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 comprehen sive 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 mathe matical 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 deve loped 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-dimen sional treatment. In order to place in its proper setting this entire development during the past quarter century, a historical sketch has been prepared by Pro fessor 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 modi fications due to viscosity and compressibility; Experiment and research, equipment and methods; Applied airfoil theory with analysis and dis cussion 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; Per formance, prediction and analysis; General view of airplane as com prising four interacting and related systems; Airships, aerodynamics and performance; Hydrodynamics of boats and floats; and the Aero dynamics 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 recommen dations 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. W. F. Durand Stanford University, California January, 1934. GENERAL LIST OF DIVISIONS WITH AUTHORS Volume I. A. JUathelllatical Aids W.F.DDRAND - 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 of America, Washington, D. C., and Technical Editor of the "Aero Digest". D. Historical Sketch 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' Aero tecnlCa . 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 Polyt echnical 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' vari able-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 Uni versity 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 of America, Washington, D. C., and Technical Editor of the "Aero Digest". R. Performance of Airships 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. Connec tion 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 102 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. Trans formation 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. 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 . • . . . 139 23. Expressions for the Calculation of the Velocity Components when the "Generalized Forces" are given p.139 -- 24. Expressions for wx, Wy, Wz 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. 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 Three Dimensional Case p. 159 - 32. Concluding Remarks.-Inverse Problem p.163. x CONTENTS CHAP. PAGE IV. AIRFOILS AND AIRFOIL SYSTEMS OF FINITE SPAN 165 1. Introduction p. 165. A. Single Wing . . . . . . . . . . . . . . . . . . . . 167 2. Case of Elliptic Loading p. 167 - 3. General Problem of the Single Wing p. 171 - Appendix to Section 3. -Evaluation of the Inte gral 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. B. Multiplane Systems . . . . . . . . . . . . . . . . • . . • . • . 201 16. Minimum Induced Drag of Multiplane Systems p.201 - 17. Closed Rectangular System p.203 -- Appendix to Section 17.-The Schwarz Christoffel 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 Geo metrical Angles of Incidence of both Wings Are Given p. 227 - 27. Refine ment 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)