Theoretical Aerodynamics

ETHIRAJAN RATHAKRISHNAN, Indian Institute of Technology Kanpur, India

• About the Author xvPreface xvii1 Basics 11.1 Introduction 11.2 Lift and Drag 11.3 Monoplane Aircraft 41.3.1 Types of Monoplane 51.4 Biplane 51.4.1 Advantages and Disadvantages 61.5 Triplane 61.5.1 Chord of a Profile 71.5.2 Chord of an Aerofoil 81.6 Aspect Ratio 91.7 Camber 101.8 Incidence 111.9 Aerodynamic Force 121.10 Scale Effect 151.11 Force and Moment Coefficients 171.12 The Boundary Layer 181.13 Summary 20Exercise Problems 21Reference 222 Essence of Fluid Mechanics 232.1 Introduction 232.2 Properties of Fluids 232.2.1 Pressure 232.2.2 Temperature 242.2.3 Density 242.2.4 Viscosity 252.2.5 Absolute Coefficient of Viscosity 252.2.6 Kinematic Viscosity Coefficient 272.2.7 Thermal Conductivity of Air 272.2.8 Compressibility 282.3 Thermodynamic Properties 282.3.1 Specific Heat 282.3.2 The Ratio of Specific Heats 292.4 Surface Tension 302.5 Analysis of Fluid Flow 312.5.1 Local and Material Rates of Change 322.5.2 Graphical Description of Fluid Motion 332.6 Basic and Subsidiary Laws 342.6.1 System and Control Volume 342.6.2 Integral and Differential Analysis 352.6.3 State Equation 352.7 Kinematics of Fluid Flow 352.7.1 Boundary Layer Thickness 372.7.2 Displacement Thickness 382.7.3 Transition Point 392.7.4 Separation Point 392.7.5 Rotational and Irrotational Motion 402.8 Streamlines 412.8.1 Relationship between Stream Function and Velocity Potential 412.9 Potential Flow 422.9.1 Two-dimensional Source and Sink 432.9.2 Simple Vortex 452.9.3 Source-Sink Pair 462.9.4 Doublet 462.10 Combination of Simple Flows 492.10.1 Flow Past a Half-Body 492.11 Flow Past a Circular Cylinder without Circulation 572.11.1 Flow Past a Circular Cylinder with Circulation 592.12 Viscous Flows 632.12.1 Drag of Bodies 652.12.2 Turbulence 702.12.3 Flow through Pipes 752.13 Compressible Flows 782.13.1 Perfect Gas 792.13.2 Velocity of Sound 802.13.3 Mach Number 802.13.4 Flow with Area Change 802.13.5 Normal Shock Relations 822.13.6 Oblique Shock Relations 832.13.7 Flow with Friction 842.13.8 Flow with Simple T0-Change 862.14 Summary 87Exercise Problems 97References 1023 Conformal Transformation 1033.1 Introduction 1033.2 Basic Principles 1033.2.1 Length Ratios between the Corresponding Elements in the Physical and Transformed Planes 1063.2.2 Velocity Ratios between the Corresponding Elements in the Physical and Transformed Planes 1063.2.3 Singularities 1073.3 Complex Numbers 1073.3.1 Differentiation of a Complex Function 1103.4 Summary 112Exercise Problems 1134 Transformation of Flow Pattern 1154.1 Introduction 1154.2 Methods for Performing Transformation 1154.2.1 By Analytical Means 1164.3 Examples of Simple Transformation 1194.4 Kutta−Joukowski Transformation 1224.5 Transformation of Circle to Straight Line 1234.6 Transformation of Circle to Ellipse 1244.7 Transformation of Circle to Symmetrical Aerofoil 1254.7.1 Thickness to Chord Ratio of Symmetrical Aerofoil 1274.7.2 Shape of the Trailing Edge 1294.8 Transformation of a Circle to a Cambered Aerofoil 1294.8.1 Thickness-to-Chord Ratio of the Cambered Aerofoil 1324.8.2 Camber 1344.9 Transformation of Circle to Circular Arc 1344.9.1 Camber of Circular Arc 1374.10 Joukowski Hypothesis 1374.10.1 The Kutta Condition Applied to Aerofoils 1394.10.2 The Kutta Condition in Aerodynamics 1404.11 Lift of Joukowski Aerofoil Section 1414.12 The Velocity and Pressure Distributions on the Joukowski Aerofoil 1444.13 The Exact Joukowski Transformation Process and Its Numerical Solution 1464.14 The Velocity and Pressure Distribution 1474.15 Aerofoil Characteristics 1554.15.1 Parameters Governing the Aerodynamic Forces 1574.16 Aerofoil Geometry 1574.16.1 Aerofoil Nomenclature 1574.16.2 NASA Aerofoils 1614.16.3 Leading-Edge Radius and Chord Line 1614.16.4 Mean Camber Line 1614.16.5 Thickness Distribution 1624.16.6 Trailing-Edge Angle 1624.17 Wing