Theory of Lift

Dr. Geordie Drummond McBain, AustraliaGeordie McBain is an engineering consultant based in Sydney, Australia. In 1995 he graduated top of his class from James Cook University with first class honours in mechanical engineering, earning him the Faculty Medal, and went on to receive his PhD there in 1999. In 2002 he was awarded a Sesquicentennial Postdoctoral Fellowship at the University of Sydney, researching fluid dynamics. During this period, he taught aerodynamics to students on the Aeronautical and Aerospace Engineering degree programmes.

• Preface xviiSeries Preface xxiiiPart One Plane Ideal Aerodynamics1 Preliminary Notions 31.1 Aerodynamic Force and Moment 31.1.1 Motion of the Frame of Reference 31.1.2 Orientation of the System of Coordinates 41.1.3 Components of the Aerodynamic Force 41.1.4 Formulation of the Aerodynamic Problem 41.2 Aircraft Geometry 51.2.1 Wing Section Geometry 61.2.2 Wing Geometry 71.3 Velocity 81.4 Properties of Air 81.4.1 Equation of State: Compressibility and the Speed of Sound 81.4.2 Rheology: Viscosity 101.4.3 The International Standard Atmosphere 121.4.4 Computing Air Properties 121.5 Dimensional Theory 131.5.1 Alternative methods 161.5.2 Example: Using Octave to Solve a Linear System 161.6 Example: NACA Report No. 502 181.7 Exercises 191.8 Further Reading 22References 222 Plane Ideal Flow 252.1 Material Properties: The Perfect Fluid 252.2 Conservation of Mass 262.2.1 Governing Equations: Conservation Laws 262.3 The Continuity Equation 262.4 Mechanics: The Euler Equations 272.4.1 Rate of Change of Momentum 272.4.2 Forces Acting on a Fluid Particle 282.4.3 The Euler Equations 292.4.4 Accounting for Conservative External Forces 292.5 Consequences of the Governing Equations 302.5.1 The Aerodynamic Force 302.5.2 Bernoulli’s Equation 332.5.3 Circulation, Vorticity, and Irrotational Flow 332.5.4 Plane Ideal Flows 352.6 The Complex Velocity 352.6.1 Review of Complex Variables 352.6.2 Analytic Functions and Plane Ideal Flow 382.6.3 Example: the Polar Angle Is Nowhere Analytic 402.7 The Complex Potential 412.8 Exercises 422.9 Further Reading 44References 453 Circulation and Lift 473.1 Powers of z 473.1.1 Divergence and Vorticity in Polar Coordinates 483.1.2 Complex Potentials 483.1.3 Drawing Complex Velocity Fields with Octave 493.1.4 Example: k = 1, Corner Flow 503.1.5 Example: k = 0, Uniform Stream 513.1.6 Example: k =−1, Source 513.1.7 Example: k =−2, Doublet 523.2 Multiplication by a Complex Constant 533.2.1 Example: w = const., Uniform Stream with Arbitrary Direction 533.2.2 Example: w = i/z, Vortex 543.2.3 Example: Polar Components 543.3 Linear Combinations of Complex Velocities 543.3.1 Example: Circular Obstacle in a Stream 543.4 Transforming the Whole Velocity Field 563.4.1 Translating the Whole Velocity Field 563.4.2 Example: Doublet as the Sum of a Source and Sink 563.4.3 Rotating the Whole Velocity Field 563.5 Circulation and Outflow 573.5.1 Curve-integrals in Plane Ideal Flow 573.5.2 Example: Numerical Line-integrals for Circulation and Outflow 583.5.3 Closed Circuits 593.5.4 Example: Powers of z and Circles around the Origin 603.6 More on the Scalar Potential and Stream Function 613.6.1 The Scalar Potential and Irrotational Flow 613.6.2 The Stream Function and Divergence-free Flow 623.7 Lift 623.7.1 Blasius’s Theorem 623.7.2 The Kutta–Joukowsky Theorem 633.8 Exercises 643.9 Further Reading 65References 664 Conformal Mapping 674.1 Composition of Analytic Functions 674.2 Mapping with Powers of ζ 684.2.1 Example: Square Mapping 684.2.2 Conforming Mapping by Contouring the Stream Function 694.2.3 Example: Two-thirds Power Mapping 694.2.4 Branch Cuts 704.2.5 Other Powers 