understood by well-informed
users. The modern subjects covered in the book are based on lecture notes on
‘Unsteady Aerodynamics’ courses offered by me since the past several years at
Istanbul Technical University.
Preface ix
The first five chapters of the book are on the classical topics, whereas the rest
covers the modern topics, and the outlook and the possible future developments
finalize the book. The examples provided in each chapter are helpful in terms of
application of relevant material, and the problems at the end of each chapter are
useful for the reader towards understanding of the subject matter and its future
usage. Themainideatobedelivered ineachchapterisgiven asaverbalsummary
at chapters’ end together with the most up-to-date references. There are ten
Appendixesthatsupplementtheformulaedrivenwithoutdistractingtheuniformity
of the text.
Ihadtheopportunitytoreuseandborrowsomematerialfromthepublicationsof
JosephKatz,AIAA,NATO-AGARD/RTOandAnnualReviewofFluidMechanics
with their kind copyright permissions. Dr. Christoph Baumann read the text and
madethenecessaryarrangementsforitspublicationbySpringer.ZelihaGülçatand
Canan Danışman provided me with their kind help in editing the entire text.
N. Thiyagarajan prepared the metadata of the book. Aydın Mısırlıoğlu and Fırat
Edis helped me in transferring the graphs into word documents. I did the typing
of the book, and obtained most of the graphs and plots despite the ‘carpal tunnel
syndrome’ caused by the intensive usage of the mouse. Furthermore, heavy con-
centration on subject matter and continuous work hours spent on the text showed
itself as developing ‘shingles’! My wife Zeliha stood by me in all these difficult
times with great patience. I would like to extend my gratitude, once more, to all
who contributed to the realization of this book.
Datça and Istanbul Ülgen Gülçat
August 2010
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Center of Pressure (X ) . . . . . . . . . . . . . . . . . . . . . . 2
cp
1.1.4 Aerodynamic Center (X ). . . . . . . . . . . . . . . . . . . . . 3
ac
1.1.5 Steady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.6 Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . 3
1.1.7 Compressible Aerodynamics . . . . . . . . . . . . . . . . . . . 3
1.1.8 Vortex Aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Generation of Lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Unsteady Lifting Force Coefficient. . . . . . . . . . . . . . . . . . . . . 6
1.4 Steady Aerodynamics of Thin Wings . . . . . . . . . . . . . . . . . . . 7
1.5 Unsteady Aerodynamics of Slender Wings . . . . . . . . . . . . . . . 8
1.6 Compressible Steady Aerodynamics . . . . . . . . . . . . . . . . . . . . 9
1.7 Compressible Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . 12
1.8 Slender Body Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Hypersonic Aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.10 The Piston Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 Modern Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.12 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.4 Acceleration Potential. . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.5 Moving Coordinate System . . . . . . . . . . . . . . . . . . . . 38
xi
xii Contents
2.2 Real Gas Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 System and Control Volume Approaches. . . . . . . . . . . 40
2.2.2 Global Continuity and the Continuity of the Species. . . 41
2.2.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.4 Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.5 Equation of Motion in General Coordinates. . . . . . . . . 46
2.2.6 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 48
2.2.7 Thin Shear Layer Navier-Stokes Equations . . . . . . . . . 50
2.2.8 Parabolized Navier-Stokes Equations . . . . . . . . . . . . . 52
2.2.9 Boundary Layer Equations. . . . . . . . . . . . . . . . . . . . . 53
2.2.10 Incompressible Flow Navier-Stokes Equations . . . . . . . 54
2.2.11 Aerodynamic Forces and Moments. . . . . . . . . . . . . . . 56
2.2.12 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.13 Initial and Boundary Conditions. . . . . . . . . . . . . . . . . 59
2.3 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Incompressible Flow About an Airfoil. . . . . . . . . . . . . . . . . . . . . . 63
3.1 Impulsive Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Unsteady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Loewy’s Problem: Returning Wake Problem . . . . . . . . . . . . . . 86
3.6 Arbitrary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.1 Arbitrary Motion and Wagner Function. . . . . . . . . . . . 88
3.6.2 Gust Problem, Küssner Function . . . . . . . . . . . . . . . . 92
3.7 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 Incompressible Flow About Thin Wings . . . . . . . . . . . . . . . . . . . . 103
4.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Unsteady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Arbitrary Motion of a Thin Wing. . . . . . . . . . . . . . . . . . . . . . 127
4.5 Effect of Sweep Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.6 Low Aspect Ratio Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.7 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5 Subsonic and Supersonic Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Subsonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 Subsonic Flow About a Thin Wing . . . . . . . . . . . . . . . . . . . . 144
5.3 Subsonic Flow Past an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4 Kernel Function Method for Subsonic Flows. . . . . . . . . . . . . . 147
5.5 Doublet—Lattice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 Arbitrary Motion of a Profile in Subsonic Flow. . . . . . . . . . . . 154