Aerodynamic Theory A General Review of Progress

Under a Grant of the Guggenheim Fund for the Promotion of Aeronautics

William Frederick Durand Editor-in-Chief

Volume II Division E

General Aerodynamic Theory-Perfect Fluids Th. von Karman and 1. M. Burgers

With 113 Figures and 4 Plates

Berlin· Julius Springer' 1935

All rights reserved

ISBN-13: 978-3-642-89628-6 e-ISBN-13: 978-3-642-91485-0 DOT: 10.1007/978-3-642-91485-0

GENERAL PREFACE During the active lile of the Guggenheim Fmld for the Promotion

of Aeronautics, provision was made for the preparation of a series of monographs on the general subject of Aerodynamic Theory. It was recognized that in its highly specialized form, as developed during the past twenty-five years, there was nowhere to be found a fairly comprehen­sive exposition of this theory, both general and in its more important applications to the problems of aeronautic design. The preparation and publication of a series of monographs on the various phases of this subject seemed, therefore, a timely undertaking, representing, as it is intended to do, a general review of progress during the past quarter century, and thus covering substantially the period since flight in heavier than air machines became an assured fact.

Such a present taking of stock should also be of value and of interest as furnishing a point of departure from which progress during coming decades may be measured.

But the chief purpose held in view in this project has been to provide for the student and for the aeronautic designer a reasonably adequate presentation of background theory. No attempt has been made to cover the domains of design itself or of construction. Important as these are, they lie quite aside from the purpose of the present work.

In order the better to suit the work to this main purpose, the first volume is largely taken up with material dealing with special mathe­matical topics and with fluid mechanics. The purpose of this material is to furnish, close at hand, brief treatments of special mathematical topics which, as a rule, are not usually included in the curricula of engineering and technical courses and thus to furnish to the reader, at least some elementary notions of various mathematical methods and resources, of which much use is made in the development of aerodynamic theory. The same material should also be acceptable to many who from long disuse may have lost facility in such methods and who may thus, close at hand, find the means of refreshing the memory regarding these various matters.

The treatment of the subject of Fluid Mechanics has been deve­loped in relatively extended form since the texts usually available to the technical student are lacking in the developments more especially of interest to the student of aerodynamic theory. The more elementary treatment by the General Editor is intended to be read easily by the average technical graduate with some help from the topics comprised in Division A. The more advanced treatment by Dr. Munk will call

IV GENERAL PREFACE

for some familiarity with space vector analysis and with more advanced mathematical methods, but will commend itself to more advanced students by the elegance of such methods and by the generality and importance of the results reached through this generalized three-dimen­sional treatment.

In order to place in its proper setting this entire development during the past quarter century, a historical sketch has been prepared by Pro­fessor Giacomelli whose careful and extended researches have resulted in a historical document which will especially interest and commend itself to the study of all those who are interested in the story of the gradual evolution of the ideas which have finally culminated in the developments which furnish the main material for the present work.

The remaining volumes of the work are intended to include the general subjects of: The aerodynamics of perfect fluids; The modi­fications due to viscosity and compressibility; Experiment and research, equipment and methods; Applied airfoil theory with analysis and dis­cussion of the most important experimental results; The non-lifting system of the airplane; The air propeller; Influence of the propeller on the remainder of the structure; The dynamics of the airplane; Per­formance, prediction and analysis; General view of airplane as com­prising four interacting and related systems; Airships, aerodynamics and performance; Hydrodynamics of boats and floats; and the Aero­dynamics of cooling.

Individual reference will be made to these various divisions of the work, each in its place, and they need not, therefore, be referred to in detail at this point.

Certain general features of the work editorially may be noted as follows:

1. Symbols. No attempt has been made to maintain, in the treatment of the various Divisions and topics, an absolutely uniform system of notation. This was found to be quite impracticable.

Notation, to a large extent, is peculiar to the special subject under treatment and must be adjusted thereto. Furthermore, beyond a few symbols, there is no generally accepted system of notation even in any one country. For the few important items covered by the recommen­dations of the National Advisory Committee for Aeronautics, symbols have been employed accordingly. Otherwise, each author has developed his system of symbols in accordance with his peculiar needs.

