THEORY OF LIFT WILEY MATLAB /OCTAVE INTRODUCTORY COMPUTATIONAL AERODYNAMICS IN. G. D. McBain,

Transcription

1 THEORY OF LIFT INTRODUCTORY COMPUTATIONAL AERODYNAMICS IN MATLAB /OCTAVE G. D. McBain, School ofaerospace, Mechanical, & Mechatronic Engineering The University of Sydney, Australia WILEY A John Wiley & Sons, Ltd., Publication

2 Contents Preface Series Preface xvii xxiii PART ONE PLANE IDEAL AERODYNAMICS 1 Preliminary Notions Aerodynamic Force and Moment Motion of the Frame ofreference Orientation of the System of Coordinates Components of the Aerodynamic Force Formulation of the Aerodynamic Problem Aircraft Geometry Wing Section Geometry Wing Geometry Velocity Properties of Air Equation of State: Compressibility and the Speed of Sound Rheology: Viscosity 10 1:4.3 The International Standard Atmosphere Computing Air Properties Dimensional Theory Alternative methods Example: Using Octave to Solve a Linear System Example: NACA Report No Exercises Further Reading Plane Ideal Flow Material Properties: The Perfect Fluid Conservation of Mass Governing Equations: Conservation Laws The Continuity Equation Mechanics: The Euler Equations Rate of Change ofmomentum 27

3 viii Contents Forces Acting on a Fluid Particle The Euler Equations Accounting for Conservative External Forces Consequences of the Governing Equations The Aerodynamic Force Bernoulli's Equation Circulation, Vorticity, and Irrotational Flow Plane Ideal Flows The Complex Velocity Review ofcomplex Variables Analytic Functions and Plane Ideal Flow Example: the PolarAngle Is Nowhere Analytic The Complex Potential Exercises Further Reading Circulation and Lift Powers of z Divergence and Vorticity in Polar Coordinates Complex Potentials Drawing Complex Velocity Fields with Octave Example: k 1, Corner Flow = Example: k = 0, Uniform Stream Example: k = 1, Source Example: k = 2, Doublet Multiplication by a Complex Constant Example: w = const., Uniform Stream with Arbitrary Direction Example: w i/z, Vortex = Example: Polar Components Linear Combinations of Complex Velocities Example: Circular Obstacle in a Stream Transforming the Whole Velocity Field Translating the Whole Velocity Field Example: Doublet as the Sum ofa Source and Sink Rotating the Whole Velocity Field Circulation and Outflow Curve-integrals in Plane Ideal Flow Example: Numerical Line-integralsfor Circulation and Outflow Closed Circuits Example: Powers ofz and Circles around the Origin More on the Scalar Potential and Stream Function The Scalar Potential and Irrotational Flow The Stream Function and Divergence-free Flow 62

4 Contents ix 3.7 Lift Blasius's Theorem The Kutta-Joukowsky Theorem Exercises Further Reading Conformal Mapping Composition of Analytic Functions Mapping with Powers of f Example: Square Mapping, Conforming Mapping by Contouring the Stream Function Example: Two-thirds Power Mapping Branch Cuts Other Powers Joukowsky's Transformation Unit Circle from a Straight Line Segment Uniform Flow and Flow over a Circle Thin Flat Plate at Nonzero Incidence Flow over the Thin Flat Plate with Circulation Joukowsky Aerofoils Exercises Further Reading 78 78, 91 5 Flat Plate Aerodynamics Plane Ideal Flow over a Thin Flat Plate Stagnation Points ie Kutta-Joukowsky Condition Lift on a Thin Flat Plate Surface Speed Distribution Pressure Distribution Distribution of Circulation Thin Flat Plate as Vortex Sheet Application of Thin Aerofoil Theory to the Flat Plate J Thin Aerofoil Theory Vortex Sheet along the Chord Changing the Variable ofintegration Glauert's Integral The Kutta-Joukowsky Condition Circulation and Lift Aerodynamic Moment Centre ofpressure and Aerodynamic Centre Exercises ,, Further Reading 91

