Advanced Fluid Mechanics (eBook)

Front Cover 1
Advanced Fluid Mechanics 4
Copyright Page 5
Table of Contents 8
Preface 15
Chapter 1 Fundamentals 18
1.1 Introduction 18
1.2 Velocity, Acceleration, and the Material Derivative 21
1.3 The Local Continuity Equation 22
1.4 Path Lines, Streamlines, and Stream Functions 24
1.4.1 Lagrange’s Stream Function for Two-Dimensional Flows 24
1.4.2 Stream Functions for Three-Dimensional Flows, Including Stokes Stream Function 28
1.5 Newton's Momentum Equation 30
1.6 Stress 31
1.7 Rates of Deformation 38
1.8 Constitutive Relations 41
1.9 Equations for Newtonian Fluids 44
1.10 Boundary Conditions 45
1.11 Vorticity and Circulation 46
1.12 The Vorticity Equation 51
1.13 The Work-Energy Equation 53
1.14 The First Law of Thermodynamics 54
1.15 Dimensionless Parameters 56
1.16 Non-Newtonian Fluids 57
1.17 Moving Coordinate Systems 58
Problems 60
Chapter 2 Inviscid Irrotational Flows 63
2.1 Inviscid Flows 63
2.2 Irrotational Flows and the Velocity Potential 64
2.2.1 Intersection of Velocity Potential Lines and Streamlines in Two Dimensions 66
2.2.2 Basic Two-Dimensional Irrotational Flows 68
2.2.3 Hele-Shaw Flows 74
2.2.4 Basic Three-Dimensional Irrotational Flows 75
2.2.5 Superposition and the Method of Images 76
2.2.6 Vortices Near Walls 78
2.2.7 Rankine Half-Body 82
2.2.8 Rankine Oval 84
2.2.9 Circular Cylinder or Sphere in a Uniform Stream 85
2.3 Singularity Distribution Methods 86
2.3.1 Two- and Three-Dimensional Slender Body Theory 86
2.3.2 Panel Methods 88
2.4 Forces Acting on a Translating Sphere 94
2.5 Added Mass and the Lagally Theorem 96
2.6 Theorems for Irrotational Flow 98
2.6.1 Mean Value and Maximum Modulus Theorems 98
2.6.2 Maximum-Minimum Potential Theorem 98
2.6.3 Maximum-Minimum Speed Theorem 99
2.6.4 Kelvin’s Minimum Kinetic Energy Theorem 99
2.6.5 Maximum Kinetic Energy Theorem 100
2.6.6 Uniqueness Theorem 101
2.6.7 Kelvin’s Persistence of Circulation Theorem 101
2.6.8 Weiss and Butler Sphere Theorems 101
Problems 102
Chapter 3 Irrotational Two-Dimensional Flows 104
3.1 Complex Variable Theory Applied to Two-Dimensional Irrotational Flow 104
3.2 Flow Past a Circular Cylinder with Circulation 108
3.3 Flow Past an Elliptical Cylinder with Circulation 110
3.4 The Joukowski Airfoil 112
3.5 Kármán-Trefftz and Jones-McWilliams Airfoils 115
3.6 NACA Airfoils 116
3.7 Lifting Line Theory 118
3.8 Kármán Vortex Street 120
3.9 Conformal Mapping and the Schwarz-Christoffel Transformation 125
3.10 Cavity Flows 127
3.11 Added Mass and Forces and Moments for Two-Dimensional Bodies 129
Problems 131
Chapter 4 Surface and Interfacial Waves 135
4.1 Linearized Free Surface Wave Theory 135
4.1.1 Infinitely Long Channel 135
4.1.2 Waves in a Container of Finite Size 139
4.2 Group Velocity 140
4.3 Waves at the Interface of Two Dissimilar Fluids 142
4.4 Waves in an Accelerating Container 144
4.5 Stability of a Round Jet 145
4.6 Local Surface Disturbance on a Large Body of Fluid—Kelvin's Ship Wave 147
4.7 Shallow-Depth Free Surface Waves—Cnoidal and Solitary Waves 149
4.8 Ray Theory of Gravity Waves for Nonuniform Depths 153
Problems 156
Chapter 5 Exact Solutions of the Navier-Stokes Equations 157
5.1 Solutions to the Steady-State Navier-Stokes Equations When Convective Acceleration Is Absent 157
5.1.1 Two-Dimensional Flow Between Parallel Plates 158
5.1.2 Poiseuille Flow in a Rectangular Conduit 159
5.1.3 Poiseuille Flow in a Round Conduit or Annulus 161
5.1.4 Poiseuille Flow in Conduits of Arbitrarily Shaped Cross-Section 162
5.1.5 Couette Flow Between Concentric Circular Cylinders 164
5.2 Unsteady Flows When Convective Acceleration Is Absent 164
5.2.1 Impulsive Motion of a Plate—Stokes’s First Problem 164
5.2.2 Oscillation of a Plate—Stokes’s Second Problem 166
5.3 Other Unsteady Flows When Convective Acceleration Is Absent 169
5.3.1 Impulsive Plane Poiseuille and Couette Flows 169
5.3.2 Impulsive Circular Couette Flow 170
5.4 Steady Flows When Convective Acceleration Is Present 171
5.4.1 Plane Stagnation Line Flow 172
5.4.2 Three-Dimensional Axisymmetric Stagnation Point Flow 175
5.4.3 Flow into Convergent or Divergent Channels 175
5.4.4 Flow in a Spiral Channel 179
5.4.5 Flow Due to a Round Laminar Jet 180
5.4.6 Flow Due to a Rotating Disk 182
Problems 185
Chapter 6 The Boundary Layer Approximation 187
6.1 Introduction to Boundary Layers 187
6.2 The Boundary Layer Equations 188
6.3 Boundary Layer Thickness 191