Geometrical Parameters 1624.18 Aerodynamic Force and Moment Coefficients 1664.18.1 Moment Coefficient 1694.19 Summary 171Exercise Problems 180Reference 1815 Vortex Theory 1835.1 Introduction 1835.2 Vorticity Equation in Rectangular Coordinates 1845.2.1 Vorticity Equation in Polar Coordinates 1865.3 Circulation 1885.4 Line (point) Vortex 1925.5 Laws of Vortex Motion 1945.6 Helmholtz’s Theorems 1955.7 Vortex Theorems 1965.7.1 Stoke’s Theorem 2005.8 Calculation of uR, the Velocity due to Rotational Flow 2045.9 Biot-Savart Law 2075.9.1 A Linear Vortex of Finite Length 2105.9.2 Semi-Infinite Vortex 2115.9.3 Infinite Vortex 2115.9.4 Helmholtz’s Second Vortex Theorem 2165.9.5 Helmholtz’s Third Vortex Theorem 2205.9.6 Helmholtz’s Fourth Vortex Theorem 2205.10 Vortex Motion 2205.11 Forced Vortex 2235.12 Free Vortex 2245.12.1 Free Spiral Vortex 2265.13 Compound Vortex 2295.14 Physical Meaning of Circulation 2305.15 Rectilinear Vortices 2355.15.1 Circular Vortex 2365.16 Velocity Distribution 2375.17 Size of a Circular Vortex 2395.18 Point Rectilinear Vortex 2395.19 Vortex Pair 2405.20 Image of a Vortex in a Plane 2415.21 Vortex between Parallel Plates 2425.22 Force on a Vortex 2445.23 Mutual action of Two Vortices 2445.24 Energy due to a Pair of Vortices 2445.25 Line Vortex 2475.26 Summary 248Exercise Problems 254References 2566 Thin Aerofoil Theory 2576.1 Introduction 2576.2 General Thin Aerofoil Theory 2586.3 Solution of the General Equation 2616.3.1 Thin Symmetrical Flat Plate Aerofoil 2626.3.2 The Aerodynamic Coefficients for a Flat Plate 2656.4 The Circular Arc Aerofoil 2696.4.1 Lift, Pitching Moment, and the Center of Pressure Location for Circular Arc Aerofoil 2716.5 The General Thin Aerofoil Section 2756.6 Lift, Pitching Moment and Center of Pressure Coefficients for a Thin Aerofoil 2786.7 Flapped Aerofoil 2836.7.1 Hinge Moment Coefficient 2866.7.2 Jet Flap 2886.7.3 Effect of Operating a Flap 2886.8 Summary 289Exercise Problems 294References 2957 Panel Method 2977.1 Introduction 2977.2 Source Panel Method 2977.2.1 Coefficient of Pressure 3007.3 The Vortex Panel Method 3027.3.1 Application of Vortex Panel Method 3027.4 Pressure Distribution around a Circular Cylinder by Source Panel Method 3057.5 Using Panel Methods 3097.5.1 Limitations of Panel Method 3097.5.2 Advanced Panel Methods 3097.6 Summary 329Exercise Problems 330Reference 3308 Finite Aerofoil Theory 3318.1 Introduction 3318.2 Relationship between Spanwise Loading and Trailing Vorticity 3318.3 Downwash 3328.4 Characteristics of a Simple Symmetrical Loading – Elliptic Distribution 3358.4.1 Lift for an Elliptic Distribution 3368.4.2 Downwash for an Elliptic Distribution 3368.4.3 Drag Dv due to Downwash for Elliptical Distribution 3388.5 Aerofoil Characteristic with a More General Distribution 3398.5.1 The Downwash for Modified Elliptic Loading 3418.6 The Vortex Drag for Modified Loading 3438.6.1 Condition for Vortex Drag Minimum 3458.7 Lancaster – Prandtl Lifting Line Theory 3478.7.1 The Lift 3498.7.2 Induced Drag 3508.8 Effect of Downwash on Incidence 3538.9 The Integral Equation for the Circulation 3558.10 Elliptic Loading 3568.10.1 Lift and Drag for Elliptical Loading 3578.10.2 Lift Curve Slope for Elliptical Loading 3598.10.3 Change of Aspect Ratio with Incidence 3598.10.4 Problem II 3608.10.5 The Lift for Elliptic Loading 3638.10.6 The Downwash Velocity for Elliptic Loading 3668.10.7 The Induced Drag for Elliptic Loading 3668.10.8 Induced Drag Minimum 3698.10.9 Lift and Drag Calculation by Impulse Method 3708.10.10 The Rectangular Aerofoil 3718.10.11 Cylindrical Rectangular Aerofoil 3728.11 Aerodynamic Characteristics of Asymmetric Loading 3728.11.1 Lift on the Aerofoil 3728.11.2 Downwash 3728.11.3 Vortex Drag 3738.11.4 Rolling Moment 3748.11.5 Yawing Moment 3768.12 Lifting Surface Theory 3788.12.1 Velocity Induced by a Lifting Line Element 3788.12.2 Munk’s Theorem of Stagger 3818.12.3 The