Aerodynamic Theory

Division N Dynamics of the Airplane.- Preface.- I. Steady Motions and Limiting Accelerations.- 1. Introduction f. 5 — 2. Symmetric Flight p 5..- A. Steady Straight Symmetric Flight.- 3. Principal Forces Involved p. 6 — 4. Dimensionless Coefficients p 7 — 5. Horizontal Flight p.7 — 6. Indicated Air Speed p.9 — 7. Gliding Flight p. 10 — 8. Some Aspects of the Landing Problem p.11—9. Climbing Flight p. 12,.- B. Accelerated Symmetric Flight.- 10. Introduction p.13 — 11. Radius of Curvature of the Flight Path p.11 — 12. Minimum Radius of Curvature p.14—13. Minimum Radius of Curvature in Terms of Stalling Speed p. 15 — 14. Minimum Radius and Rapid Manoeuvres p.15 — 15. Apparent Gravity and Load Factors p 16 — 16. Measurement of Load Factors p.16 — 17. Maximum Load Factor p. 17..- C. Straight Asymmetric Flight.- 18. Steady Straight Side-slip p.18 — 19. Relation of Side-force to Side-slip p. 18 — 20. Relation of Side-slip to “Bank” p. 19..- D. Circling Fight.- 21. The Flat Turn p. 19 — 22. Side-slip in the Flat Turn p. 20 — 23. The True-banked Turn p. 20 — 24. Stalling Speed in a True- banked Turn p. 21 — 25. Power Required in a True-banked Turn p. 21 — 26. Horizontal Turn of Minimum Radius p. 21 — 27. Aero-dynamic Moments p. 22..- II. Symmetric or Pitching Moments.- 1. Pitching Moments and Static Stability p. 22 — 2. The Geo-metric Mean Chord p. 23 — 3. The Air-Reactions on an Isolated Wing p. 23 — 4. Functions of the Tail p.24 — 5. Fixed Tail and Elevators p. 25 — 6. Effect of C.G. Position on Static Stability p. 26 — 7. Dimensionless Coefficients p. 27 — 8. C.G. Positions for Neutral Equilibrium p. 27 — 9. A Relation Between Control and Stability p. 28 — 10. Experiments on Complete Models p. 29..- A. Contributions of Separate Parts of the Aeroplane to Pitching Moments.- 11. Introduction p. 29 — 12. Pitching Moments from Tail and Wings—C.G. on Wing Chord p. 30 — 13. Joukowski’s Theoretical Values for [dkM/dkL] p. 31 — 14. Pitching Moments When the C.G. is not on Wing Chord p. 32 — 15. Direct Contribution of the Airscrew to Pitching Moments p. 34 — 16. Numerical Values—Airscrew Contribution p. 35 — 17. Contribution of the Body and Minor Parts to Pitching Moments p. 36..- B. Forces on Tail and Interference Factors.- 18. Statement of Problem p. 36 — 19. Interference Between Aeroplane and Tail p. 37 — 20. Hinge Moments and Free Elevators p. 38 — 21. Wing-tail Interference p. 39 — 22. Airscrew-tail Interferences p. 40 — 23. Body-tail Interference p. 40 — 24. Airscrew- wing Interference p. 40 — 25. Interference Problems Discussed p. 41 — 26. Numerical Illustrations p. 41 — 27. Pitching Moments at Very High Speeds p 42 — 28. Pitching Moments in Stalled Flight p. 43..- C. Pitching Moments in Circling Flight (Mq).- 29. Introduction p. 44 — 30. The Tail in Circling Flight p 44 — 31. The Derivative Mqp. 45 — 32. The Dimensionless Coefficient (kmq) p. 45 — 33. Contribution of the Wings to M g p. 46 — 34. Direct Contribution of the Screw to Mqp. 46 — 35. Influence of Airscrew Slip-Stream on Tail p. 47 — 36. Numerical Illustration p. 47..- D. Experimental Methods of Measuring Mq.- 37. Whirling Arm Experiment p. 48 — 3S. The Free Oscillation Method p. 48 — 39. The Forced Oscillation Method p. 49 — 40. Relations between ?, Mq, and M ? p. 49—41. Experimental Separation of Mq and M à p. 51 — 42. Influence of the Screw on M ? p 52..- E. Experimental Results for Mq.- 43. Normal Flight p. 52 — 44. Mq in Stalled Flight p 53..- F. The Geometric Mean Chord.