ELEMENTS OF GASDYNAMICS
Part I Basic Concepts and Equations of Fluid Dynamics 1 1 Introduction to the Fluid Model 3 11 The Fluid State 4 12 Description of the Flow-Field 5 13 Volume Forces and Surface Forces 7 14 Relative Motion Near a Point 10 15 Stress–Strain Relations 13 2 Equations of Fluid Flows 15 21 The Transport Theorem 16 22 The Material Derivative 18 23 The Law of Conservation of Mass 18 24 Equation of Motion 19 25 The Energy Equation 19 26 The Equation of Vorticity 22 27 The Incompressible Fluid 23 28 Boundary Conditions 24 29 A Program for Analysis of the Governing Equations 25 3 Hamiltonian Formulation of Fluid-Flow Problems 27 31 Hamiltonian Dynamics of Continuous Systems 28 32 Three-Dimensional Incompressible Flows 32 33 Two-Dimensional Incompressible Flows 35 4 Surface Tension Effects 39 41 Shape of the Interface between Two Fluids 39 42 Capillary Rises in Liquids 41 Part II Dynamics of Incompressible Fluid Flows 45 5 Fluid Kinematics and Dynamics 47 51 Stream Function 47 52 Equations of Motion 50 53 Integrals of Motion 50 54 CapillaryWaves on a Spherical Drop 51 55 Cavitation 54 56 Rates of Change of Material Integrals 55 57 The Kelvin Circulation Theorem 57 58 The Irrotational Flow 58 59 Simple-Flow Patterns 62 (i) The Source Flow 62 (ii) The Doublet Flow 63 (iii) The Vortex Flow 66 (iv) Doublet in a Uniform Stream 66 (v) Uniform Flow Past a Circular Cylinder with Circulation 67 6 The Complex-Variable Method 71 61 The Complex Potential 71 62 Conformal Mapping of Flows 74 63 Hydrodynamic Images 82 64 Principles of Free-Streamline Flow 84 (i) Schwarz-Christoffel Transformation 84 (ii) Hodograph Method 93 7 Three-Dimensional Irrotational Flows 99 71 Special Singular Solutions 99 (i) The Source Flow 99 (ii) The Doublet Flow 101 72 d’Alembert’s Paradox 104 73 Image of a Source in a Sphere 105 74 Flow Past an Arbitrary Body 107 75 Unsteady Flows 109 76 Renormalized (or Added) Mass of Bodies Moving through a Fluid 111 8 Vortex Flows 115 81 Vortex Tubes 115 82 Induced Velocity Field 117 83 Biot-Savart’s Law 117 84 von Kármán Vortex Street 121 85 Vortex Ring 124 86 Hill’s Spherical Vortex 129 87 Vortex Sheet 131 88 Vortex Breakdown: Brooke Benjamin’s Theory 135 9 Rotating Flows 143 91 Governing Equations and Elementary Results 143 92 Taylor-Proudman Theorem 144 93 Propagation of InertialWaves in a Rotating Fluid 146 94 Plane InertialWaves 147 95 ForcedWavemotion in a Rotating Fluid 150 (i) The Elliptic Case 153 (ii) The Hyperbolic Case 154 96 Slow Motion along the Axis of Rotation 155 97 RossbyWaves 160 10 Water Waves 167 101 Governing Equations 168 102 A Variational Principle for SurfaceWaves 169 103 WaterWaves in a Semi-Infinite Fluid 171 104 WaterWaves in a Fluid Layer of Finite Depth 172 105 Shallow-WaterWaves 174 (i) Analogy with Gas Dynamics 175 (ii) Breaking ofWaves 176 106 Water Waves Generated by an Initial Displacement over a Localized Region 176 107 Waves on a Steady Stream 182 (i) One-Dimensional GravityWaves 183 (ii) One-Dimensional Capillary-GravityWaves 184 (iii) ShipWaves 185 108 GravityWaves in a Rotating Fluid 188 109 Theory of Tides 193 1010 Hydraulic Jump 195 (i) Tidal Bores 195 (ii) The Dam-Break Problem 199 1011 Nonlinear Shallow-WaterWaves 202 (i) SolitaryWaves 206 (ii) Periodic CnoidalWaves 208 (iii) Interacting SolitaryWaves 214 (iv) StokesWaves 219 (v) Modulational Instability and Envelope Solutions 220 1012 Nonlinear Capillary-GravityWaves 230 (i) Resonant Three-Wave Interactions 230 (ii) Second-Harmonic Resonance 235 11 Applications to Aerodynamics 241 111 Airfoil Theory: Method of Complex Variables 242 (i) Force and Moments on an Arbitrary Body 242 (ii) Flow Past an Arbitrary Cylinder 245 (iii) Flow Around a Flat Plate 248 (iv) Flow Past an Airfoil 250 (v) The Joukowski Transformation 253 112 Thin Airfoil Theory 259 (i) Thickness Problem 262 (ii) Camber Problem 264 (iii) Flat Plate at an Angle of Attack 269 (iv) Combined Aerodynamic