Fundamentals of Modern Unsteady Aerodynamics

347 Pages·2011·6.55 MB·English

by  Ülgen Gülçat (auth.)

Fundamentals of Modern
Unsteady Aerodynamics
Ülgen Gülçat
Fundamentals of Modern
Unsteady Aerodynamics
123
Prof.Dr. ÜlgenGülçat
Faculty ofAeronautics and Astronautics
IstanbulTechnical University
34469Maslak Istanbul
Turkey
[email protected]
ISBN 978-3-642-14760-9 e-ISBN 978-3-642-14761-6
DOI 10.1007/978-3-642-14761-6
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010933355
(cid:2)Springer-VerlagBerlinHeidelberg2010
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Preface
The flying animate objects were present in earth’s atmosphere about hundreds of
millionyearsbeforetheexistanceofhumankindonearth.Onlyatthebeginningof
twentiethcentury,theproperanalysisoftheliftingforcewasmadetoprovidethe
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 late
seventeenth century, his theory was a failure due to calculation of insufficient lift
generation and 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, then 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, that delayed the true analysis of the lift generation. The proper
analysisofliftingforce,ontheotherhand,wasindependentlymadeattheonsetof
twentieth century by the theoretical aerodynamicists Martin Kutta and Nicolai
Joukowski of Central and Eastern Europe respectively. At about the same years,
theWrightbrothers,whoseeffortsonpoweredflightwereridiculedbyauthorities
of their time, were able to fly a short distance. Thereafter, in a time interval little
more than a century, which is a considerably short span compared to the dawn of
civilization, we see not only tens of thousands of aircrafts flying in earth’s
atmosphere at a given moment but we also witnessed unmanned or manned
missions to the moon, missions to almost every planet in our solar system and to
deeper space to let the existence of life on earth be known by the other possible
intelligent life forms.
The foundation of the century old discipline of aeronautics and astronautics
undoubtedly lies in the progress made in aerodynamics. The improvement made
on the aerodynamics of wings, based on satisfying the Kutta condition at the
trailingedgetogiveacirculationnecessaryforliftgeneration,wassorapidthatin
less than a quarter century it led to the breaking of the sound barrier and to the
discoveryofsustainablesupersonicflight,whichwasunprecedentedinnatureand
v
vi Preface
once thought to be not possible! In many engineering applications involving
motion we encounter either forced or velocity induced oscillatory motion at high
speeds.Ifthechanges intheexcitations arerapid,theresponse ofthesystemlags
considerably. Similarly, the response of the aerodynamic systems cannot be
determined using steady aerodynamics for rapidly changing excitations. The
unsteady aerodynamics, on the other hand, has sufficient tools to give accurately
the phase lag between the rapid motion change and the response of the aerody-
namic system.Asweobserve theperformancesofperfect aerodynamicstructures
of nature, we understand the effect of unsteady phenomena to such an extent that
liftcanbegeneratedwithapparentmassevenwithoutafreestream.Insomecases,
when the classical unsteady aerodynamics does not suffice, we go beyond the
conventionalconcepts,withobservingnatureagain,toutilizetheextraliftcreated
by the suction force of strong vorticies shed from the sharp leading edge of low
aspect ratio wings at high angles of attack. We implement this fact in designing
highly maneuverable aircrafts at high angles of attack and low free stream
velocities. If we go to angles of attack higher than this, we observe aerodynami-
cally induced but undesirable unsteady phenomena called wing rock. In addition,
quite recently the progress made in unsteady aerodynamics integrated with elec-
tronics 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 the project engi-
neerswiththefoundationsaswellastheknowledgeneededaboutthemostrecent
developments involvingunsteady aerodynamics. This need emerges from the fact
thatthedesignandtheanalysistoolsusedbytheresearchengineersaretreatedas
blackboxesprovidingresultswithinadequateinformationaboutthetheoryaswell
as practice. In addition, the models of complex aerodynamic flows and their
solution methodologies are provided with examples, and enhanced withproblems
and questions asked at the end of each chapter. Unlike this full text, the recent
developments made in unsteady aerodynamics together with the fundamentals
havenotappearedasatextbookexceptinsomechaptersofbooksonaeroelasticity
or helicopter dynamics!