- 45. Statement of Problem p. 54 — 46. The Geometric Mean Chord Defined p. 54 — 47. Example of Application to a Biplane p. 56 — 48. Wings of Non-Rectangular Plan Form and with Dihedral Angle p. 56 — 49. Twisted and Tapered Wings p. 57..- III. The Asymmetric or Lateral Moments.- 1. Introduction p. 57—2. Axes p.57—3. Symbols p. 57—4. Dimensionless Coefficients p. 58—5. Controls p. 59—6. Independent Variables Which Govern Asymmetric Moments p. 59..- A. Effects of Side-Slip.- 7. Dihedral Angle p. 59—8. Theoretical Estimates of Rolling Moments Due to Side-Slip p. 60—9. Theoretical Estimate of Yawing Moment Due to Side-Slip p. 61—10. Conversion to Chord Axes p. 61—11. Omissions in the Assumptions p. 61— 12. Experimental Measurements of the Effects of Side-Slip on an Isolated Wing p. 62—13. klv in Stalled Flight p. 62—14. Effect of Side-Slip on Body, Fin and Rudder p. 63—15. Experimental Values of klv and knv for Complete Model Aeroplanes p. 65—16. Effects of Large Side-Slips p. 67..- B. Theoretial Estimates of Moments due to Angular Velocities of Roll and Yaw.- 17. Moments Due to Rate of Yaw p 67—18. The Derivatives klr and knrp69—19. Moments Due to Rate of Roll (p) p. 70—20. Approximate Formulae for the Rotary Derivatives p 71 —21. Change of Axes p. 71 —22. Calculations of Rotary Derivatives Using Chord Axes Throughout p 72 —23. Rotary Derivatives Due to Body, Fin and Rudder p 73..- C. Experimental Methods of Measuring the Rotary Derivatives.- 24. Oscillation Methods p. 73 — 25. Continuous Rotation Methods p 73..- D. Discussion of the Results of Experiments on the Rotary Derivatives at Incidences of Normal Flight.- 26. Comparison Between Oscillation and Continuous Rotation Experiments p. 74 — 27. Comparison Between Experiment and “Strip” Calculation p. 74 — 28. Influence of Body, Fin and Rudder p. 75 — 29. Comparison Between Model and Full Scale Experiments p. 75 — 30. Effect of Large Angular Velocities p. 75 — 31. Summary- Rotary Derivatives in Normal Flight p 75..- E. Rotary Derivatives in Stalled Flight.- 32. Introduction p. 76 — 33. Theoretical Estimate of Moments Due to Finite Rates of Roll p 76 — 34. Comparison Between Strip Hypothesis and of Experiments with Continuous Finite Rates of Roll p. 78 — 35. Comparison Between Strip Hypothesis and Experiments with Continuous Yawing p. 79 — 36. Comparison Between Continuous Rotation and Oscillation Experiments p. 80 — 37. Auto- Rotation and Wing Tip Slots p. 81 — 38. Collection of Typical Values for the Six Asymmetric Moment Derivatives p. 81..- F. The Asymmetric Controls.- 39. Balanced Controls p. 82 — 40. Aileron Rolling Moments p. 83 — 41. Aileron Yawing Moments p 83 — 42. Ailerons and Wing Tip Slots p. 84 — 43. The Rudder in Normal Fight p. 84 — 44. The Rudder in Stalled Flight p. 85..- IV. Free Flight-Simple Discussion.- 1. Introduction p. 85 — 2. Lanchester’s Idealized Flight Path p. 86 — 3. Equation of the Fight Path p. 87 — 4. Forms of the Flight Path p. 88 — 5. Comparison with the Path of a Real Aeroplane p. 89..- A. The Slow Oscillation of Small Amplitude.- 6. The Ideally Simple Oscillation p. 90 — 7. Form of the Oscillation of Small Amplitude p. 90— 8. Comparison with the Oscillation of a Real Aeroplane p. 91 — 9. The Effect of Mq on the Period p. 91 — 10. The Damping of the Oscillation p. 94 — 11. Influence of the Airscrew p. 95 — 12. Other Factors which Influence the Oscillation p. 95 — 13. Gliding and Climbing Flight p 95 — 14. The Unstable Aeroplane p. 95..- B. The Rapid Incidence Adjustement.