Ebook: Low speed aerodynamics

Low-speed aerodynamics is important in the design and operation of aircraft flying at low Mach number, and ground and marine vehicles. This text offers a modern treatment of both the theory of inviscid, incompressible, and irrotational aerodynamics, and the computational techniques now available to solve complex problems. A unique feature is that the computational approach--from a single vortex element to a three-dimensional panel formulation--is interwoven throughout. This second edition features a new chapter on the laminar boundary layer (emphasis on the viscous-inviscid coupling), the latest versions of computational techniques, and additional coverage of interaction problems. The authors include a systematic treatment of two-dimensional panel methods and a detailed presentation of computational techniques for three-dimensional and unsteady flows
1.1 Description of Fluid Motion 1 -- 1.2 Choice of Coordinate System 2 -- 1.3 Pathlines, Streak Lines, and Streamlines 3 -- 1.4 Forces in a Fluid 4 -- 1.5 Integral Form of the Fluid Dynamic Equations 6 -- 1.6 Differential Form of the Fluid Dynamic Equations 8 -- 1.7 Dimensional Analysis of the Fluid Dynamic Equations 14 -- 1.8 Flow with High Reynolds Number 17 -- 1.9 Similarity of Flows 19 -- 2 Fundamentals of Inviscid, Incompressible Flow 21 -- 2.1 Angular Velocity, Vorticity, and Circulation 21 -- 2.2 Rate of Change of Vorticity 24 -- 2.3 Rate of Change of Circulation: Kelvin's Theorem 25 -- 2.4 Irrotational Flow and the Velocity Potential 26 -- 2.5 Boundary and Infinity Conditions 27 -- 2.6 Bernoulli's Equation for the Pressure 28 -- 2.7 Simply and Multiply Connected Regions 29 -- 2.8 Uniqueness of the Solution 30 -- 2.9 Vortex Quantities 32 -- 2.10 Two-Dimensional Vortex 34 -- 2.11 The Biot-Savart Law 36 -- 2.12 The Velocity Induced by a Straight Vortex Segment 38 -- 2.13 The Stream Function 41 -- 3 General Solution of the Incompressible, Potential Flow Equations 44 -- 3.1 Statement of the Potential Flow Problem 44 -- 3.2 The General Solution, Based on Green's Identity 44 -- 3.3 Summary: Methodology of Solution 48 -- 3.4 Basic Solution: Point Source 49 -- 3.5 Basic Solution: Point Doublet 51 -- 3.6 Basic Solution: Polynomials 54 -- 3.7 Two-Dimensional Version of the Basic Solutions 56 -- 3.8 Basic Solution: Vortex 58 -- 3.9 Principle of Superposition 60 -- 3.10 Superposition of Sources and Free Stream: Rankine's Oval 60 -- 3.11 Superposition of Doublet and Free Stream: Flow around a Cylinder 62 -- 3.12 Superposition of a Three-Dimensional Doublet and Free Stream: Flow around a Sphere 67 -- 3.13 Some Remarks about the Flow over the Cylinder and the Sphere 69 -- 3.14 Surface Distribution of the Basic Solutions 70 -- 4 Small-Disturbance Flow over Three-Dimensional Wings: Formulation of the Problem 75 -- 4.1 Definition of the Problem 75 -- 4.2 The Boundary Condition on the Wing 76 -- 4.3 Separation of the Thickness and the Lifting Problems 78 -- 4.4 Symmetric Wing with Nonzero Thickness at Zero Angle of Attack 79 -- 4.5 Zero-Thickness Cambered Wing at Angle of Attack-Lifting Surfaces 82 -- 4.6 The Aerodynamic Loads 85 -- 4.7 The Vortex Wake 88 -- 4.8 Linearized Theory of Small-Disturbance Compressible Flow 90 -- 5 Small-Disturbance Flow over Two-Dimensional Airfoils 94 -- 5.1 Symmetric Airfoil with Nonzero Thickness at Zero Angle of Attack 94 -- 5.2 Zero-Thickness Airfoil at Angle