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, thankstothe software allowingsymbolic operations andversatile
numerical techniques, is quite simple to solve and analyze even on our PCs.
Although the problems become more challenging and demanding by time, how-
ever,thedevelopmentofnoveltechnologiesandmethodsrenderthempossibleto
solve provided that the fundamentals are well taught and understood by well
informedusers.Themodernsubjectscoveredinthebookarebasedonthelecture
notesof‘UnsteadyAerodynamics’coursesofferedbymeforthepastseveralyears
at Istanbul Technical University.
Preface vii
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 at 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.Themainideatobedeliveredineachchapterisgivenasaverbalsummary
at chapters’ end together with the most up to date references. There are ten
Appendixes appearing to supplement the formulae driven without distracting the
uniformity of the text.
I had the opportunity of reusing and borrowing some material from the pub-
lications of Joseph Katz, AIAA, NATO-AGARD/RTO and Annual Review of
Fluid Mechanics with their kind copyright permissions. Dr. Christoph Baumann
readthetextandmadethenecessaryarrangementsforitspublicationbySpringer.
Zeliha Gülçat and Canan Danı(cid:2)smam provided me with their kind help in editing
the entire text. N. Thiyagarajan from Scientific Publishing Services prepared the
metadata of the book. Aydın Mısırlıog˘lu and Fırat Edis helped me in transferring
thegraphsintoworddocuments.Ididthetypingofthebook,andobtainedmostof
the graphs and plots despite the ‘carpal tunnel syndrome’ caused by the intensive
usage of the mouse. Furthermore, heavy concentration on subject matter and
continuous work hours spent on the text showed itself as developing ‘shingles’!
MywifeZelihastoodbymeinallthesedifficulttimeswithgreatpatience.Iwould
liketoextendmygratitude,oncemore,toallwhocontributedtotherealizationof
this book.
Datça and I_stanbul, August, 2010 Ülgen Gülçat
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. . . . . . . . . . . . . . . . . . . . . 5
1.4 Steady Aerodynamics of Thin Wings . . . . . . . . . . . . . . . . . . . 7
1.5 Unsteady Aerodynamics of Slender Wings . . . . . . . . . . . . . . . 8
1.6 Compressible Steady Aerodynamics. . . . . . . . . . . . . . . . . . . . 8
1.7 Compressible Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . 12
1.8 Slender Body Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Hypersonic Aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.10 The Piston Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 Modern Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.12 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.4 Acceleration Potential. . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.5 Moving Coordinate System. . . . . . . . . . . . . . . . . . . . 35
ix
x Contents
2.2 Real Gas Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 System and Control Volume Approaches . . . . . . . . . . 37
2.2.2 GlobalContinuityandtheContinuityoftheSpecies . . . 37
2.2.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.4 Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.5 Equation of Motion in General Coordinates. . . . . . . . . 42
2.2.6 Navier–Stokes Equations. . . . . . . . . . . . . . . . . . . . . . 44
2.2.7 Thin Shear Layer Navier–Stokes Equations. . . . . . . . . 47
2.2.8 Parabolized Navier–Stokes Equations . . . . . . . . . . . . . 49
2.2.9 Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . 50
2.2.10 Incompressible Flow Navier–Stokes Equations. . . . . . . 51
2.2.11 Aerodynamic Forces and Moments. . . . . . . . . . . . . . . 52
2.2.12 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.13 Initial and Boundary Conditions. . . . . . . . . . . . . . . . . 55
2.3 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Incompressible Flow About an Airfoil. . . . . . . . . . . . . . . . . . . . . . 59
3.1 Impulsive Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Steady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Unsteady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 Loewy’s Problem: Returning Wake Problem. . . . . . . . . . . . . . 81
3.6 Arbitrary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Arbitrary Motion and Wagner Function . . . . . . . . . . . . . . . . . 82
3.8 Gust Problem, Küssner Function . . . . . . . . . . . . . . . . . . . . . . 87