At the head of each Division, however, will be found a table giving the most frequently employed symbols with their meaning. Symbols in general are explained or defined when first introduced·.

2. General Plan of Construction. The work as a whole is made up of Divisions, each one dealing with a special topic or phase of the general

GENERAL PREFACE v

subject. These are designated by letters of the alphabet in accordance with the table on a following page.

The Divisions are then divided into chapters and the chapters into sections and occasionally subsections. The Chapters are designated by Roman numerals and the Sections by numbers in bold face.

The Chapter is made the unit for the numbering of sections and the section for the numbering of equations. The latter are given a double number in parenthesis, thus (13.6) of which the number at the left of the point designates the section and that on the right the serial number of the equation in that section.

Each page carries at the top, the chapter and section numbers.

Stanford University, California January, 1934.

W. F. Durand

GENERAL LIST OF DIVISIONS WITH AUTHORS

A. JUathelllatical Aids W.F.DDRAND

Volume I.

- Professor (Emeritus) of Mechanical Engineering, Stanford University, Calif., Member of the National Advisory Committee for Aeronautics.

B. Fluill ]}Iechanics, Part I W. F. DURAND

C. Fluid JUechanics, Part n MAX 1\'I. MUNK - Lecturer in Aerodynamics at the Catholic University

D. Historical Sketch

of America, Washington, D. C., and Technical Editor of the "Aero Digest".

R. GIACOMELLI - Lecturer in History of Mechanics at the University of Rome, Italy, and Editor of "L' Aerotecnica".

with the collaboration of E. PISTOLESI - Professor of Mechanics at the Royal School of Engi­

neer~g"at Pisa, Italy, and Editor-in-Chief of "L' Aero­tecnlCa .

Volume II. E. General Aerotlynalllic Theory-Perfect Fluids

TH. VON KARMAN - Director of the Guggenheim Aeronautics Laboratory, California Institute of Technology, Pasadena, Calif., and formerly Director of the Aerodynamic Institute. Aachen, Germany.

J". 1\'1. BURGERS - Professor of Aero- and Hydrodynamics at the Tech-nische Hoogeschool at Delft, Holland.

Volume III. I". The Theory of Single Burbling

C. WITOSZYNSKI - Professor of Aerodynamics at the Warsaw Poly technical School and Director of the Warsaw Aerodynamic Institute, Poland.

M. ,T. THOMPSON - Assistant Professor of Aeronautical Engincering at the University of Michigan, Ann Arbor, Mich.

G. 'l'he lliechanics of Viscous Fluids L. PRANDTL - Professor in Applied Mechanics at the University of

G6ttingen, Germany, and Director of the Kaiser Wilhelm Institute for Fluid Research.

H. The llIechanics of Comllressible Fluids G. I. TAYLOR - Yarrow Rescarch Professor of the Royal Society,

Fellow of Trinity College, Cambridge, England. J. 'V. MACCOLL - Research Officer, Department of External Ballistics,

Ordnance Committee, Woolwich, England. I. Expel'illlental lliethods-Wind Tunnels

A. TOUSSAINT --- Director of the Acrodynamic Laboratory, Saint-Cyr-l' Ecole, France.

E. JACOBS --- Associate Aeronautical Engineer, in charge of the National Advisory Committee for Aeronautics' vari­able-density wind tunnel, Langley Field, Virginia.

GENERAL LIST OF DIVISIONS WITH AUTHORS VII

Volume IV. J. Applied Airfoil Theory

A. BETZ - Professor at the University and Director of the Aero-dynamic Research Institute at G6ttingen, Germany.

K. Airplane Body (Non-Lifting System) Drag and Influence on Lifting System C. WIESELSBERGER - Professor of Aerodynamics and Director of the Aero­

dynamic Institute, Technische Hochsehule, Aachen, Germany.

L. Airplane Propellers H. GLAUERT t - Past Fellow of Trinity College, Cambridge, England;

Principal Scientific Officer at the Royal Aircraft Establishment, Farnborough.