- 15. Simplifying Assumptions p. 96 — 16. Sudden Change of Incidence p. 96 — 17. Sudden Application of Elerators p. 98 — 18. Solutions with More Complete Assumptions p. 99 — 19. Discussion of the Solutions p. 101— 20. Unstable Aeroplanes p. 102 — 21. Stable Aeroplanes p. 103 — 22. Dimensionless Forms of the Quadratic for ?p. 103..- C. Asymmetric Motions.- 23. Statement of the Problem p. 104— 24. The True-banked Steady Turn p. 105 — 25. The Steady Turn with Side-Slip p 107 —26. Ascending and Descending Steady Turns p 108..- D. Small Disturbances from Straight Flight.- 27. Introduction p. 109 — 28. The Rapid Damping of Rolling Motions p. 110 — 29. The Slow Spiral Motion p. 111 — 30. More Accurate Estimate of the Spiral Motion p. 113 — 31. The Oscillation of a Statically Stable Aeroplane p. 113 — 32. The Oscillation of a Statically Neutral Aeroplane p. 116 — 33. The Oscillation of a Statically Unstable Aeroplane p. 117 — 34. The Two Possible Forms of Instability p 117..- E. The Stalled Aeroplane.- 35. The Stalled Aeroplane-Symmetric Motion p. 118 —36. The Stalled Aeroplane-Asymmetric Motion p. 119..- V. The Equations of Motion with Solutions for small Disturbances from Steady Symmetric Flight.- A. Axes, Symbols and Equations of Motion.- 1. Axes p. 121 — 2. Orientation p. 122 — 3. Symbols p. 122 — 4. Effect of Gravity p. 123 — 5. The Equations of Motion p. 123 — 6. Dependence of the Air-Reactions on the Velocity Components p. 124 — 7. Influence of W on M p. 125 — 8. Step-by-Step Methods of Solution p. 125 — 9. Steady Motions p 126..- B. Small Disturbances from Steady Symmetric Flight.- 10. Historical p. 126 — 11. Moderate Finite Disturbances p. 127 — 12. Force and Moment Derivatives p. 127 — 13. Shortened Notation p. 128 — 14. The Applied Forces p. 128 — 15. The Mass Accelerations p. 129 — 16. Separation into Symmetric and Asymmetric Groups p. 129 — 17. The Equations of Motion Rearranged p. 130..- C. Conversion to Dimensionless Form.- 18. Introduction p. 130 — 19. The Parameter p. 131 — 20. The Dimensionless System Explained p. 132 — 21. The Equations of Motion in Dimensionless Form p 134..- D. The Solutions of the Equations of Motion.- 22. Introductory p. 135 — 23. The Complementary Function p. 135—24. The Quartic Equation for A p 136 — 25. The Ratios u1:w1:q1p. 136 — 26. The Four Arbitrary Constants p. 137 —27. General Form of the Complementary Function p. 137 — 28. Initial Conditions p. 137 — 29. Solutions of the Asymmetric Group p. 138 — 30. Complex Roots p 138 — 31. Complex Roots in the Asymmetric Equations p. 141..- E. Complete Solution with Constant Applied Control Couples.- 32. No Roots Zero—Symmetric Group p. 141 — 33. No Roots Zero—The Asymmetric Group 14:2 — 34. One Root Zero—The Symmetric Group p. 142 — 35. One Root Zero—The Asymmetric Group p. 143 — 36. Two Roots Zero p. 144 — 37. Case of Equal Roots p. 144..- VI. Numerical Solution of the Symmetric Equations of Chapter V.- 1. Introduction p. 144 — 2. Notation and Axes p. 144 — 3. Evaluation of the Derivatives p. 145 — 4. The Force-Velocity Derivatives p. 146 — 5. Force-Rotary Derivatives p. 147 — 6. Moment-Velocity Derivatives p. 148 — 7. Moment-Rotary Derivatives p. 148 — 8. The Derivative mwp. 148 — 9. Choice of Numerical Examples p. 149 — 10. The Derivatives p 149 — 11. The Examples Specified p. 150 — 12. The Constants Tabulated p. 151..- A. The Complementary Function.- 13. The Equations of Motion p. 152 — 14. The Quartic for ? p 152 — 15. Information from Inspection of the Quartic for ? p. 152 — 16. Values of ? and of the Ratio u1:w1:q1p. 154..- B. Complete Solutions and Initial Conditions.