3.9 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Incompressible Flow About Thin Wings . . . . . . . . . . . . . . . . . . . . 95
4.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Steady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.1 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.2 Weissinger’s L-Method. . . . . . . . . . . . . . . . . . . . . . . 105
4.2.3 Low Aspect Ratio Wings . . . . . . . . . . . . . . . . . . . . . 107
4.3 Unsteady Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Reissner’s Approach. . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4 Arbitrary Motion of a Thin Wing. . . . . . . . . . . . . . . . . . . . . . 119
4.5 Effect of Sweep Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Low Aspect Ratio Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.7 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Contents xi
5 Subsonic and Supersonic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1 Subsonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Subsonic Flow about a Thin Wing. . . . . . . . . . . . . . . . . . . . . 135
5.3 Subsonic Flow Past an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . 137
5.4 Kernel Function Method for Subsonic Flows. . . . . . . . . . . . . . 139
5.5 Doublet–Lattice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 Arbitrary Motion of a Profile in Subsonic Flow. . . . . . . . . . . . 144
5.7 Supersonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.8 Unsteady Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.9 Supersonic Flow About a Profile . . . . . . . . . . . . . . . . . . . . . . 152
5.10 Supersonic Flow About Thin Wings. . . . . . . . . . . . . . . . . . . . 154
5.11 Mach Box Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.12 Supersonic Kernel Method . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.13 Arbitrary Motion of a Profile in Supersonic Flow . . . . . . . . . . 161
5.14 Slender Body Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.15 Munk’s Airship Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.16 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6 Transonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1 Two Dimensional Transonic Flow, Local Linearization. . . . . . . 171
6.2 Unsteady Transonic Flow, Supersonic Approach . . . . . . . . . . . 176
6.3 Steady Transonic Flow, Non Linear Approach. . . . . . . . . . . . . 177
6.4 Unsteady Transonic Flow: General Approach . . . . . . . . . . . . . 180
6.5 Transonic Flow around a Finite Wing. . . . . . . . . . . . . . . . . . . 184
6.6 Unsteady Transonic Flow Past Finite Wings . . . . . . . . . . . . . . 187
6.7 Wing–Fuselage Interactions at Transonic Regimes. . . . . . . . . . 189
6.8 Problems and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7 Hypersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.1 Newton’s Impact Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.2 Improved Newton’s Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.3 Unsteady Newtonian Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.4 The Piston Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.5 Improved Piston Theory: M2s2 = O(1). . . . . . . . . . . . . . . . . . 203
7.6 Inviscid Hypersonic Flow: Numerical Solutions. . . . . . . . . . . . 205
7.7 Viscous Hypersonic Flow and Aerodynamic Heating . . . . . . . . 213
7.8 High Temperature Effects in Hypersonic Flow. . . . . . . . . . . . . 220
7.9 Hypersonic Viscous Flow: Numerical Solutions. . . . . . . . . . . . 231
7.10 Hypersonic Plane: Waverider. . . . . . . . . . . . . . . . . . . . . . . . . 236
7.11 Problems and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
xii Contents
8 Modern Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.1 Static Stall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Dynamic Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.3 The Vortex Lift (Polhamus Theory) . . . . . . . . . . . . . . . . . . . . 258
8.4 Wing Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.5 Flapping Wing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8.6 Flexible Airfoil Flapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.7 Finite Wing Flapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
8.8 Problems and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
9 Aerodynamics: The Outlook for the Future . . . . . . . . . . . . . . . . . 307
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337