JU. Influence of the Propeller on other Parts of the Airplane Structure C. KONING - Rijks-Studiedienst voor de Luchtvaart, Amsterdam,

Holland.

Volume V. N. Dynamics of the Airplane

B. MELVILL JONES - Professor of Aeronautical Engineering in the Uni­versity of Cambridge, England, Member of the Aeronautical Research Committee of Great Britain.

O. Airplane Performance L. V. KERBER - Former Chief Aerodynamics Branch Materiel Division,

U. S. Army Air Corps, and former Chief, Engineering Section Aeronautics Branch, Department of Commerce.

Volume VI. P. Airplane as a Whole-General View of Mutual Intel'actions Among Constituent

Systems M. P ANETTI - Professor of Applied Mechanics and Director of the

School of Aeronautics in the R. Politecnico di Torino, Italy.

Q. Aerodynamic Theory of Airships MAx M. MUNK - Lecturer in Aerodynamics at the Catholic University

R. Performance of Airships

of America, Washington, D. C., and Technical Editor of the "Aero Digest".

K. ARNSTEIN - Chief Engineer of the Goodyear Zeppelin Company, Akron, Ohio.

W. KLEMPERER - Research Engineer of the Goodyear Zeppelin Com-pany, Akron, Ohio.

S. Hydrodynamics of Boats and Floats E. G. BARRILLON - Director of the Naval Experimental Tank, Paris, France.

T. Aerodynamics of Cooling H. L. DRYDEN - Physicist in the United States Bureau of Standards,

Chief of the Aerodynamics Section, Washington, D. C.

CONTENTS

DIVISIONE

GENERAL AERODYNAMIC THEORY-PERFECT FLUIDS

By Th. von Karman, Director of the Guggenheim Aeronautics Laboratory, California Institute of

Technology, Pasadena, Calif., and formerly Director of the Aerodynamic Institute, Aachen, Germany

and J. M. Burgers, Professor of Aero- and Hydrodynamics at the Technische Hoogeschool at Delft,

Holland CHAP. PAGE

1. BASIC IDEAS OF WING THEORY: FLOW AROUND AN AIRFOIL 1 1. Introductory Remarks p. 1 - 2. Principle Data Characterizing an

Airfoil p. 2 - 3. Reaction of the Air upon an Airfoil p. 3 - 4. Moment of the Reaction of the Air upon an Airfoil p.5 - 5. The Circulatory Flow around an Airfoil p. 6 -~~ 6. The Kutta-Joukowski Theorem p. 8 -7. Vortex System Connected with the Circulatory Motion around the Airfoil p. 9 -- 8. Origin of the Circulation around the Airfoil p. 11 -9. Equivalence of an Airfoil and a System of Vortices p.14 - lO. Connec­tion between Equation (9.8) and the Kutta-Jonkowski Theorem p. 17 -11. General Expression for the Induced Resistance p.19 - 12. Reduction Formulae p.21 - 13. Concluding Remarks. Program for the Following Chapters p. 23.

II. THEORY OF AIRPLANE WINGS OF INFINITE SPAN ..... 24 1. Introduction p. 24.

A. Vortex Systems and their Application in the Theory of Thin Airfoils 25 2. Forces Acting on a Fluid in Two-Dimensional Motion p. 25 - 3. Forces on a System of Vortex Filaments p. 26 -~ 4. Calculation of the Forces Acting on a Vortex System by the Method of Complex Variables p. 30 -5. Vortex Sheets p. 33 -- 6. The Velocity Field of the Vortex Sheet in the Complex Form p. 35 -- 7. The Plane Airfoil p. 37 - 8. Theory of Thin Wing Sections (Thin Airfoils) p. 39 - 9. Munk's Integral Formulae for the Lift and Moment of a Thin Airfoil p. 43 - 10. Simple Types of Thin Airfoils. General Discussion p. 48 - 11. Airfoil with Flap p. 53 -12. Two-Dimensional Approximate Biplane Theory p.56.

