Theoretical and applied aerodynamics

625 Pages·2015·8.13 MB·English

by  Chattot J.J.

J.J. Chattot · M.M. Hafez
Theoretical
and Applied
Aerodynamics
and Related Numerical Methods
Theoretical and Applied Aerodynamics
J.J. Chattot M.M. Hafez
(cid:129)
Theoretical and Applied
Aerodynamics
and Related Numerical Methods
123
J.J. Chattot M.M. Hafez
Department of Mechanical Department of Mechanical
andAerospace Engineering andAerospace Engineering
Universityof California Universityof California
Davis,CA Davis,CA
USA USA
ISBN 978-94-017-9824-2 ISBN 978-94-017-9825-9 (eBook)
DOI 10.1007/978-94-017-9825-9
LibraryofCongressControlNumber:2015932665
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
The purpose of this book is to expose students to the classical theories of aero-
dynamics to enable them to apply the results to a wide range of projects, from
aircraft to wind turbines and propellers. Most of the tools are analytical, but
computercodesarealsoavailableandareusedbythestudentstocarryoutsevento
eightprojectsduringthecourseofaquarter.Thesecomputertoolscanbefoundat
http://mae.ucdavis.edu/chattot/EAE127/ along with the project statements.
The main focus is on aircraft and the theories and codes that help in estimating
the forces and moments acting on profiles, wings, wing-tail and fuselage configu-
rations,appropriatetotheflowregime,i.e.,subsonic,transonic,supersonic,viscous
or inviscid, depending on the Mach number and Reynolds number.
Thebookculminateswithastudyofthelongitudinalequilibriumofagliderand
its static stability, a topic that is not usually found in an aerodynamics but in a
stabilityandcontrolsbook.Thischapterreflectstheexpertiseofoneoftheauthors
(JJC), who has been involved for several years in the SAE Aero Design West
competition, as faculty advisor for a student team, (http://students.sae.org/
competitions/aerodesign/west/) and has developed the tools and capabilities
enablingstudentstodeveloptheirowndesignsandperformwellinthecompetition.
Asallairplanemodelersknow,placingthecenterofgravity inthecorrect location
is critical to the viability of an aircraft, and a statically stable remote controlled
model is a requirement for human piloting.
The material is presented in a progressive way, starting with plane, two-
dimensional flow past cylinders of various cross sections and then by mid-quarter,
movingtothree-dimensionalflowspastfinitewingsandslenderbodies.Inasimilar
fashion, inviscid incompressible flow is followed by compressible flow and tran-
sonic flow, the latter requiring the numerical solution of the nonlinear transonic
small disturbance equation (TSD). Viscous effects are discussed and also, due to
nonlinear governing equations, numerical simulation is emphasized.
A set of problems with solutions is placed in Part III. It corresponds to final
examinations given over the last 10 years or so that the students have 2 hours to
complete.
vii
viii Preface
Finally, the reader is assumed to have the basic knowledge in fluid mechanics
that can be found in standard textbooks on this topic, in particular as concerns the
physical properties of fluids (density, pressure, temperature, equation of state,
viscosity,etc.)andtheconservationtheoremsusingcontrolvolumes.Thereaderis
also assumed to master undergraduate mathematics (calculus of single variables,
vector calculus, linear algebra, and differential equations). Three appendices are
included in the book, summarizing the material relevant to the subject of interest.
Aerodynamics has a long history and it has reached a mature status during the
lastcentury.Thereareatleast20bookswrittenonaerodynamicsinthelast20years
(seereferences).Someoftheseareexcellenttextbooksandsomeareoutdatedorout
ofprint.Alloftheexistingtextsarebased,however,onsmalldisturbancetheories.
These theories are essential to gain understanding of the physical phenomena
involvedandthecorrespondingstructureoftheflowfields.Theyalsoprovidegood
approximations for some simple cases. For practical problems, however, there is a
demand for accurate solutions using modern computer simulation. Small distur-
bance theories can still provide special solutions to test the computer codes. More
important perhaps, they can provide a guideline to construct accurate and efficient
algorithmsforpracticalflowsimulations.Theyarealsousedtodevelopthefarfield
behavior required for the numerical solution of the boundary value problems. In
general,thelinearizedboundaryconditionsandtherestrictiontoCartesiangridsare
no longer sufficient. Grid generation algorithms for complete airplanes, although
still a major task in a simulation, are nowadays used routinely in industry. Hence,
smalldisturbanceapproximationsarenolongernecessaryandindeedfullnonlinear
potentialflowcodes,developedoverthelasttwodecades,areavailableeverywhere.
While it is argued that the corrections to potential flow solutions due to vorticity
generated at the shocks can be ignored for cruising speed at design conditions, the
viscous effects are definitely important to assess. Again, boundary layer approxi-
mations can be useful as a guideline to construct effective viscous/inviscid inter-
action procedures.
