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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 ©SpringerScience+BusinessMediaDordrecht2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Thereareinstanceswherewehavebeenunabletotraceorcontactthecopyrightholder.Ifnotifiedthe publisherwillbepleasedtorectifyanyerrorsoromissionsattheearliestopportunity. Printedonacid-freepaper SpringerScience+BusinessMediaB.V.DordrechtispartofSpringerScience+BusinessMedia (www.springer.com) To our families for their love and support 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

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