B. Application of the Theory of Conformal Transformation to the Investiga-tion of the Flow around Airfoil Profiles . . . . . . . . . . . . . 58

13. Conformal Transformation p. 58 - 14. General Expressions for Lift and Moment p. 60 - 15. Metacentric Parabola p. 64 - 16. The Joukowski Transformation. Classification of Airfoil Families p. 65 - 17. The Jonkowski Family of Airfoils p. 68 - 18. Graphical Method for Plotting Jonkowski Airfoils and Computing Velocity Distribution p. 71 - 19. The Karman-Trefftz Family of Airfoils p.74 - 20. The Mises Family of Airfoils p. 77 - 21. Aerodynamic Characteristics of Given Airfoils p. 80 -

CONTENTS IX CHAP. PAGE

22. The Theory of Biplanes p. 83 - 23. Flow through a Lattice Composed of Airfoils p. 91 - 24. Some Examples of the Application of Conformal Transformation to Problems Connected with Airfoils p.96.

III. MATHEMATICAL FOUNDATION OF THE THEORY OF WINGS WITH FINITE SPAN. . . . . . . . . . . . . . . . . . . . . . 100

1. Equations of Motion of the Fluid p. 100.

A. Motion of a Perfect Fluid Produced by External Forces

2. Motion Produced by Impulsive Forces p. 102 - 3. Generation of a Vortex Ring by an Impulsive Pressure Acting over a Circular Area p. 103 - 4. Action of Continuous Forces p. 104 - 5. Forces Directed Perpendicular to the Original Motion of the Fluid p.l06 - 6. Steady Motion under the Action of Forces Independent of the Time. Trans­formation of the Hydrodynamic Equations p.l08 - 7. Solution of the Equations by Successive Approximations p. 110 - 8. Solution of the System of Equations (6.2), (6.7).--Determination of q p. 112 --9. Determination of the Components of the Velocity p.113 - Appendix to Section 9.-·Remark in Connection with Bernoulli's Theorem p.l14 -10. Discussion of the Result Obtained-Vorticity p. 115 - 11. Forces Parallel to the Direction of the Original Motion p. 116 - 12. Forces Directed Normal to the Original Motion-Loaded Line with Uniform Lift Distribution p. 117 - 13. Loaded Line in Arbitrary Position and with Variable Lift Distribution p.120 - 14. Introduction of the "Induced Forces" (Second Order Forces) (/x, (/y, (/z p.122 -.- 15. Continuation. Influence of the "Second Order Forces" (/ in the Wake p.124.

102

B. Wake Energy and Induced Drag •................ 125

16. Energy Expended in Producing the Flow Pattern p.125 - 17. Case of Generalized Forces all Parallel to 0 z p.128 - 18. Reduction of the Integral for the Induced Resistance.-Munk's Theorems p. 131 -19. General Case of Forces Perpendicular to the Axis 0 x p.134 -20. Problems of Minimum Induced Resistance p.135 - 21. Distribution of Generalized Forces Giving a Constant Value of Wz 00, Wy co over a Perpendicular Section of the Wake p.137 -. 22. Example. Case of the Single Wing p. 138.

C. The Field of Induced Velocities . • . . .

23. Expressions for the Calculation of the Velocity Components when the "Generalized Forces" are given p.139 -- 24. Expressions for wx, Wy, W z in the Case of Uniform Loading p.141 - 25. Approximate Calculation of Induced Velocities (Reduced Span) p.143 - 26. Full Expression for the DOWllwash at Infinity in the Case of Elliptic Loading p.146 - 27. Calculation of the DOWllwash at the Points of the Load System-Wing Replaced by Loaded Line p. 149 - 28. Case of a Loaded Surface of Arbitrary Form p.153 - 29. Remark in Connection with Equations (28.8) and (27.6) p.155.