714.3 Joukowsky’s Transformation 714.3.1 Unit Circle from a Straight Line Segment 714.3.2 Uniform Flow and Flow over a Circle 724.3.3 Thin Flat Plate at Nonzero Incidence 734.3.4 Flow over the Thin Flat Plate with Circulation 744.3.5 Joukowsky Aerofoils 754.4 Exercises 754.5 Further Reading 78References 785 Flat Plate Aerodynamics 795.1 Plane Ideal Flow over a Thin Flat Plate 795.1.1 Stagnation Points 805.1.2 The Kutta–Joukowsky Condition 805.1.3 Lift on a Thin Flat Plate 815.1.4 Surface Speed Distribution 825.1.5 Pressure Distribution 835.1.6 Distribution of Circulation 845.1.7 Thin Flat Plate as Vortex Sheet 855.2 Application of Thin Aerofoil Theory to the Flat Plate 875.2.1 Thin Aerofoil Theory 875.2.2 Vortex Sheet along the Chord 875.2.3 Changing the Variable of Integration 885.2.4 Glauert’s Integral 885.2.5 The Kutta–Joukowsky Condition 895.2.6 Circulation and Lift 895.3 Aerodynamic Moment 895.3.1 Centre of Pressure and Aerodynamic Centre 905.4 Exercises 905.5 Further Reading 91References 916 Thin Wing Sections 936.1 Thin Aerofoil Analysis 936.1.1 Vortex Sheet along the Camber Line 936.1.2 The Boundary Condition 936.1.3 Linearization 946.1.4 Glauert’s Transformation 956.1.5 Glauert’s Expansion 956.1.6 Fourier Cosine Decomposition of the Camber Line Slope 976.2 Thin Aerofoil Aerodynamics 986.2.1 Circulation and Lift 986.2.2 Pitching Moment about the Leading Edge 996.2.3 Aerodynamic Centre 1006.2.4 Summary 1016.3 Analytical Evaluation of Thin Aerofoil Integrals 1016.3.1 Example: the NACA Four-digit Wing Sections 1046.4 Numerical Thin Aerofoil Theory 1056.5 Exercises 1096.6 Further Reading 109References 1097 Lumped Vortex Elements 1117.1 The Thin Flat Plate at Arbitrary Incidence, Again 1117.1.1 Single Vortex 1117.1.2 The Collocation Point 1117.1.3 Lumped Vortex Model of the Thin Flat Plate 1127.2 Using Two Lumped Vortices along the Chord 1147.2.1 Postprocessing 1167.3 Generalization to Multiple Lumped Vortex Panels 1177.3.1 Postprocessing 1177.4 General Considerations on Discrete Singularity Methods 1177.5 Lumped Vortex Elements for Thin Aerofoils 1197.5.1 Panel Chains for Camber Lines 1197.5.2 Implementation in Octave 1217.5.3 Comparison with Thin Aerofoil Theory 1227.6 Disconnected Aerofoils 1237.6.1 Other Applications 1247.7 Exercises 1257.8 Further Reading 125References 1268 Panel Methods for Plane Flow 1278.1 Development of the CUSSSP Program 1278.1.1 The Singularity Elements 1278.1.2 Discretizing the Geometry 1298.1.3 The Influence Matrix 1318.1.4 The Right-hand Side 1328.1.5 Solving the Linear System 1348.1.6 Postprocessing 1358.2 Exercises 1378.2.1 Projects 1388.3 Further Reading 139References 1398.4 Conclusion to Part I: The Origin of Lift 139Part Two Three-dimensional Ideal Aerodynamics9 Finite Wings and Three-Dimensional Flow 1439.1 Wings of Finite Span 1439.1.1 Empirical Effect of Finite Span on Lift 1439.1.2 Finite Wings and Three-dimensional Flow 1439.2 Three-Dimensional Flow 1459.2.1 Three-dimensional Cartesian Coordinate System 1459.2.2 Three-dimensional Governing Equations 1459.3 Vector Notation and Identities 1459.3.1 Addition and Scalar Multiplication of Vectors 1459.3.2 Products of Vectors 1469.3.3 Vector Derivatives 1479.3.4 Integral Theorems for Vector Derivatives 1489.4 The Equations Governing Three-Dimensional Flow 1499.4.1 Conservation of Mass and the Continuity Equation 1499.4.2 Newton’s Law and Euler’s Equation 1499.5 Circulation 1509.5.1 Definition of Circulation in Three Dimensions 1509.5.2 The Persistence of Circulation 1519.5.3 Circulation and Vorticity 1519.5.4 Rotational Form of Euler’s Equation 1539.5.5 Steady Irrotational Motion 1539.6 Exercises 1549.7 Further Reading 155References 