5 X Contents 6 Thin Wing Sections Thin Aerofoil Analysis Vortex Sheet along the Camber Line The Boundary Condition Linearization Glauert's Transformation Glauert's Expansion Fourier Cosine Decomposition ofthe Camber Line Slope Thin Aerofoil Aerodynamics Circulation and Lift Pitching Moment about the Leading Edge Aerodynamic Centre Summary Analytical Evaluation of Thin Aerofoil Integrals Example: the NACA Four-digit Wing Sections Numerical Thin Aerofoil Theory Exercises Further Reading Lumped Vortex Elements The Thin Rat Plate at Arbitrary Incidence, Again Single Vortex The Collocation Point Lumped Vortex Model of the Thin Flat Plate Using Two Lumped Vortices along the Chord Postprocessing Generalization to Multiple Lumped Vortex Panels Postprocessing General Considerations on Discrete Singularity Methods Lumped Vortex Elements for Thin Aerofoils Panel Chainl for Camber Lines Implementation in Octave Comparison with Thin Aerofoil Theory Disconnected Aerofoils Other Applications Exercises Further Reading Panel Methods for Plane Flow Development of the CUSSSP Program The Singularity Elements Discretizing the Geometry The Influence Matrix The Right-hand Side 132

6 Contents xi Solving the Linear System Postprocessing Exercises Projects Further Reading Conclusion to Part I: The Origin of Lift 139 PART TWO THREE-DIMENSIONAL IDEAL AERODYNAMICS 9 Finite Wings and Three-Dimensional Flow Wings of Finite Span Empirical Effect of Finite Span on Lift Finite Wings and Three-dimensional Flow Three-Dimensional Flow Three-dimensional Cartesian Coordinate System Three-dimensional Governing Equations Vector Notation and Identities Addition and Scalar Multiplication of Vectors Products of Vectors Vector Derivatives Integral Theorems for Vector Derivatives The Equations Governing Three-Dimensional Flow Conservation of Mass and the Continuity Equation Newton's Law and Euler's Equation Circulation Definition of Circulation in Three Dimensions The Persistence of Circulation Circulation and Vorticity Rotational Form ofeuler's Equation Steady Irrotational Motion Exercises Further Reading Vorticity and Vortices Streamlines, Stream Tubes, and Stream Filaments Streamlines Stream Tubes and Stream Filaments Vortex Lines, Vortex Tubes, and Vortex Filaments Strength oj: Vortex Tubes and Filaments Kinematic Properties ofvortex Tubes Helmholtz's Theorems 'Vortex Tubes Move with the Flow' 159 " 'The Strength of a Vortex Tube is Constant' 160

7 xu Contents 10.4 Line Vortices The Two-dimensional Vortex Arbitrarily Oriented Rectilinear Vortex Filaments Segmented Vortex Filaments The BiotSavart Law Rectilinear Vortex Filaments Finite Rectilinear Vortex Filaments Infinite Straight Line Vortices Semi-infinite Straight Line Vortex Truncating Infinite Vortex Segments Implementing Line Vortices in Octave Exercises Further Reading Lifting Line Theory Basic Assumptions of Lifting Line Theory The Lifting Line, Horseshoe Vortices, and the Wake Deductionsfrom Vortex Theorems Deductions from the Wing Pressure Distribution The Lifting Line Model ofair Flow Horseshoe Vortex Continuous Trailing Vortex Sheet The Form ofthe Wake The Effect of Downwash Effect on the Angle of Incidence: Induced Incidence Effect on the Aerodynamic Force: Induced Drag The Lifting Line Equation Glauert's Solution of the Lifting Line Equation Wing Properties in Terms of Glauert's Expansion The Elliptic Lift Loading Properties of the Elliptic Lift Loading Lift-Incidence Relation Linear Lift-Incidence Relation Realizing the Elliptic Lift Loading Corrections to the Elliptic Loading Approximation Exercises Further Reading Nonelliptic Lift Loading Solving the Lifting Line Equation The Sectional Lift-Incidence Relation Linear Sectional Lift-Incidence Relation Finite Approximation: Truncation and Collocation 185