6.4 Falkner-Skan Solutions for Flow Past a Wedge 192
6.4.1 Boundary Layer on a Flat Plate 193
6.4.2 Stagnation Point Boundary Layer Flow 195
6.4.3 General Case 195
6.5 The Integral Form of the Boundary Layer Equations 196
6.6 Axisymmetric Laminar Jet 199
6.7 Flow Separation 200
6.8 Transformations for Nonsimilar Boundary Layer Solutions 201
6.8.1 Falkner Transformation 202
6.8.2 von Mises Transformation 203
6.8.3 Combined Mises-Falkner Transformation 204
6.8.4 Crocco’s Transformation 204
6.8.5 Mangler’s Transformation for Bodies of Revolution 205
6.9 Boundary Layers in Rotating Flows 205
Problems 208
Chapter 7 Thermal Effects 210
7.1 Thermal Boundary Layers 210
7.2 Forced Convection on a Horizontal Flat Plate 212
7.2.1 Falkner-Skan Wedge Thermal Boundary Layer 212
7.2.2 Isothermal Flat Plate 212
7.2.3 Flat Plate with Constant Heat Flux 213
7.3 The Integral Method for Thermal Convection 214
7.3.1 Flat Plate with a Constant Temperature Region 215
7.3.2 Flat Plate with a Constant Heat Flux 216
7.4 Heat Transfer Near the Stagnation Point of an Isothermal Cylinder 217
7.5 Natural Convection on an Isothermal Vertical Plate 218
7.6 Natural Convection on a Vertical Plate with Uniform Heat Flux 219
7.7 Thermal Boundary Layer on Inclined Flat Plates 220
7.8 Integral Method for Natural Convection on an Isothermal Vertical Plate 220
7.9 Temperature Distribution in an Axisymmetric Jet 221
Problems 222
Chapter 8 Low Reynolds Number Flows 224
8.1 Stokes Approximation 224
8.2 Slow Steady Flow Past a Solid Sphere 226
8.3 Slow Steady Flow Past a Liquid Sphere 227
8.4 Flow Due to a Sphere Undergoing Simple Harmonic Translation 229
8.5 General Translational Motion of a Sphere 231
8.6 Oseen's Approximation for Slow Viscous Flow 231
8.7 Resolution of the Stokes/Whitehead Paradoxes 233
Problems 234
Chapter 9 Flow Stability 235
9.1 Linear Stability Theory of Fluid Flows 235
9.2 Thermal Instability in a Viscous Fluid—Rayleigh-Bénard Convection 236
9.3 Stability of Flow Between Rotating Circular Cylinders—Couette-Taylor Instability 243
9.4 Stability of Plane Flows 245
Problems 248
Chapter 10 Turbulent Flows 250
10.1 The Why and How of Turbulence 250
10.2 Statistical Approach—One-Point Averaging 251
10.3 Zero-Equation Turbulent Models 257
10.4 One-Equation Turbulent Models 259
10.5 Two-Equation Turbulent Models 259
10.6 Stress-Equation Models 260
10.7 Equations of Motion in Fourier Space 261
10.8 Quantum Theory Models 263
10.9 Large Eddy Models 265
10.10 Phenomenological Observations 266
10.11 Conclusions 267
Chapter 11 Computational Methods—Ordinary Differential Equations 268
11.1 Introduction 268
11.2 Numerical Calculus 279
11.3 Numerical Integration of Ordinary Differential Equations 284
11.4 The Finite Element Method 289
11.5 Linear Stability Problems—Invariant Imbedding and Riccati Methods 291
11.6 Errors, Accuracy, and Stiff Systems 296
Problems 298
Chapter 12 Multidimensional Computational Methods 300
12.1 Introduction 300
12.2 Relaxation Methods 301
12.3 Surface Singularities 305
12.4 One-Step Methods 314
12.4.1 Forward Time, Centered Space—Explicit 314
12.4.2 Dufort-Frankel Method—Explicit 315
12.4.3 Crank-Nicholson Method—Implicit 315
12.4.4 Boundary Layer Equations—Crank-Nicholson 316
12.4.5 Boundary Layer Equation—Hybrid Method 320
12.4.6 Richardson Extrapolation 320
12.4.7 Further Choices for Dealing with Nonlinearities 321
12.4.8 Upwind Differencing for Convective Acceleration Terms 321
12.5 Multistep, or Alternating Direction, Methods 322
12.5.1 Alternating Direction Explicit (ADE) Method 322
12.5.2 Alternating Direction Implicit (ADI) Method 322
12.6 Method of Characteristics 323
12.7 Leapfrog Method—Explicit 326
12.8 Lax-Wendroff Method—Explicit 327
12.9 MacCormack's Methods 328
12.9.1 MacCormack’s Explicit Method 329
12.9.2 MacCormack’s Implicit Method 329
12.10 Discrete Vortex Methods (DVM) 330
12.11 Cloud in Cell Method (CIC) 331
Problems 332
Appendix 335
A.1 Vector Differential Calculus 335
A.2 Vector Integral Calculus 337
A.3 Fourier Series and Integrals 340
A.4 Solution of Ordinary Differential Equations 342
A.4.1 Method of Frobenius 342
A.4.2 Mathieu Equations 343
A.4.3 Finding Eigenvalues—The Riccati Method 344
A.5 Index Notation 346
A.6 Tensors in Cartesian Coordinates 350
A.7 Tensors in Orthogonal Curvilinear Coordinates 354
A.7.1 Cylindrical Polar Coordinates 356
A.7.2 Spherical Polar Coordinates 357
A.8 Tensors in General Coordinates 358
References 363
Index 373