Aerodynamic Theory

Division E General Aerodynamic Theory Perfect Fluids.- I. Basic Ideas of Wing Theory: Flow Around an Airfoil.- 1. Introductory Remarks.- 2. Principle Data Characterizing an Airfoil.- 3. Reaction of the Air upon an Airfoil.- 4. Moment of the Reaction of the Air upon an Airfoil.- 5. The Circulatory Flow around an Airfoil.- 6. The Kutta-Joukowski Theorem.- 7. Vortex System Connected with the Circulatory Motion around the Airfoil.- 8. Origin of the Circulation around the Airfoil.- 9. Equivalence of an Airfoil and a System of Vortices.- 10. Connection between Equation (9.8) and the Kutta-Joukowski Theorem.- 11. General Expression for the Induced Resistance.- 12. Reduction Formulae.- 13. Concluding Remarks. Program for the Following Chapters.- II. Theory of Airplane Wings of Infinite Span.- 1. Introduction.- A. Vortex Systems and their Application in the Theory of Thin Airfoils.- 2. Forces Acting on a Fluid in Two-Dimensional Motion.- 3. Forces on a System of Vortex Filaments.- 4. Calculation of the Forces Acting on a Vortex System by the Method of Complex Variables.- 5. Vortex Sheets.- 6. The Velocity Field of the Vortex Sheet in the Complex Form.- 7. The Plane Airfoil.- 8. Theory of Thin Wing Sections (Thin Airfoils).- 9. Munk’s Integral Formulae for the Lift and Moment of a Thin Airfoil.- 10. Simple Types of Thin Airfoils. General Discussion.- 11. Airfoil with Flap.- 12. Two-Dimensional Approximate Biplane Theory.- B. Application of the Theory of Conformai Transformation to the Investigation of the Flow around Airfoil Profiles.- 13. Conformai Transformation.- 14. General Expressions for Lift and Moment.- 15. Metacentric Parabola.- 16. The Joukowski Transformation. Classification of Airfoil Families.- 17. The Joukowski Family of Airfoils.- 18. Graphical Method for Plotting Joukowski Airfoils and Computing Velocity Distribution.- 19. The Kármán-Trefftz Family of Airfoils.- 20. The Mises Family of Airfoils.- 21. Aerodynamic Characteristics of Given Airfoils.- 22. The Theory of Biplanes.- 23. Flow through a Lattice Composed of Airfoils.- 24. Some Examples of the Application of Conformai Transformation to Problems Connected with Airfoils.- III. Mathematical Foundation of the Theory of Wings with Finite Span.- 1. Equations of Motion of the Fluid.- A. Motion of a Perfect Fluid Produced by External Forces.- 2. Motion Produced by Impulsive Forces.- 3. Generation of a Vortex Ring by an Impulsive Pressure Acting over a Circular Area.- 4. Action of Continuous Forces.- 5. Forces Directed Perpendicular to the Original Motion of the Fluid.- 6. Steady Motion under the Action of Forces Independent of the Time. Transformation of the Hydrodynamic Equations.- 7. Solution of the Equations by Successive Approximations.- 8. Solution of the System of Equations (6.2), (6.7).—Determination of q.- 9. Determination of the Components of the Velocity.- Appendix to Section 9.—Remark in Connection with Bernoulli’s Theorem.- 10. Discussion of the Result Obtained—Vorticity.- 11. Forces Parallel to the Direction of the Original Motion.- 12. Forces Directed Normal to the Original Motion—Loaded Line with Uniform Lift Distribution.- 13. Loaded Line in Arbitrary Position and with Variable Lift Distribution.- 14. Introduction of the “Induced Forces” (Second Order Forces) gx, gy, gz.- 15. Continuation. Influence of the “Second Order Forces” g in the Wake.- B. Wake Energy and Induced Drag.- 16. Energy Expended in Producing the Flow Pattern.- 17. Case of Generalized Forces all Parallel to O z.- 18. Reduction of the Integral for the Induced Resistance.—Munk’s Theorems.- 19. General Case of Forces Perpendicular to the Axis O x.- 20. Problems of Minimum Induced Resistance.- 21. Distribution of Generalized Forces Giving a Constant Value of wz ?, wy ? over a Perpendicular Section of the Wake.- 22. Example. Case of the Single Wing.- C. The Field of Induced Velocities.- 23. Expressions for the Calculation of the Velocity Components when the “Generalized Forces” are given.- 24. Expressions for wx, wy, wz in the Case of Uniform Loading.- 25. Approximate Calculation of Induced Velocities (Reduced Span).- 26. Full Expression for the Downwash at Infinity in the Case of Elliptic Loading.- 27. Calculation of the Downwash at the Points of the Load System—Wing Replaced by Loaded Line.- 28. Case of a Loaded Surface of Arbitrary Form.- 29. Remark in Connection with Equations (28.8) and (27.6).- D. The Kutta-Joukowski Theorem.- 30. The Kutta-Joukowski Theorem for Wings of Infinite Span.- 31. The Application of the Kutta-Joukowski Theorem to the Three-Dimensional Case.- 32. Concluding Remarks.—Inverse Problem.- IV. Airfoils and Airfoil Systems of Finite Span.- 1. Introduction.- A. Single Wing.- 2. Case of Elliptic Loading.- 3. General Problem of the Single Wing.- Appendix to Section 3..- Evaluation of the Integral In.- 4. General Relations Expressed with the Aid of the Fourier Coefficients An.- 5. Rectangular Wing of Constant Profile and Constant Angle of Incidence.