8 Contents xiii Computer Implementation Example: a Rectangular Wing 12.2 Numerical Convergence 12.3 Symmetric Spanwise Loading A Example: Exploiting Symmetry Exercises Lumped Horseshoe Elements 13.1 A Single Horseshoe Vortex Induced Incidence of the Lumped Horseshoe Element 13.2 Multiple Horseshoes along the Span A Finite-step Lifting Line in Octave 13.3 An Improved Discrete Horseshoe Model 13.4 Implementing Horseshoe Vortices in Octave Example: Yawed Horseshoe Vortex Coefficients 13.5 Exercises 13.6 Further Reading The Vortex Lattice Method 14.1 Meshing the Mean Lifting Surface of a Wing Plotting the Mesh of a Mean Lifting Surface 14.2 A Vortex Lattice Method The Vortex Lattice Equations UnitNormals to the Vortex-lattice Spanwise Symmetry Postprocessing Vortex Lattice Methods 14.3 Examples of Vortex Lattice Calculations Campbell's Flat Swept Tapered Wing Bertin's Flat Swept Untapered Wing Spanwise and Chordwise Refinement 14.4 Exercises 14.5 Further Reading Three-dimensional Panel Methods PART THREE NONTDEAL FLOW IN AERODYNAMICS 15 Viscous Flow 15.1 Cauchy's First Law of Continuum Mechanics 15.2 Rheological Constitutive Equations Perfect Fluid Linearly Viscous Fluid 15.3 The Navier-Stokes Equations

9 15.4 The No-Slip Condition and the Viscous Boundary Layer 15.5 Unidirectional Flows Plane Couette and Poiseuille Flows 15.6 A Suddenly Sliding Plate Solution by Similarity Variable The Diffusion of Vorticity 15.7 Exercises 15.8 Further Reading 16 Boundary Layer Equations 16.1 The Boundary Layer over a Flat Plate Scales in the Conservation ofmass Scales in the Streamwise Momentum Equation The Reynolds Number Pressure in the Boundary Layer The Transverse Momentum Balance The Boundary Layer Momentum Equation Pressure and External Tangential Velocity Application to Curved Surfaces 16.2 Momentum Integral Equation 16.3 Local Boundary Layer Parameters The Displacement and Momentum Thicknesses The Skin Friction Coefficient Example: Three Boundary Layer Profiles 16.4 Exercises 16.5 Further Reading 17 Laminar Boundary Layers 17.1 Boundary Layer Profile Curvature Pressure Gradient and Boundary Layer Thickness 17.2 Pohlhausen's Quartic Profiles 17.3 Thwaites's Method for Laminar Boundary Layers 77J.7 F(X) « k Correlations for Shape Factor and Skin Friction Example: Zero Pressure Gradient Example: Laminar Separation from a Circular Cylinder 17.4 Exercises 17.5 Further Reading 18 Compressibility 18.1 Steady-State Conservation of Mass 18.2 Longitudinal Variation of Stream Tube Section The Design of Supersonic Nozzles

10 Contents xv 18.3 Perfect Gas Thermodynamics Thermal and Caloric Equations of State The First Law of Thermodynamics The Isochoric and Isobaric Specific Heat Coefficients Isothermal and Adiabatic Processes Adiabatic Expansion The Speed of Sound and Temperature The Speed of Sound and the Speed Thermodynamic Characteristics of Air Example: Stagnation Temperature Exercises Further Reading Linearized Compressible Flow The Nonlinearity of the Equation for the Potential Small Disturbances to the Free-Stream The Uniform Free-Stream The Disturbance Potential Prandtl-Glauert Transformation Fundamental Linearized Compressible Flows The Speed of Sound Application of the Prandtl-Glauert Rule Transforming the Geometry Computing Aerodynamical Forces The Prandlt-Glauert Rule in Two Dimensions The Critical Mach Number Sweep Exercises Further Reading Appendix A Notes on Octave Programming 287 A.l Introduction 287 A.2 Vectorization 287 A.2.1 Iterating Explicitly 288 A.2.2 Preallocating Memory 288 A.2.3 Vectorizing Function Calls 288 A.2.4 Many FunctionsAct Elementwise on Arrays 289 A.2.5 Functions Primarily Defined forarrays 289 A.2.6 Elementwise Arithmetic with Single Numbers 289 A.2.7 Elementwise Arithmetic between Arrays 290 A.2.8 Vector and Matrix Multiplication 290 A.3 Generating Arrays 290 A.3.1 Creating Tables with bsxfun 290

11 xvi A.4 Indexing A.4.1 Indexing by Logical Masks A.4.2 Indexing Numerically A.5 Just-in-Time Compilation A.6 Further Reading Glossary Nomenclature Index