InthebookweadoptthisviewincontrasttoacompleteCFDapproachbasedon
the solution of the Navier-Stokes equations everywhere in the field for more than
one reason: it is more attractive, from an educational viewpoint, to use potential
flowmodel andviscouscorrection.Itisalso more practical,since Euler and hence
Navier-Stokes codes are more expensive and subject to errors due to artificial
viscosity as a result of the discrete approximations. A simple example is the
accuratecapturingofthewakeofawingandthecalculationofinduceddrag,stilla
challenge today; for the same reasons, the simulations of propellers and helicopter
rotor flows are in continuing development, let alone, the problem of turbulence.
In the text, the formulation and the numerics are developed progressively to
allow for both small disturbances and full nonlinear potential flows with viscous/
inviscid interactions. Onlyafew existingbooks(twoorthree)addressthese issues
and we hope to cover this material in a thorough and simple manner.
Preface ix
The book contains an extensive list of references on aerodynamics including
textbooks, advanced and specialized books, classical and old books, flight
mechanics books as well as references cited in the text.
Davis, California J.J. Chattot
November 2013 M.M. Hafez
Contents
Part I Fundamental Aerodynamics
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Definitions and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Discussion of Mathematical Models. . . . . . . . . . . . . . . . . . . 6
1.3 Description of the Book Content. . . . . . . . . . . . . . . . . . . . . 9
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Inviscid, Incompressible Flow Past Circular Cylinders
and Joukowski Airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 Other Formulations . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Elementary Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Uniform Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Source and Sink . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Potential Vortex. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Superposition of Elementary Solutions. . . . . . . . . . . . . . . . . 21
2.3.1 Global Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Example of Superposition: Semi-infinite
Obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Flow Past a Circular Cylinder. . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Flow Past Arbitrary Airfoils . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Kutta-Joukowski Lift Theorem . . . . . . . . . . . . . . . 25
2.5.2 The d’Alembert Paradox. . . . . . . . . . . . . . . . . . . . 29
2.6 The Kutta-Joukowski Condition . . . . . . . . . . . . . . . . . . . . . 30
xi
xii Contents
2.7 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7.1 Center of Pressure—Aerodynamic Center . . . . . . . . 31
2.7.2 Results for the Circular Cylinder . . . . . . . . . . . . . . 31
2.8 Special Cases of Joukowski Airfoils . . . . . . . . . . . . . . . . . . 33
2.8.1 The Ellipse at Zero Incidence . . . . . . . . . . . . . . . . 33
2.8.2 The Ellipse at Incidence . . . . . . . . . . . . . . . . . . . . 35
2.8.3 The Flat Plate at Incidence . . . . . . . . . . . . . . . . . . 37
2.8.4 Circular Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8.5 Joukowski Airfoil at Incidence . . . . . . . . . . . . . . . 44
2.9 Summary of Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Inviscid, Incompressible Flow Past Thin Airfoils . . . . . . . . . . . . . 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Definition of a Thin Airfoil. . . . . . . . . . . . . . . . . . 51
3.1.2 Profile at Incidence . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Examples of Camber and Thickness
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Small Disturbance Linearization Method . . . . . . . . . . . . . . . 55
3.2.1 Linearization of the Tangency Condition. . . . . . . . . 56
3.2.2 Linearization of the Pressure Coefficient. . . . . . . . . 57
3.3 Decomposition into Symmetric and Lifting Problems. . . . . . . 58
3.4 The Symmetric Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Lifting Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.1 Solution of the Fundamental Integral Equation . . . . 65
3.5.2 Example: Flat Plate . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.3 Example: Parabolic Plate . . . . . . . . . . . . . . . . . . . 68
3.5.4 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.5 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.6 Center of Pressure . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.7 Aerodynamic Center. . . . . . . . . . . . . . . . . . . . . . . 74
3.5.8 Example of Design Problem . . . . . . . . . . . . . . . . . 75
3.6 A Family of Profiles with Minimum Pressure Gradient . . . . . 77
3.7 Numerical Solution of the Fundamental Integral Equation. . . . 82
3.8 Summary of Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Inviscid, Compressible Flow Past Thin Airfoils . . . . . . . . . . . . . . 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Linearized Compressible Flow Potential Equation . . . . . . . . . 93
4.4 Prandtl-Glauert Transformation. . . . . . . . . . . . . . . . . . . . . . 95