139

D. The Kutta-Joukowski Theorem .........•........ 157

30. The Kutta-Joukowski Theorem for Wings of Infinite Span p.157 -31. The Application of the Kutta-Joukowski Theorem to the Three­Dimensional Case p. 159 - 32. Concluding Remarks.-Inverse Problem p.163.

x CONTENTS CHAP. PAGE

IV. AIRFOILS AND AIRFOIL SYSTEMS OF FINITE SPAN 1. Introduction p. 165.

165

A. Single Wing . . . . . . . . . . . . . . . . . . . .

2. Case of Elliptic Loading p. 167 - 3. General Problem of the Single Wing p. 171 - Appendix to Section 3. -Evaluation of the Inte­gral In p. 173 - 4. General Relations Expressed with the Aid of the Fourier Coefficients An p.174 - 5. Rectangular Wing of Constant Profile and Constant Angle of Incidence p.177 6. Effective Angle of Incidence. Induced Resistance p.179 - 7. Comparison with Other Calculations p.182 - 8. Tapered Airfoils p.182 - 9. Twisted Airfoils p. 184 - 10. Influence of Sweep-Back on Pitching Moment p. 185 -n. Airfoil with Ailerons Moved out of Neutral Position. Discontinous Change of Angle of Incidence at Certain Points of the Span p.186 -12. Iteration Method proposed by Irmgard Lotz p. 188 - 13. Airfoils of Moderate or Small Aspect Ratio.-Summary of Blenk's Theory for the Rectangular Airfoil p.192 - 14. Application to the Inverse Problem. Calculation of the Distribution of the Lift for a Given Airfoil p. 194 --15. Application of Equation III (28.8) to the Calculation of 'Wz.-~Formulae for Yawed Rectangular Airfoil p.195.

167

B. Multiplane Systems . . . . . . . . . . . . . . . . • . . • . • . 201

16. Minimum Induced Drag of Multiplane Systems p.201 - 17. Closed Rectangular System p.203 -- Appendix to Section 17.-The Schwarz­Christoffel Theorem p. 206 - 18. Biplane System with Equal Span for Both Wings p. 208 ~- 19. Single Wing with Shields at Ends p. 211 -20. Airfoil with Gap p. 212 - 21. Direct Method for the Calculation of Biplane Systems p. 214- 22. Elliptic Distribution of the Generalized Load for Both Wings 1). 216 - 23. Final Expression for the Induced Resistance p. 219 - 24. Induced Resistance of Triplane Systems p. 221 -25. Detailed Investigation of the Forces Acting on the Wings of a Biplane System.-Mean Values of the Velocity Components along the Wings p. 222 -- 26. Continuation. Calculation of L1 and L2 when the Geo­metrical Angles of Incidence of both Wings Are Given p. 227 - 27. Refine­ment of the Theory.-Correction for Curvature of Stream-Lines p. 231 ~-28. Further Refinement of the Theory p. 234.

C. Influence of Boundaries in the Field of Motion around Airfoil Systems 236

29. General Considerations Concerning the Influence of Boundaries p. 236 -30. Example.-Image of a System with Respect to a Single Plane Boundary p. 239 - 31. General Treatment of the Influence of a Plane Boundary p. 242 - 32. Disturbing Velocities Experienced by the Original System p.245 -- 33. Case of a Plane Boundary Perpendicular to the Axis 0 y p. 247 - 34. Boundaries Composed of Systems of Plane Surfaces p.249 - 35. Case of Four Boundaries Forming a Rectangular Prism p. 252 - 36. General Considerations on the Influence of Cylindrical or Prismatic Boundaries p. 256 - 37. Extension of the Theorem of III 16 p. 258 - 38. Equation for the Induced Resistance p. 259 -39. Image of a Vortex System with Respect to a Circular Boundary in Two-Dimensional Motion p.261 - 40. Application to the Case of an Airfoil with Uniform Loading p. 263 - 41. Symmetrical Biplane p. 265 -42. Calculation of the Windchannel Corrections at an Arbitrary Point of the Field p. 266 - 43. Application to a Special Case p. 269 - 44. Case of a Channel with Fixed Cylindrical Boundary (Closed Working Section)

CONTENTS XI CHAP. PAGE

p.271 - 45. Influence of an Internal Cylindrical Boundary upon the Field of Motion around a Loaded Line p.273 - 46. The Problem of :Minimum Induced Resistance for a Loaded Line Connected with an Infinite Cylinder p. 276.