Characteristics 271 (v) The Leading-Edge Problem of a Thin Airfoil 271 113 Slender-Body Theory 275 114 Prandtl’s Lifting-Line Theory for Wings 277 115 Oscillating Thin-Airfoil Problem: Theodorsen’s Theory 282 Part III Dynamics of Compressible Fluid Flows 297 12 Review of Thermodynamics 299 121 Thermodynamic System and Variables of State 299 122 The First Law of Thermodynamics and Reversible and Irreversible Processes 300 123 The Second Law of Thermodynamics 303 124 Entropy 304 125 Liquid and Gaseous Phases 307 13 Isentropic Fluid Flows 309 131 Applications of Thermodynamics to Fluid Flows 309 132 Linear SoundWave Propagation 310 133 The Energy Equation 310 134 Stream-Tube Area and Flow Velocity Relations 312 14 Potential Flows 317 141 Governing Equations 317 142 Streamline Coordinates 319 143 Conical Flows: Prandtl-Meyer Flow 320 144 Small Perturbation Theory 324 145 Characteristics 326 (i) Compatibility Conditions in Streamline Coordinates 328 (ii) A Singular-Perturbation Problem for Hyperbolic Systems 331 15 Nonlinear Theory of Plane Sound Waves 343 151 Riemann Invariants 343 152 SimpleWave Solutions 344 153 Nonlinear Propagation of a SoundWave 352 154 Nonlinear Resonant Three-Wave Interactions of Sound Waves 355 155 Burgers Equation 361 16 Shock Waves 371 161 The Normal ShockWave 371 162 The Oblique ShockWave 384 163 Blast Waves: Taylor’s Self-similarity and Sedov’s Exact Solution 387 17 The Hodograph Method 393 171 The Hodograph Transformation of Potential Flow Equations 393 172 The Chaplygin Equation 394 173 The Tangent-Gas Approximation 396 174 The Lost Solution 401 175 The Limit Line 402 18 Applications to Aerodynamics 411 181 Thin Airfoil Theory 411 (i) Thin Airfoil in Linearized Supersonic Flows 411 (ii) Far-Field Behavior of Supersonic Flow Past a Thin Airfoil 414 (iii) Thin Airfoil in Transonic Flows 417 182 Slender Bodies of Revolution 420 183 Oscillating Thin Airfoil in Subsonic Flows: Possio’s Theory 427 184 Oscillating Thin Airfoils in Supersonic Flows: Stewartson’s Theory 435 Part IV Dynamics of Viscous Fluid Flows 439 19 Exact Solutions to Equations of Viscous Fluid Flows 441 191 Channel Flows 442 192 Decay of a Line Vortex: The Lamb-Oseen Vortex 443 193 Line Vortex in a Uniform Stream 446 194 Diffusion of a Localized Vorticity Distribution 446 195 Burgers Vortex 451 196 Flow Due to a Suddenly Accelerated Plane 453 197 The Round Laminar Jet: Landau-Squire Solution 456 198 Ekman Layer at a Free Surface in a Rotating Fluid 459 199 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution 462 1910 Shock Structure: Becker’s Solution 464 1911 Couette Flow of a Gas 467 20 Flows at Low Reynolds Numbers 469 201 Dimensional Analysis 469 202 Stokes’ Flow Past a Rigid Sphere: Stokes’ Formula 470 203 Stokes’ Flow Past a Spherical Drop 474 204 Stokes’ Flow Past a Rigid Circular Cylinder: Stokes’ Paradox 478 205 Oseen’s Flow Past a Rigid Sphere 479 206 Oseen’s Approximation for Periodically Oscillating Wakes 483 21 Flows at High Reynolds Numbers 489 211 Prandtl’s Boundary-Layer Concept 489 212 The Method of Matched Asymptotic Expansions 490 213 Location and Nature of the Boundary Layers 497 214 Incompressible Flow Past a Flat Plate 500 (i) The Outer Expansion 501 (ii) The Inner Expansion 502 (iii) Flow Due to Displacement Thickness 507 215 Separation of Flow in a Boundary Layer: Landau’s Theory 509 216 Boundary Layers in Compressible Flows 512 (i) Crocco’s Integral 514 (ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation 516 217 Flow in a Mixing Layer between Two Parallel Streams 517 (i) Geometrical Characteristics of the Mixing Flow 520 218 Narrow Jet: Bickley’s Solution 521 219 Wakes 524 2110 Periodic Boundary Layer Flows 524 22 Jeffrey-Hamel Flow 529 221 The Exact Solution 529 (i) Only ??1 Is Real and Positive 531 (ii) ??1, ??2, and ??3 Are Real and Distinct 532 222 Flows at Low Reynolds Numbers 535 223 Flows at High Reynolds Numbers 541 References 545