Introduction to Transonic Aerodynamics

561 Pages·2015·21.21 MB·English

by  Roelof Vos

Fluid Mechanics and Its Applications
Roelof Vos
Saeed Farokhi
Introduction
to Transonic
Aerodynamics
Fluid Mechanics and Its Applications
Volume 110
Series editor
André Thess, German Aerospace Center, Institute of Engineering
Thermodynamics, Stuttgart, Germany
Founding Editor
René Moreau, Ecole Nationale Supérieure d’Hydraulique de Grenoble,
Saint Martin d’Hères Cedex, France
Aims and Scope of the Series
Thepurposeofthisseriesistofocusonsubjectsinwhichfluidmechanicsplaysa
fundamental role.
Aswellasthemoretraditionalapplicationsofaeronautics,hydraulics,heatand
masstransferetc.,bookswillbepublisheddealingwithtopicswhicharecurrently
in a state of rapid development, such as turbulence, suspensions and multiphase
fluids, super and hypersonic flows and numerical modeling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive
intense scientific attention, bringing them to the forefront of technological
advancement. Fluidshave the ability to transport matter and itsproperties as well
astotransmitforce,thereforefluidmechanicsisasubjectthatisparticularlyopen
to cross fertilization with other sciences and disciplines of engineering. The
subject of fluid mechanics will be highly relevant in domains such as chemical,
metallurgical, biological and ecological engineering. This series is particularly
open to such new multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts
are monographs defining the current state of a field; others are accessible to final
year undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/5980
Roelof Vos Saeed Farokhi
(cid:129)
Introduction to Transonic
Aerodynamics
123
RoelofVos Saeed Farokhi
Faculty ofAerospace Engineering Department of Aerospace Engineering
Delft Universityof Technology The Universityof Kansas
Delft Lawrence, KS
The Netherlands USA
ISSN 0926-5112 ISSN 2215-0056 (electronic)
FluidMechanics andIts Applications
ISBN 978-94-017-9746-7 ISBN 978-94-017-9747-4 (eBook)
DOI 10.1007/978-94-017-9747-4
LibraryofCongressControlNumber:2015930202
SpringerDordrechtHeidelbergNewYorkLondon
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This book is dedicated to our wives,
Carola and Mariam.
Preface
Within aerodynamics, the transonic Mach range is a much-studied topic among
industrial and academic institutions. Many of today’s airliners cruise at Mach
numbers where both subsonic and supersonic flow exist. Terms such as drag
divergence, buffet, and transonic dip are all associated with transonic flow condi-
tions.Transonicaerodynamicshas,toalargedegree,dominatedtheexteriordesign
ofhigh-subsonicaircraftforthepast60years.However,textbooksoncompressible
aerodynamicsoftenfocusonthesupersonicMachrangeandlessonthephenomena
in the transonic Mach regime. Therefore, a comprehensive course on transonic
aerodynamics is seldom part of an aerospace engineering curriculum.
This discrepancy between the importance of transonic aerodynamics in practice
and the lack of a comprehensive course on the subject is often explained by the
complexity of the subject matter. Indeed, the governing equations that include all
the relevant aerodynamic phenomena cannot be easily reduced to something
equivalent to a thin airfoil theory or a lifting-line theory. Also, many correction
factors such as the well-known Prandt-Glauert compressibility correction do not
holdinthetransonicregime.Inordertopredicttheperformanceofawingorwing
section, one therefore often relies on computer programs that solve a numerical
implementationofthegoverningequationsofmotion.Butteachingstudentshowto
operateapieceofsoftwareisnotthesameasteachingstudentsthephysicsofhow
and why certain phenomena occur when a body is subjected to transonic flow
conditions. Therefore, this textbook was written to teach students about the nature
of transonic flow and how it can be captured in mathematical equations.
Thistextbookservesasanintroductiontothesubjectoftransonicaerodynamics.
In eight chapters we present a quantitative and qualitative assessment of subsonic,
supersonic,andtransonicflowaboutbodiesintwoandthreedimensions.Wehave
included relevant analytical analysis methods that allow students to practice with
thesubjectmatter.Thebookcontainsnumerousexamplesandeverychaptercloses
with a list of problems. Some subjects are treated more from a numerical per-
spective(e.g.,shockandexpansiontheory),whileothersarediscussedmorefroma
qualitative point of view (e.g., shock-boundary-layer interaction). Where possible,
numericalexamplesandmethodshavebeenincludedtoenhancetheunderstanding
vii
viii Preface
of each subject. The book contains 60 examples and more than 200 practice
problems.
Thistextbookisintendedprimarilyforseniorundergraduatestudentsorgraduate
students with prior knowledge in aerodynamics. Although we repeat the funda-
mental equations and flow characteristics in the beginning of the book, we do
assumethatthestudenthashadacourse onsubsonicaerodynamicsandisfamiliar
with its fundamentals. Even though knowledge of transonic aerodynamics is
important for many internal flow applications (e.g., turbo machinery, engine
intakes, exhausts, etc.) the present textbook primarily focuses on external aerody-
namics with limited applications to internal flows. Examples are targeted mainly
toward wings and bodies exposed to a transonic flow field. Many of the examples
reference real aircraft or wing components. Therefore, a strong connection is
present between the content of this textbook and the subject of transonic aircraft