- 17. Choice of Initial Conditions p. 157 —18. Complete Solutions in Dimensionless Form p. 160 — 19. Conditions to which the Solutions Apply p. 160 — 20. The Solutions Discussed p. 161 — 21. Conversion to Ordinary Units p. 162 — 22. The Solutions In Terms of Other Variables p 163 —23. Graphs of the Solutions Plotted Against Time p. 163..- C. The Graphs Discussed.- 24. Introductory p. 164 — 25. Cruising Speed—Elevator Movement p. 164 — 26. Cruising Speed—Vertical Gust p. 165 — 27. Cruising Speed—Horizontal Gust p. 165 — 28. Slow Speed—Unstalled p. 165 — 29. Stalled Flight p. 168..- D. Approximations Applicable to Normal Flight Only.- 30. Statement of the Problem p. 168 — 31. The Quartic of ? p. 169 — 32. Equations of Motion Referred to Wind Axes p. 170 — 33. Gliding Flight—Engine Off p 171 — 34. Numerical Values p. 172 — 35. The Large Quadratic Root p. 172 — 36. The Small Quadratic Root p. 173..- E. Level Flight.- 37. Introductory p. 174 — 38. Effect of Airscrew on mwp. 175 — 39. Effect of Airscrew on mq and mwp. 175 — 40. The Effect of the Airscrew on xw, zu, zwp. 176 — 41. The Value of dT/dV for an Airscrew p. 176 —42. The Effect of Airscrew on xup. 177 — 43. Effect of Airscrew on mu p. 178 — 44. Effect of Airscrew on the Coefficients B, C and E in the Quartic for ? p. 179 — 45. Effect of Airscrew on the Damping of the Slow Motion p. 180 — 46. Numerical Values p. 180 — 47. The Effect of Freeing the Elevators p. 181..- VII. Numerical Solutions of the Asymmetric Equations.- I. Introduction p. 182 — 2. Notation and Axes p. 182 — 3. Choice of Illustrative Examples p. 183 — 4. Cruising Speed p. 183 — 5. Slow Speed p. 184 — 6. Stalled Flight p 184 — 7. Table of Dimensionless Constants p. 185..- A. Complementary Function.- 8. The Equations of Motion p 185 — 9. The Quartic for ? p 185 — 10. Values of ? and of the Ratios v: p: rp. 185 — 11. The Large Real Root p 187 — 12. The Small Real Root p. 187 — 13. The Complex Root p. 187..- B. Complete Solutions and Initial Conditions.- 14. Choice of Initial Conditions p. 188 — 15. Complete Solutions in Dimensionless Form p. 188 — 16. Conditions to which the Solutions are Applicable p. 188 — 17. Conversion to Ordinary Units p 189 — 18. Graphs of the Solutions Plotted Against Time p. 191..- C. The Graphs Discussed.- 19. Initial Rate of Roll p. 192 — 20. Applied Control Moments p. 193 — 21. Applied Rolling Moment p. 193 — 22. Applied Yawing Moments p. 196 — 23. Rolling and Yawing Moments Applied Simultaneously p. 196 — 24. Stalled Flight p. 197 — 25. Uncertainty of the Values of the Derivatives in Stalled Flight p 198 26. Scale Effect on a Thick Wing in Stalled Flight p 199..- D. Information by Inspection of the Quartic for ?.- 27. Condition for Complete Stability p 199 — 28. The Large Root p. 199 — 29. The Small Root p. 200 — 30. Approximate Values of the Complex Roots p. 200 — 31. Level and Climbing Flight p 201..- E. Changes of Dihedral Angle and Fin Area.- 32. Introduction p. 202 — 33. Spiral Instability p. 202 — 34. Increasing Oscillations p. 202 — 35. Spinning Instability p. 203 — 36. Summary p. 203..- VIII. The Spin.- 1. Introduction p. 204 — 2. Experimental Methods p. 205 — 3. Theoretical Calculations p. 206 — 4. Notation and the Equations of Motion p. 207..- A. Autorotation.- 5. Rotation about a Fixed Axis p. 208 — 6. Autorotation Defined p. 209 — 7. Latent Autorotation p 209 8. Autorotation Rates p. 210..- B. Pitching Moments.- 9. The Centrifugal Pitching Moment p. 210—10. The Origin of the Centrifugal Pitching Moment p 211 — 11. Balance of Pitching Moments p. 213..