15510 Vorticity and Vortices 15710.1 Streamlines, Stream Tubes, and Stream Filaments 15710.1.1 Streamlines 15710.1.2 Stream Tubes and Stream Filaments 15810.2 Vortex Lines, Vortex Tubes, and Vortex Filaments 15910.2.1 Strength of Vortex Tubes and Filaments 15910.2.2 Kinematic Properties of Vortex Tubes 15910.3 Helmholtz’s Theorems 15910.3.1 ‘Vortex Tubes Move with the Flow’ 15910.3.2 ‘The Strength of a Vortex Tube is Constant’ 16010.4 Line Vortices 16010.4.1 The Two-dimensional Vortex 16010.4.2 Arbitrarily Oriented Rectilinear Vortex Filaments 16010.5 Segmented Vortex Filaments 16110.5.1 The Biot–Savart Law 16110.5.2 Rectilinear Vortex Filaments 16210.5.3 Finite Rectilinear Vortex Filaments 16410.5.4 Infinite Straight Line Vortices 16410.5.5 Semi-infinite Straight Line Vortex 16410.5.6 Truncating Infinite Vortex Segments 16510.5.7 Implementing Line Vortices in Octave 16510.6 Exercises 16610.7 Further Reading 167References 16711 Lifting Line Theory 16911.1 Basic Assumptions of Lifting Line Theory 16911.2 The Lifting Line, Horseshoe Vortices, and the Wake 16911.2.1 Deductions from Vortex Theorems 16911.2.2 Deductions from the Wing Pressure Distribution 17011.2.3 The Lifting Line Model of Air Flow 17011.2.4 Horseshoe Vortex 17011.2.5 Continuous Trailing Vortex Sheet 17111.2.6 The Form of the Wake 17211.3 The Effect of Downwash 17311.3.1 Effect on the Angle of Incidence: Induced Incidence 17311.3.2 Effect on the Aerodynamic Force: Induced Drag 17411.4 The Lifting Line Equation 17411.4.1 Glauert’s Solution of the Lifting Line Equation 17511.4.2 Wing Properties in Terms of Glauert’s Expansion 17611.5 The Elliptic Lift Loading 17811.5.1 Properties of the Elliptic Lift Loading 17911.6 Lift–Incidence Relation 18011.6.1 Linear Lift–Incidence Relation 18111.7 Realizing the Elliptic Lift Loading 18211.7.1 Corrections to the Elliptic Loading Approximation 18211.8 Exercises 18211.9 Further Reading 183References 18312 Nonelliptic Lift Loading 18512.1 Solving the Lifting Line Equation 18512.1.1 The Sectional Lift–Incidence Relation 18512.1.2 Linear Sectional Lift–Incidence Relation 18512.1.3 Finite Approximation: Truncation and Collocation 18512.1.4 Computer Implementation 18712.1.5 Example: a Rectangular Wing 18712.2 Numerical Convergence 18812.3 Symmetric Spanwise Loading 18912.3.1 Example: Exploiting Symmetry 19112.4 Exercises 192References 19213 Lumped Horseshoe Elements 19313.1 A Single Horseshoe Vortex 19313.1.1 Induced Incidence of the Lumped Horseshoe Element 19513.2 Multiple Horseshoes along the Span 19513.2.1 A Finite-step Lifting Line in Octave 19713.3 An Improved Discrete Horseshoe Model 20013.4 Implementing Horseshoe Vortices in Octave 20313.4.1 Example: Yawed Horseshoe Vortex Coefficients 20513.5 Exercises 20613.6 Further Reading 207References 20714 The Vortex Lattice Method 20914.1 Meshing the Mean Lifting Surface of a Wing 20914.1.1 Plotting the Mesh of a Mean Lifting Surface 21014.2 A Vortex Lattice Method 21214.2.1 The Vortex Lattice Equations 21314.2.2 Unit Normals to the Vortex-lattice 21514.2.3 Spanwise Symmetry 21514.2.4 Postprocessing Vortex Lattice Methods 21514.3 Examples of Vortex Lattice Calculations 21614.3.1 Campbell’s Flat Swept Tapered Wing 21614.3.2 Bertin’s Flat Swept Untapered Wing 21814.3.3 Spanwise and Chordwise Refinement 21914.4 Exercises 22014.5 Further Reading 22114.5.1 Three-dimensional Panel Methods 222References 222Part Three Nonideal Flow in Aerodynamics15 Viscous Flow 22515.1 Cauchy’s First Law of Continuum Mechanics 22515.2 Rheological Constitutive Equations 22715.2.1 Perfect Fluid 22715.2.2 Linearly Viscous Fluid 22715.3 The Navier–Stokes Equations 22815.4 The No-Slip Condition and the