V. PROBLEMS OF NON-UNIFORM AND OF CURVILINEAR MOTION 280

A. Problems of Non-Uniform Motion ................ 280 1. Introduction.-Vortex System Associated with the Variations of

the Circulation around an Airfoil p. 280 - 2. Equations for the Motion of a Fluid under the Influence of External Forces, if both the Latter and the General Velocity V are Functions of the Time p. 283 -~ 3. Con­tinuation. Equation for the Vorticity p. 285 - 4. Circulation around an Airfoil in the Presence of Free Vortices p. 289 -- 5. Accelerated Rectilinear Motion, Starting from Rest at t = 0 p. 292 -- 6. Airfoil Moving with Constant Velocity Describing Harmonic Oscillations p.293 -7. Expressions for the Force and the Moment Acting upon the Airfoil p. 295 -- Appendix to Section 7.-Calculation of the Integrals I and J p. 298 -- 8. Calculation of the Force Experienced by the Airfoil p. 301 -9. Energy Expended in Producing the Vortex System p. 304 - Appendix to Section 9.-Calculation of the Coefficient C in the Expression y = 2 c/vx0 for the Vorticity in the Neighborhood of the Leading Edge p.309.

B. CUl'vilinear Motion of an Airfoil 10. General Remarks Concerning the Vortex System in the Case of Curvilinear Flight p. 310 - II. The Downward Velocity at the Airfoil, Due to Slightly Curved Vortices p.311 - 12. Determination of the Distribution of the Lift over the Span p. 313.

VI. THE DEVELOPMENT OF THE VORTEX SYSTEM DOWNSTREAM

310

OF THE AIRFOIL . . . . . . . . . . . . . . . . . . . . . . . 315 1. Introductory Considerations p. 315 - 2. Continuation p. 318 -

Appendix to Section 2.-0n the Influence of Higher Approximations in the Case of a Continuous Distribution of Vorticity p.319 - 3. The Rolling up of the Vortex Sheet behind an Airfoil p. 320 -- 4. Continua­tion.-FUl'ther Approximations p.323 - 5. Application to the Vortex Sheet behind an Airfoil p. 324 - Appendix to Section 5.-Impulse of a System of Vortices p. 325 - 6. On the Calculation of the Downward Velocity Experienced by a Tailplane Placed behind a Single Airfoil p.326 - 7. Conclusion p.328 -- Appendix to Section 7.-Energy of a Vortex Pair p.329.

VII. THEORY OF THE WAKE . 1. Introductory Remarks p. 330 -~ 2. The Method of Discontinuous

Potential Motion p. 331 - 3. Discontinuous Potential Motion in the Case of a Straight Airfoil p. 332 - 4. Extension of the Theory of the Discontinuous Potential Motion to Curved Boundaries. Method of Levi­Civita p. 336 - 5. The Instability of Vortex Sheets p. 340 - 6. Stability of Double Rows of Vortices p. 342 - 7. The Expression for the Drag p.346 - 8. Oseen's Theory of the Wake p. 349.

330

BIBLIOGRAPHY 353

INDEX. . . . . 363

NOTATION The following table comprises a list of the principal notations employed in

the present Division, with their more usual meanings. Notations not listed are either so well understood as to render mention unnecessary, or are only rarely employed and are explained as introduced. Where occasionally a symbol is em­ployed with more than one meaning, the local context will make the significance clear.

x, Y, Z X, Y x, y, z ~, 1), ,

z

x, w, fJ b c

Space axes Two-dimensional axes, sometimes forces along X and Y Space coordinates Supplementary space coordinates Used for the complex (x + iy) Used for the complex (~+ i1)) Cylindrical coordinates, IV 42 Half span Chord

