Basic Fluid Mechanics

Fifth Edition

by

David С. Wilcox

feDCWjl Is Industries SP

Contents

1 Introduction 1 1.1 Aiming for the Stars 1 1.2 Dimensions and Units 6

1.2.1 Independent Dimensions 6 1.2.2 Common Systems of Units 6 1.2.3 Converting Dimensional Quantities 10

1.3 Definition of a Fluid 12 1.3.1 Simplistic Definition of a Fluid 12 1.3.2 Rigorous Definition of a Fluid 12

1.4 Continuum Approximation 13 1.5 Microscopic and Macroscopic Views 15

1.5.1 Density 16 1.5.2 Temperature 17 1.5.3 Pressure 17 1.5.4 Vapor Pressure 19

1.6 Thermodynamic Properties of Gases 21 1.7 Compressibility 22 1.8 Surface Tension 25 1.9 Viscosity 30 1.10 Examples of Viscosity Dominated Flows 32

1.10.1 CouetteFlow 32 1.10.2 Hagen-Poiseuille Flow 34

1.11 Fluid-Flow Regimes 35 1.12 Brief History of Fluid Mechanics 36

Chapter Summary 38 Problems 39

Dimensional Analysis 53 2.1 Basic Premises 54 2.2 Buckingham П Theorem 57

2.2.1 Finding Dimensionless Groupings 57 2.2.2 Interpretation of Results 62 2.2.3 Nonuniqueness of Results 65 2.2.4 E. S. Taylor's Method 66

2.3 Common Dimensionless Groupings 68 2.4 Dynamic Similitude 69

2.4.1 Similitude for an Airplane 70

Vll

Vlll CONTENTS

2.4.2 Similitude for a Ship 72 Chapter Summary 76 Problems 77

3 Effects of Gravity on Pressure 97 3.1 Pressure on an Infinitesimal Element 97 3.2 The Hydrostatic Relation 100 3.3 Atmospheric Pressure Variation 102 3.4 Manometry 105 3.5 Hydrostatic Forces on Plane Surfaces 107 3.6 Hydrostatic Forces on Curved Surfaces 117 3.7 Equivalent Pressure Fields and Superposition 121 3.8 Buoyancy 123

Chapter Summary 126 Problems 127

4 Kinematics 147 4.1 Eulerian Versus Lagrangian Description 147 4.2 Steady and Unsteady Flows 153 4.3 Vorticity and Circulation 154 4.4 Streamlines, Streaklines and Pathlines 159 4.5 Extensive and Intensive Properties 163 4.6 Surface Fluxes 164 4.7 Reynolds Transport Theorem 166

4.7.1 Finite-Sized Control Volume 166 4.7.2 Differential-Sized Control Volume 170

4.8 Vector Calculus and Multiple Integrals 173 Chapter Summary 176 Problems 177

5 Mass and Momentum Principles 187 5.1 Mass Principle 188

5.1.1 Integral Form 188 5.1.2 Differential Form 189

5.2 Momentum Principle 190 5.2.1 Integral Form 190 5.2.2 Differential Form 191

5.3 Summary of the Mass and Momentum Principles 192 5.4 Mass Principle for Incompressible Flows 193 5.5 Euler's Equation 194

5.5.1 Rotating Tank 196 5.5.2 Galilean Invariance of Euler's Equation 198

5.6 Bernoulli's Equation 200 5.7 Velocity-Measurement Techniques 205

5.7.1 Stagnation Points 205 5.7.2 PitotTube 205 5.7.3 Pitot-Static Tube 207 Chapter Summary 208 Problems 209

CONTENTS ix

6 Control-Volume Method 219 6.1 Historical Perspective 219 6.2 Preliminaries 221

6.2.1 Overview of the Method 221 6.2.2 Useful Control-Volume Theorem 225 6.2.3 Fluid Statics Revisited 226

6.3 Mass Principle Applications 227 6.3.1 Steady Flow in a Pipe 228 6.3.2 Sphere Falling in a Cylinder 230 6.3.3 Deforming Control Volume 234