- 6. Effective Angle of Incidence. Induced Resistance.- 7. Comparison with Other Calculations.- 8. Tapered Airfoils.- 9. Twisted Airfoils.- 10. Influence of Sweep-Back on Pitching Moment.- 11. Airfoil with Ailerons Moved out of Neutral Position. Discontinous Change of Angle of Incidence at Certain Points of the Span.- 12. Iteration Method proposed by Irmgard Lotz.- 13. Airfoils of Moderate or Small Aspect Ratio.—Summary of Blenk’s Theory for the Rectangular Airfoil.- 14. Application to the Inverse Problem. Calculation of the Distribution of the Lift for a Given Airfoil.- 15. Application of Equation III (28.8) to the Calculation of wz.—Formulae for Yawed Rectangular Airfoil.- B. Multiplane Systems.- 16. Minimum Induced Drag of Multiplane Systems.- 17. Closed Rectangular System.- Appendix to Section.- The Schwarz-Christoffel Theorem.- 18. Biplane System with Equal Span for Both Wings.- 19. Single Wing with Shields at Ends.- 20. Airfoil with Gap.- 21. Direct Method for the Calculation of Biplane Systems.- 22. Elliptic Distribution of the Generalized Load for Both Wings.- 23. Final Expression for the Induced Resistance.- 24. Induced Resistance of Triplane Systems.- 25. Detailed Investigation of the Forces Acting on the Wings of a Biplane System.—Mean Values of the Velocity Components along the Wings.- 26. Continuation. Calculation of L1 and L2 when the Geometrical Angles of Incidence of both Wings Are Given.- 27. Refinement of the Theory.—Correction for Curvature of Stream-Lines.- 28. Further Refinement of the Theory.- C. Influence of Boundaries in the Field of Motion around Airfoil Systems.- 29. General Considerations Concerning the Influence of Boundaries.- 30. Example.—Image of a System with Respect to a Single Plane Boundary.- 31. General Treatment of the Influence of a Plane Boundary.- 32. Disturbing Velocities Experienced by the Original System.- 33. Case of a Plane Boundary Perpendicular to the Axis O y.- 34. Boundaries Composed of Systems of Plane Surfaces.- 35. Case of Four Boundaries Forming a Rectangular Prism.- 36. General Considerations on the Influence of Cylindrical or Prismatic Boundaries.- 37. Extension of the Theorem of III 16.- 38. Equation for the Induced Resistance.- 39. Image of a Vortex System with Respect to a Circular Boundary in Two-Dimensional Motion.- 40. Application to the Case of an Airfoil with Uniform Loading.- 41. Symmetrical Biplane.- 42. Calculation of the Windchannel Corrections at an Arbitrary Point of the Field.- 43. Application to a Special Case.- 44. Case of a Channel with Fixed Cylindrical Boundary (Closed Working Section).- 45. Influence of an Internal Cylindrical Boundary upon the Field of Motion around a Loaded Line.- 46. The Problem of Minimum Induced Resistance for a Loaded Line Connected with an Infinite Cylinder.- V. Problems of Non-Uniform and of Curvilinear Motion.- A. Problems of Non-Uniform Motion.- 1. Introduction.- Vortex System Associated with the Variations of the Circulation around an Airfoil.- 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.- 3. Continuation. Equation for the Vorticity.- 4. Circulation around an Airfoil in the Presence of Free Vortices.- 5. Accelerated Rectilinear Motion, Starting from Rest at t = 0.- 6. Airfoil Moving with Constant Velocity Describing Harmonic Oscillations.- 7. Expressions for the Force and the Moment Acting upon the Airfoil.- Appendix to Section 7..- Calculation of the Integrals I and J.- 8. Calculation of the Force Experienced by the Airfoil.- 9. Energy Expended in Producing the Vortex System.- Appendix to Section 9..- Calculation of the Coefficient C in the Expression $$ \gamma = 2\;C/\sqrt {x + c} $$ for the Vorticity in the Neighborhood of the Leading Edge.- B. Curvilinear Motion of an Airfoil.- 10. General Remarks Concerning the Vortex System in the Case of Curvilinear Flight.- 11. The Downward Velocity at the Airfoil, Due to Slightly Curved Vortices.- 12. Determination of the Distribution of the Lift over the Span.- VI. The Development of the Vortex System Downstream of the Airfoil.- 1. Introductory Considerations.- 2. Continuation.- Appendix to Section 2..- On the Influence of Higher Approximations in the Case of a Continuous Distribution of Vorticity.- 3. The Rolling up of the Vortex Sheet behind an Airfoil.- 4. Continuation.—Further Approximations.- 5. Application to the Vortex Sheet behind an Airfoil.- Appendix to Section 5..- Impulse of a System of Vortices.- 6. On the Calculation of the Downward Velocity Experienced by a Tailplane Placed behind a Single Airfoil.- 7. Conclusion.- Appendix to Section 7..- Energy of a Vortex Pair.- VII. Theory of the Wake.- 1. Introductory Remarks.- 2. The Method of Discontinuous Potential Motion.- 3. Discontinuous Potential Motion in the Case of a Straight Airfoil.- 4. Extension of the Theory of the Discontinuous Potential Motion to Curved Boundaries. Method of Levi-Civita.- 5. The Instability of Vortex Sheets.- 6. Stability of Double Rows of Vortices.- 7. The Expression for the Drag.- 8. Oseen’s Theory of the Wake.