S Area of airfoil, or area in general ()( Geometrical angle of incidence ~ Effective angle of incidence fJ Angle of inclination, usually to axis of X f3 Angle between X and 1st axis, II 13, 14 T, Y Special angles in conformal transformation, II 13, 14 x Exponent used in conformal transformation, II 19 vx, Vy, Vz Component velocities along axes of x, y, z, III 1 wx, Wy, W z Component "added" velocities along axes of x, y, z, III 6 V Velocity in general; often initial or undisturbed velocity W Added velocity, usually downward w Used for complex velocity u - iv r Circulation; strength of vortex y Vorticity y Strength of vortex sheet Yx' yy' yz Components of vorticity, III 1 cp, (jJ Usually potential 'P, rp Usually stream function X' t, F Usually complex function cp + i 'P Ix, Iy , Iz Component real forces, III 6 gx, gy, gz Component induced forces, III 6 kx, ky, kz Component generalized forces, III 6 K Resultant of generalized forces

Qx, Qy, Qz I I I f 1· f . h III 9 Q- Q- -Q f ntegra s 0 genera Ized orces WIt respect to x or ~,

x, y, z D Drag Di Induced drag L Lift l Lift per unit span A Generalized force per unit span M Moment

p OJ R R E, T W GL GD GDi GDO GMO },

a t5, T

E,K,F Res. Re. 1m.

(! t

Pressure Impulsive pressure, Resultant air force

NOTATION

III 3

Complex force resultant (X - i Y), Kinetic energy Work Lift coefficient Drag coefficient Coefficient of induced drag Profile drag coefficient Moment coefficient for zero lift Aspect ratio Special coefficient, IV (22.6), Special coefficients, IV (4.11), Elliptic integrals Residue (Cauchy) Real part Imaginary part

II (4.8)

(22.10), (6.6),

V - 1 or sometimes the i th term of a series Density Time

(25.10) (6.10)

XIII

EDITOR'S PREFACE (DIVISION E) The present Division of this work is intended to present the general

mathematical foundation of the mechanics of perfect fluids-incom­pressible and non-viscous-with such specialized developments as are of particular importance for application to the problems of aerodynamics.

Chapter I is in considerable degree descriptive of the phenomena involved in the flow of a fluid about a solid body and in particular in the flow of air about an airplane wing. This descriptive material is supplemented by certain general statements regarding the laws connect­ing these phenomena together, the whole intended to serve as a form of introduction to the more formal and mathematical treatments of the following chapters.

This more formal treatment begins with Chapter II, in which, however, the paths of relative fluid motion are assumed to be limited to planes, thus constituting so-called two-dimensional flow, and permitting of the application of certain specialized agencies-conformal transformation in particular-for their investigation.

Chapters III and IV then follow, the first dealing in more fundamental mathematical fashion with the laws of three-dimensional fluid flow and the second with several special and important applications to the problems of aerodynamics as furnishing the ground work for all rational aeronautic design.

Chapter V follows with a development of the more important aspects of the mechanics of non-uniform and of curvilinear motion. This is followed in Chapters VI and VII by a discussion of certain phenomena occurring in the wake formed downstream from a body around which a fluid is flowing. The two chapters deal with somewhat different aspects of this general problem and together serve to indicate the more important lines of development relating to this phase of the subject.

Chapters V, VI and VII are intended to present the important aspects of the more recent advances in aerodynamic theory and to give also indications of the lines along which further work on these subjects should be directed.

The notation required to present adequately the somewhat extended mathematical developments of this Division is, of necessity, of con­siderable complexity and multiple duty has been required of several symbols. As in other Divisions of the work, the more important symbols are listed together with their meanings and where needful, a reference to where introduced or defined. In cases of multiple use of the same symbol, the local context will usually serve to indicate which particular

EDITOR'S PREFACE xv

meaning is intended. The Editor trusts that the occasional references to the more elementary material presented in Division A and B together with the notes and appendices will serve to make somewhat easier the reading of this Division by those who may be less familiar with the more formal and mathematical methods here of necessity employed.

In the preparation of this volume, the general plan of the work having been agreed upon through discussion between both authors, Dr. von Karman has primarily contributed Chapters II and VII, and Dr. Burgers the main part of Chapters I, III, IV, V and VI. Each author, however, has had the advantage of the reading of the other's manuscript and of suggestions and criticism based thereon, so that the volume represents their joint work.

Stanford University, California July, 1934

w.]'. Duraud