6.4 Mass and Momentum Principle Applications 236 6.4.1 Channel Flow With Suction 236 6.4.2 Indirect Force Computation—Rotational Flow 240 6.4.3 Indirect Force Computation—Irrotational Flow 243 6.4.4 Reaction Forces 245 6.4.5 Nonuniform Cross-Sectional Velocity Profiles 248

6.5 Accelerating Control Volume 250 6.5.1 Inertial and Noninertial Coordinate Frames 250 6.5.2 Leaky Wheelbarrow 252 Chapter Summary 256 Problems 257

7 Energy Principle 289 7.1 Thermodynamics 289

7.1.1 Fundamental Concepts 290 7.1.2 The First Law of Thermodynamics 293 7.1.3 The Second Law of Thermodynamics 293 7.1.4 Combined First and Second Laws 294 7.1.5 The First Law for a Moving Fluid 295

7.2 Integral Form of the Energy Equation 296 7.3 Differential Form for Adiabatic, Inviscid Flow 299 7.4 Entropy Generation 301 7.5 Relation to Bernoulli's Equation 302

7.5.1 Flow of a Liquid 302 7.5.2 Flow of a Gas 303

7.6 Approximate Form of the Energy Equation 305 7.7 Flow in Pipes 310

7.7.1 Friction and Head Loss in Straight Pipes 310 7.7.2 Minor Losses and Non-Circular Cross Sections 315 7.7.3 Multiple-Pipe Systems 327

7.8 Open-Channel Flow 330 7.8.1 Uniform Flow 334 7.8.2 The Equations of Chezy and Manning 336 7.8.3 Surface Wave Speed and Flow Classification 338 7.8.4 Specific Energy 343 7.8.5 Hydraulic Jump 346 Chapter Summary 348 Problems 350

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8 One-Dimensional Compressible Flow 373 8.1 Classification 374 8.2 Thermodynamic Properties of Air 375 8.3 Speed of Sound 376 8.4 Subsonic Versus Supersonic Flow 379 8.5 Analysis of a Streamtube 380 8.6 Isentropic Flow 384 8.7 Normal Shock Waves 386 8.8 Directionality 388 8.9 Laval Nozzle 390

Chapter Summary 397 Problems 398

9 Turbomachinery 409 9.1 Classification 409 9.2 Angular-Momentum Principle 411

9.2.1 Derivation 411 9.2.2 Lawn Sprinkler 412

9.3 The Euler Turbomachine Equations 414 9.4 Application to a Centrifugal Pump 420

9.4.1 Velocity Triangles 420 9.4.2 Computing Pump Properties 423 9.4.3 Efficiency 425 9.4.4 Performance Curves 426 9.4.5 Matching Pumps to System Properties 427 9.4.6 Cavitation 430

9.5 Dimensional Considerations 430 9.5.1 Primary Dimensionless Groupings 431 9.5.2 Specific Speed 433

9.6 Common Turbomachines 434 9.6.1 Pumps and Compressors 434 9.6.2 Reaction and Impulse Turbines 437 Chapter Summary 441 Problems 443

10 Vorticity, Viscosity, Lift and Drag 451 10.1 The Vortex Force 452 10.2 Helmholtz's Theorem and d'Alembert's Paradox 453 10.3 Boundary Conditions at a Solid Boundary 455 10.4 Viscous Effects and Vorticity Generation 456 10.5 Diffusion of Vorticity 459 10.6 Boundary Layers 459 10.7 Lift and Drag of Common Objects 460