of Attack 100 -- 5.3 Classical Solution of the Lifting Problem 104 -- 5.4 Aerodynamic Forces and Moments on a Thin Airfoil 106 -- 5.5 The Lumped-Vortex Element 114 -- 5.6 Summary and Conclusions from Thin Airfoil Theory 120 -- 6 Exact Solutions with Complex Variables 122 -- 6.1 Summary of Complex Variable Theory 122 -- 6.2 The Complex Potential 125 -- 6.3 Simple Examples 126 -- 6.3.1 Uniform Stream and Singular Solutions 126 -- 6.3.2 Flow in a Corner 127 -- 6.4 Blasius Formula, Kutta-Joukowski Theorem 128 -- 6.5 Conformal Mapping and the Joukowski Transformation 128 -- 6.5.1 Flat Plate Airfoil 130 -- 6.5.2 Leading-Edge Suction 131 -- 6.5.3 Flow Normal to a Flat Plate 133 -- 6.5.4 Circular Arc Airfoil 134 -- 6.5.5 Symmetric Joukowski Airfoil 135 -- 6.6 Airfoil with Finite Trailing-Edge Angle 137 -- 6.7 Summary of Pressure Distributions for Exact Airfoil Solutions 138 -- 6.8 Method of Images 141 -- 6.9 Generalized Kutta-Joukowski Theorem 146 -- 7 Perturbation Methods 151 -- 7.1 Thin-Airfoil Problem 151 -- 7.2 Second-Order Solution 154 -- 7.3 Leading-Edge Solution 157 -- 7.4 Matched Asymptotic Expansions 160 -- 7.5 Thin Airfoil between Wind Tunnel Walls 163 -- 8 Three-Dimensional Small-Disturbance Solutions 167 -- 8.1 Finite Wing: The Lifting Line Model 167 -- 8.1.1 Definition of the Problem 167 -- 8.1.2 The Lifting-Line Model 168 -- 8.1.3 The Aerodynamic Loads 172 -- 8.1.4 The Elliptic Lift Distribution 173 -- 8.1.5 General Spanwise Circulation Distribution 178 -- 8.1.6 Twisted Elliptic Wing 181 -- 8.1.7 Conclusions from Lifting-Line Theory 183 -- 8.2 Slender Wing Theory 184 -- 8.2.1 Definition of the Problem 184 -- 8.2.2 Solution of the Flow over Slender Pointed Wings 186 -- 8.2.3 The Method of R. T. Jones 192 -- 8.2.4 Conclusions from Slender Wing Theory 194 -- 8.3 Slender Body Theory 195 -- 8.3.1 Axisymmetric Longitudinal Flow Past a Slender Body of Revolution 196 -- 8.3.2 Transverse Flow Past a Slender Body of Revolution 198 -- 8.3.3 Pressure and Force Information 199 -- 8.3.4 Conclusions from Slender Body Theory 201 -- 8.4 Far Field Calculation of Induced Drag 201 -- 9 Numerical (Panel) Methods 206 -- 9.1 Basic Formulation 206 -- 9.2 The Boundary Conditions 207 -- 9.3 Physical Considerations 209 -- 9.4 Reduction of the Problem to a Set of Linear Algebraic Equations 213 -- 9.5 Aerodynamic Loads 216 -- 9.6 Preliminary Considerations, Prior to Establishing Numerical Solutions 217 -- 9.7 Steps toward Constructing a Numerical Solution 220 -- 9.8 Example: Solution of Thin Airfoil with the Lumped-Vortex Element 222 -- 9.9 Accounting for Effects of Compressibility and Viscosity 226 -- 10 Singularity Elements and Influence Coefficients 230 -- 10.1 Two-Dimensional Point Singularity Elements 230 -- 10.1.1 Two-Dimensional Point Source 230 -- 10.1.2 Two-Dimensional Point Doublet 231 -- 10.1.3 Two-Dimensional Point Vortex 231 -- 10.2 Two-Dimensional Constant-Strength Singularity Elements 232 -- 10.2.1 Constant-Strength Source Distribution 233 -- 10.2.2 Constant-Strength Doublet Distribution 235 -- 10.2.3 Constant-Strength Vortex Distribution 236 -- 10.3 Two-Dimensional Linear-Strength Singularity Elements 237 -- 10.3.1 Linear Source Distribution 238 -- 10.3.2 Linear Doublet Distribution 239 -- 10.3.3 Linear Vortex Distribution 241 -- 10.3.4 Quadratic Doublet Distribution 242 -- 10.4 Three-Dimensional Constant-Strength Singularity Elements 244 -- 10.4.1 Quadrilateral