Induced Lift 3828.12.4 Blenk’s Method 3838.12.5 Rectangular Aerofoil 3848.12.6 Calculation of the Downwash Velocity 3858.13 Aerofoils of Small Aspect Ratio 3878.13.1 The Integral Equation 3888.13.2 Zero Aspect Ratio 3908.13.3 The Acceleration Potential 3908.14 Lifting Surface 3918.15 Summary 394Exercise Problems 4019 Compressible Flows 4059.1 Introduction 4059.2 Thermodynamics of Compressible Flows 4059.3 Isentropic Flow 4099.4 Discharge from a Reservoir 4119.5 Compressible Flow Equations 4139.6 Crocco’s Theorem 4149.6.1 Basic Solutions of Laplace’s Equation 4189.7 The General Potential Equation for Three-Dimensional Flow 4189.8 Linearization of the Potential Equation 4209.8.1 Small Perturbation Theory 4209.9 Potential Equation for Bodies of Revolution 4239.9.1 Solution of Nonlinear Potential Equation 4259.10 Boundary Conditions 4259.10.1 Bodies of Revolution 4279.11 Pressure Coefficient 4289.11.1 Bodies of Revolution 4299.12 Similarity Rule 4299.13 Two-Dimensional Flow: Prandtl-Glauert Rule for Subsonic Flow 4299.13.1 The Prandtl-Glauert Transformations 4299.13.2 The Direct Problem-Version I 4319.13.3 The Indirect Problem (Case of Equal Potentials): P-G Transformation – Version II 4349.13.4 The Streamline Analogy (Version III): Gothert’s Rule 4359.14 Prandtl-Glauert Rule for Supersonic Flow: Versions I and II 4369.14.1 Subsonic Flow 4369.14.2 Supersonic Flow 4369.15 The von Karman Rule for Transonic Flow 4399.15.1 Use of Karman Rule 4409.16 Hypersonic Similarity 4429.17 Three-Dimensional Flow: The Gothert Rule 4449.17.1 The General Similarity Rule 4449.17.2 Gothert Rule 4469.17.3 Application to Wings of Finite Span 4479.17.4 Application to Bodies of Revolution and Fuselage 4489.17.5 The Prandtl-Glauert Rule 4509.17.6 The von Karman Rule for Transonic Flow 4549.18 Moving Disturbance 4559.18.1 Small Disturbance 4569.18.2 Finite Disturbance 4579.19 Normal Shock Waves 4579.19.1 Equations of Motion for a Normal Shock Wave 4579.19.2 The Normal Shock Relations for a Perfect Gas 4589.20 Change of Total Pressure across a Shock 4629.21 Oblique Shock and Expansion Waves 4639.21.1 Oblique Shock Relations 4649.21.2 Relation between β and θ 4669.21.3 Supersonic Flow over a Wedge 4699.21.4 Weak Oblique Shocks 4719.21.5 Supersonic Compression 4739.21.6 Supersonic Expansion by Turning 4759.21.7 The Prandtl-Meyer Function 4779.21.8 Shock-Expansion Theory 4779.22 Thin Aerofoil Theory 4799.22.1 Application of Thin Aerofoil Theory 4809.23 Two-Dimensional Compressible Flows 4859.24 General Linear Solution for Supersonic Flow 4869.24.1 Existence of Characteristics in a Physical Problem 4889.24.2 Equation for the Streamlines from Kinematic Flow Condition 4899.25 Flow over a Wave-Shaped Wall 4919.25.1 Incompressible Flow 4919.25.2 Compressible Subsonic Flow 4929.25.3 Supersonic Flow 4939.25.4 Pressure Coefficient 4949.26 Summary 495Exercise Problems 509References 51210 Simple Flights 51310.1 Introduction 51310.2 Linear Flight 51310.3 Stalling 51410.4 Gliding 51610.5 Straight Horizontal Flight 51810.6 Sudden Increase of Incidence 52010.7 Straight Side-Slip 52110.8 Banked Turn 52210.9 Phugoid Motion 52310.10 The Phugoid Oscillation 52510.11 Summary 529Exercise Problems 531Further Readings 533Index 535

"Theoretical Aerodynamics is a user-friendly text for a full course on theoretical aerodynamics.... Presented in an easy-to-read style making full use of figures and illustrations to enhance understanding, and moves well simpler to more advanced topics." (Expofairs.com, 20 June 2013)"The main objective of the book is to cover the classical theory for inviscid flow using exact solutions of the linear equations or approximations to the equations with, for example, panel methods and thin aerofoil theory. This provides a good grounding for the student in the basic properties of the fluid flow and results can be achieved by simple calculation." (The Aeronautical Journal, 2015)