Viscous Boundary Layer 22815.5 Unidirectional Flows 22915.5.1 Plane Couette and Poiseuille Flows 22915.6 A Suddenly Sliding Plate 23015.6.1 Solution by Similarity Variable 23015.6.2 The Diffusion of Vorticity 23315.7 Exercises 23415.8 Further Reading 234References 23516 Boundary Layer Equations 23716.1 The Boundary Layer over a Flat Plate 23716.1.1 Scales in the Conservation of Mass 23716.1.2 Scales in the Streamwise Momentum Equation 23816.1.3 The Reynolds Number 23916.1.4 Pressure in the Boundary Layer 23916.1.5 The Transverse Momentum Balance 23916.1.6 The Boundary Layer Momentum Equation 24016.1.7 Pressure and External Tangential Velocity 24116.1.8 Application to Curved Surfaces 24116.2 Momentum Integral Equation 24116.3 Local Boundary Layer Parameters 24316.3.1 The Displacement and Momentum Thicknesses 24316.3.2 The Skin Friction Coefficient 24316.3.3 Example: Three Boundary Layer Profiles 24416.4 Exercises 24816.5 Further Reading 249References 24917 Laminar Boundary Layers 25117.1 Boundary Layer Profile Curvature 25117.1.1 Pressure Gradient and Boundary Layer Thickness 25217.2 Pohlhausen’s Quartic Profiles 25217.3 Thwaites’s Method for Laminar Boundary Layers 25417.3.1 F(λ) ≈ 0.45 − 6λ 25517.3.2 Correlations for Shape Factor and Skin Friction 25617.3.3 Example: Zero Pressure Gradient 25617.3.4 Example: Laminar Separation from a Circular Cylinder 25717.4 Exercises 26017.5 Further Reading 261References 26218 Compressibility 26318.1 Steady-State Conservation of Mass 26318.2 Longitudinal Variation of Stream Tube Section 26518.2.1 The Design of Supersonic Nozzles 26618.3 Perfect Gas Thermodynamics 26618.3.1 Thermal and Caloric Equations of State 26618.3.2 The First Law of Thermodynamics 26718.3.3 The Isochoric and Isobaric Specific Heat Coefficients 26718.3.4 Isothermal and Adiabatic Processes 26718.3.5 Adiabatic Expansion 26818.3.6 The Speed of Sound and Temperature 26918.3.7 The Speed of Sound and the Speed 26918.3.8 Thermodynamic Characteristics of Air 27018.3.9 Example: Stagnation Temperature 27018.4 Exercises 27018.5 Further Reading 271References 27119 Linearized Compressible Flow 27319.1 The Nonlinearity of the Equation for the Potential 27319.2 Small Disturbances to the Free-Stream 27419.3 The Uniform Free-Stream 27519.4 The Disturbance Potential 27519.5 Prandtl–Glauert Transformation 27619.5.1 Fundamental Linearized Compressible Flows 27719.5.2 The Speed of Sound 27819.6 Application of the Prandtl–Glauert Rule 27919.6.1 Transforming the Geometry 27919.6.2 Computing Aerodynamical Forces 28019.6.3 The Prandlt–Glauert Rule in Two Dimensions 28219.6.4 The Critical Mach Number 28419.7 Sweep 28419.8 Exercises 28519.9 Further Reading 285References 286Appendix A Notes on Octave Programming 287A. 1 Introduction 287A. 2 Vectorization 287A.2. 1 Iterating Explicitly 288A.2. 2 Preallocating Memory 288A.2. 3 Vectorizing Function Calls 288A.2. 4 Many Functions Act Elementwise on Arrays 289A.2. 5 Functions Primarily Defined for Arrays 289A.2. 6 Elementwise Arithmetic with Single Numbers 289A.2. 7 Elementwise Arithmetic between Arrays 290A.2. 8 Vector and Matrix Multiplication 290A. 3 Generating Arrays 290A.3. 1 Creating Tables with bsxfun 290A. 4 Indexing 291A.4. 1 Indexing by Logical Masks 291A.4. 2 Indexing Numerically 291A. 5 Just-in-Time Compilation 291A. 6 Further Reading 292References 292Glossary 293Nomenclature 305Index 309

“This book is a very useful digest of key points from the literature, carefully structured and presented with helpful pointers as to how the successive aerodynamical models can be implemented in the ‘now so readily available interactive matrix computation systems.”  (Aeronautical Journal, 1 August 2013)