design. To understand why a modern high-subsonic aircraft is designed the way it
is, requires one to understand the subject matter of this textbook.
Acknowledgments
Writing this book has been a privilege. Naturally, we would not have been able to
dosowithoutthesupportofourrespectiveuniversities.Wethereforewouldliketo
thanktheUniversityofKansasandDelftUniversityofTechnologyforprovidingus
with the time and resources to write this book. We would also like to thank the
publisher,who,basedona135-pagesummary,trustedthatwewouldextendittoits
current form. We would also like to acknowledge Dr. Luca Guadani and Dr. Ali
Elham who proofread various chapters and performed some of the numerical cal-
culations to support the examples in this book. A special thank you to emeritus
professor Egbert Torenbeek for providing us with meticulous feedback on the
content of various chapters. Finally, we would like to thank the following persons
on both sides of the Atlantic, who over the course of multiple years, aided in the
preparation of this document: Mr. Thomas Statsny, Ms. Lisanne van Veen,
Mr. Maarten Broekhuizen, Mr. Kevin Haagen, Ms. Maaike Weerdesteyn, and
Mr. Amool Raina.
Delft, The Netherlands, 2014 Roelof Vos
Lawrence, KS, USA Saeed Farokhi
Contents
1 Introduction and Historic Perspective . . . . . . . . . . . . . . . . . . . . . 1
1.1 From Subsonic to Supersonic Flight . . . . . . . . . . . . . . . . . . . 1
1.2 Definition of the Transonic Flow Domain . . . . . . . . . . . . . . . 4
1.3 Transonic Wind Tunnel Experiments. . . . . . . . . . . . . . . . . . . 5
1.4 Transonic Aerodynamics of Wings and Bodies. . . . . . . . . . . . 8
1.5 Transonic Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Outline of Present Textbook. . . . . . . . . . . . . . . . . . . . . . . . . 16
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Review of Fundamental Equations. . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Review of Partial Differential Equations . . . . . . . . . . . . . . . . 22
2.2.1 One-Dimensional Wave Equation and Solution
by D’Alembert. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 One-Dimensional Heat Equation and Solution
by Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Conservation Form of PDEs. . . . . . . . . . . . . . . . . . . 33
2.2.4 Classification of Partial Differential Equations . . . . . . 35
2.3 Review of Vector Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 Vectors, Vector Fields, and Scalar Fields. . . . . . . . . . 38
2.3.2 Gradient of a Scalar Field . . . . . . . . . . . . . . . . . . . . 40
2.3.3 Divergence of a Vector Field. . . . . . . . . . . . . . . . . . 41
2.3.4 Curl of a Vector Field. . . . . . . . . . . . . . . . . . . . . . . 42
2.3.5 Relation Between Volume, Surface,
and Line Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Review of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.1 Perfect Gas Relations . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2 First Law of Thermodynamics . . . . . . . . . . . . . . . . . 46
2.4.3 Second Law of Thermodynamics . . . . . . . . . . . . . . . 48
2.4.4 Isentropic Relations. . . . . . . . . . . . . . . . . . . . . . . . . 50
ix
x Contents
2.5 Equations of Fluid Motion. . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.2 Conservation of Linear Momentum. . . . . . . . . . . . . . 53
2.5.3 Conservation of Energy. . . . . . . . . . . . . . . . . . . . . . 56
2.5.4 Conservation Form of the Navier-Stokes Equations. . . 60
2.6 Reynolds-Averaged Navier-Stokes Equations . . . . . . . . . . . . . 62
2.6.1 Incompressible Reynolds-Averaged Equations
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.2 Compressible Reynolds-Averaged Equations
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.3 Turbulence Modeling: The k-Epsilon Model. . . . . . . . 67
2.7 Equations of Motion for Inviscid Flows. . . . . . . . . . . . . . . . . 69
2.7.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.7.2 Potential Flow Equation. . . . . . . . . . . . . . . . . . . . . . 72
2.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 Transonic Similarity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Linearized Compressibility Corrections . . . . . . . . . . . . . . . . . 82
3.2.1 2-D Subsonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Other Subsonic Compressibility Corrections. . . . . . . . 89
3.2.3 2-D Supersonic Flow. . . . . . . . . . . . . . . . . . . . . . . . 99
3.2.4 The Principle of Superposition . . . . . . . . . . . . . . . . . 103
3.2.5 Slender Bodies of Revolution in Subsonic
and Supersonic Flow—Linear Theory . . . . . . . . . . . . 106
3.3 Transonic Small Disturbance Theory. . . . . . . . . . . . . . . . . . . 114
3.4 Transonic Similarity Parameters . . . . . . . . . . . . . . . . . . . . . . 117
3.4.1 Other Transonic Similarity Parameters. . . . . . . . . . . . 120
3.5 3-D Planar and Axisymmetric Slender Bodies . . . . . . . . . . . . 123
3.6 Hodograph Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.7 Empirical Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.8 Approximate Location of Detached Shocks . . . . . . . . . . . . . . 130
3.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4 Shock-Expansion Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2 Lift and Wave Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.3 Bi-Convex Airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4 Axisymmetric and Slender Bodies. . . . . . . . . . . . . . . . . . . . . 183
4.5 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . . 188