- C. The Remaining Degrees of Freedom.- 12. Freedom to Slide along the Axis of Rotation p. 214 — 13. Freedom to Leave the Axis of Rotation p 214 — 14. Two Final Degrees of Freedom p. 215..- D. Effects of Yaw on the Spin.- 15. The Meaning of Yaw p. 215 — 16. The Relation between Yaw and Side-Slip p. 216 — 17. The Effects of Yaw on Rate of Spin p. 216 — 18. The Air Yawing Moment p. 217 — 19. The Centrifugal Yawing Moment p. 218..- E. The Free Spin of Real Aeroplanes.- 20. Rotation Rate and Incidence p. 219 — 21. Radius and Rate of Descent p. 219 — 22. Load Factor and Incidence p. 220 — 23. Critical Nature of Transition to Flat Spin p. 220 — 24. Reasons for Failure of Ailerons and Rudder p. 221..- References.- Division O Airplane Performance.- Editor’s Preface.- I. Fundamental Relations.- 1. Introduction p. 224 — 2. Aerodynamics p. 225 — 3. Engine Power p. 226 — 4. Standard Atmosphere p. 226..- II. Basic Computations.- A. Power Required.- 1. Data Necessary p. 227 — 2. Lift, Lift Coefficient and Speed p. 227 — 3. Induced Drag and Equivalent Monoplane Aspect Ratio p 230 — 4. Profile Drag of Airfoils and Its Variation p. 232 —5. Structural Drag and Its Variation p. 233 — 6. Total Drag and Power Required at Sea Level and at Altitude p. 239..- B. Power Available.- 7. Power Curves of Engines p. 243 — 8. Choice of Propeller Diameter p. 244 — 9. Design v/n and Maximum Efficiency p. 245 — 10. Variation of Engine Speed and Power at Sea Level (Full Throttle) p. 246 — 11. Variation in Propeller Efficiency p. 247 — 12. Power Available at Sea Level and at Altitude (First Method) p. 257 — 13. Power Available at Sea Level and Altitude (Second Method) p. 248..- III. Performance from Power Graphs.- 1. High Speed p. 250 — 2. Rate of Climb p. 252 — 3. Ceilings p. 255 — 4. Angle of Ceilings p. 255 — 5. Time of Climb p. 256 6. Power Required p. 256 (A. Power Required by Summarizing Component Drags p. 257 — B. Power Required by Estimating A Directly p. 258) — 7. Power Available p. 261 (A. Power Available from Variation of R.P.M. with Speed p. 261 — B. Power Available from ratio of Thrust Power p. 263) — 8. Performance p. 264..- IV. Bairstow’s Method.- 1. General Outline of Method p. 267 — 2. Speed of Climb p. 269..- V. Lesley-Beid Method.- 1. Introduction p. 270 — 2. Data Required p. 270 — 3. Description of Method. Level Flight p. 271 — 4. Example of Performance Calculations p. 274..- VI. The Oswald Method.- 1. Parameters p. 280 — 2. Assumptions p. 281 — 3. Theory p. 282 — 4. Charts p. 285..- VII. Empirical-Theoretical Method.- VIII. Logarithmic Diagrams.- 1. Introduction p. 296 — 2. Theory p. 297 — 3. Determination of Performance p. 303..- IX. Range and Endurance.- 1. Range p. 305 — 2. Endurance p 311 — 3. Breguet’s Formulas p 312..- X. Influence of Principle Factors on Performance.- 1. Weight p. 315 — 2. Power p. 316 — 3. Wing Area p. 317 — 4. Span of Equivalent Monoplane p. 318 — 5. Airfoil Characteristics p. 318 — 6. Parasite Resistance p. 319 — 7. Propeller Characteristics p. 319 — 8. Supercharging and Throttling p. 320 — 9. Choice of Altitude Propeller p. 320 — 10. Power Absorbed by Any Propeller p. 321 — 11. Power Available at Sea Level from Special Propeller p. 323 — 12. Example p. 323..- XI. Limits of Performance.- 1. Speed Range p. 324— 2. High Speed p 328 — 3. Initial Rate of Climb p. 329 — 4. Absolute Ceiling p. 330 — 5. Range p. 332 — 6. Endurance p. 333..- Appendix I. The Induced Drag of any Biplane 333 Equivalent Monoplane Span for any Biplane p. 333..- Appendix II. Biplane Wing Lift Coefficients.- References.