Source 245 -- 10.4.2 Quadrilateral Doublet 247 -- 10.4.3 Constant Doublet Panel Equivalence to Vortex Ring 250 -- 10.4.4 Comparison of Near and Far Field Formulas 251 -- 10.4.5 Constant-Strength Vortex Line Segment 251 -- 10.4.6 Vortex Ring 255 -- 10.4.7 Horseshoe Vortex 256 -- 10.5 Three-Dimensional Higher Order Elements 258 -- 11 Two-Dimensional Numerical Solutions 262 -- 11.1 Point Singularity Solutions 262 -- 11.1.1 Discrete Vortex Method 263 -- 11.1.2 Discrete Source Method 272 -- 11.2 Constant-Strength Singularity Solutions (Using the Neumann B.C.) 276 -- 11.2.1 Constant Strength Source Method 276 -- 11.2.2 Constant-Strength Doublet Method 280 -- 11.2.3 Constant-Strength Vortex Method 284 -- 11.3 Constant-Potential (Dirichlet Boundary Condition) Methods 288 -- 11.3.1 Combined Source and Doublet Method 290 -- 11.3.2 Constant-Strength Doublet Method 294 -- 11.4 Linearly Varying Singularity Strength Methods (Using the Neumann B.C.) 298 -- 11.4.1 Linear-Strength Source Method 299 -- 11.4.2 Linear-Strength Vortex Method 303 -- 11.5 Linearly Varying Singularity Strength Methods (Using the Dirichlet B.C.) 306 -- 11.5.1 Linear Source/Doublet Method 306 -- 11.5.2 Linear Doublet Method 312 -- 11.6 Methods Based on Quadratic Doublet Distribution (Using the Dirichlet B.C.) 315 -- 11.6.1 Linear Source/Quadratic Doublet Method 315 -- 11.6.2 Quadratic Doublet Method 320 -- 11.7 Some Conclusions about Panel Methods 323 -- 12 Three-Dimensional Numerical Solutions 331 -- 12.1 Lifting-Line Solution by Horseshoe Elements 331 -- 12.2 Modeling of Symmetry and Reflections from Solid Boundaries 338 -- 12.3 Lifting-Surface Solution by Vortex Ring Elements 340 -- 12.4 Introduction to Panel Codes: A Brief History 351 -- 12.5 First-Order Potential-Based Panel Methods 353 -- 12.6 Higher Order Panel Methods 358 -- 12.7 Sample Solutions with Panel Codes 360 -- 13 Unsteady Incompressible Potential Flow 369 -- 13.1 Formulation of the Problem and Choice of Coordinates 369 -- 13.2 Method of Solution 373 -- 13.3 Additional Physical Considerations 375 -- 13.4 Computation of Pressures 376 -- 13.5 Examples for the Unsteady Boundary Condition 377 -- 13.6 Summary of Solution Methodology 380 -- 13.7 Sudden Acceleration of a Flat Plate 381 -- 13.7.1 The Added Mass 385 -- 13.8 Unsteady Motion of a Two-Dimensional Thin Airfoil 387 -- 13.8.1 Kinematics 388 -- 13.8.2 Wake Model 389 -- 13.8.3 Solution by the Time-Stepping Method 391 -- 13.8.4 Fluid Dynamic Loads 394 -- 13.9 Unsteady Motion of a Slender Wing 400 -- 13.9.1 Kinematics 401 -- 13.9.2 Solution of the Flow over the Unsteady Slender Wing 401 -- 13.10 Algorithm for Unsteady Airfoil Using the Lumped-Vortex Element 407 -- 13.11 Some Remarks about the Unsteady Kutta Condition 416 -- 13.12 Unsteady Lifting-Surface Solution by Vortex Ring Elements 419 -- 13.13 Unsteady Panel Methods 433 -- 14 The Laminar Boundary Layer 448 -- 14.1 The Concept of the Boundary Layer 448 -- 14.2 Boundary Layer on a Curved Surface 452 -- 14.3 Similar Solutions to the Boundary Layer Equations 457 -- 14.4 The von Karman Integral Momentum Equation 463 -- 14.5 Solutions Using the von Karman Integral Equation 467 -- 14.5.1 Approximate Polynomial Solution 468 -- 14.5.2 The Correlation Method of Thwaites 469 -- 14.6 Weak Interactions, the Goldstein Singularity, and Wakes 471 -- 14.7 Two-Equation Integral Boundary Layer Method 473 -- 14.8 Viscous-Inviscid Interaction Method 475