Chapter Summary 466 Problems 467

CONTENTS xi

11 Potential Flow 475 11.1 Differential Equations of Motion 476 11.2 Mathematical Foundation 477

11.2.1 Cylindrical Coordinates 478 11.2.2 Velocity-Potential Representation 479 11.2.3 Streamfunction Representation 480

11.3 Streamlines and Equipotential Lines 481 11.4 Bernoulli's Equation 484 11.5 Fundamental Solutions 486

11.5.1 Uniform Flow 486 11.5.2 Potential Vortex 487 11.5.3 Source and Sink 489 11.5.4 Doublet 490 11.5.5 Comments on the Use of Fundamental Solutions 494

11.6 Flow Past a Cylinder 496 11.6.1 The Basic Solution 496 11.6.2 Force on the Cylinder 499 11.6.3 Adding a Potential Vortex 502 11.6.4 Force Computation Redone 504

11.7 Other Interesting Solutions 508 11.7.1 Accelerating Cylinder 508 11.7.2 Rankine Oval 511 11.7.3 Wedge 514 11.7.4 Stagnation-Point Flow 516 11.7.5 The Method of Images 518

11.8 Airfoil Flow 523 11.8.1 The Kutta Condition 523 11.8.2 Source and Vortex Sheets 524 11.8.3 Linear Airfoil Theory 527

11.9 Computational Fluid Dynamics 531 11.9.1 Discretization Approximations 533 11.9.2 Relaxation Methods 536 11.9.3 Flow Past a Vertical Plate 537 11.9.4 Solution Convergence and Grid Sensitivity 539 11.9.5 Richardson Extrapolation 540 11.9.6 Grid Convergence Index 542 Chapter Summary 543 Problems 545

12 Viscous Effects 563 12.1 Molecular Transport of Momentum 564 12.2 Kinematics of a Fluid Particle 567

12.2.1 Basic Formulation 567 12.2.2 Volume Distortion 569 12.2.3 Angular Distortion and Rotation 570 12.2.4 Strain-Rate Tensor and Vorticity Vector 571 12.2.5 Incompressibility and Irrotationality 573

12.3 The Viscous Stress Tensor 576 12.4 Integral Form of the Momentum Principle 580

xii CONTENTS

12.4.1 Derivation 580 12.4.2 Control-Volume Example 581

12.5 Navier's Equation 584 12.5.1 Derivation 584 12.5.2 Symmetry of the Stress Tensor 587 12.5.3 Understanding Surface-Force Balances 588

12.6 Stokes' Postulate 590 12.7 Navier-Stokes Equation 593 12.8 The Vorticity Equation 595

12.8.1 Interaction With Forces on a Fluid Particle 596 12.8.2 Vortex Stretching 596

12.9 Computational Fluid Dynamics 599 12.9.1 Explicit Time-Marching Methods 600 12.9.2 von Neumann Stability Analysis 601 12.9.3 MacCormack's Method 603 12.9.4 Flow Over a Plate with Uniform Suction 605 Chapter Summary 610 Problems 611

13 Navier-Stokes Solutions 621 13.1 Couette-Poiseuille Flow 622

13.1.1 Couette-Poiseuille Flow Solution 623 13.1.2 Lubrication Theory 627

13.2 Channel Flow 629 13.3 Pipe Flow 632 13.4 Stokes' First Problem 640 13.5 Stokes' Second Problem 648 13.6 Stagnation-Point Flow 651 13.7 Computational Fluid Dynamics 658

13.7.1 Implicit Time-Marching Methods 658 13.7.2 Crank-Nicolson Method 659 13.7.3 Thomas' Algorithm 661 13.7.4 Stagnation-Point Flow Revisited 663 Chapter Summary 666 Problems 667

14 Boundary Layers 681 14.1 Importance of Friction 682 14.2 Prandtl's Boundary Layer 685

14.2.1 High Reynolds Number Flow Near a Surface 686 14.2.2 Boundary-Layer Equations 687 14.2.3 Momentum-Integral Equation 693 14.2.4 Blasius Solution 698 14.2.5 Effects of Pressure Gradient 701 14.2.6 Boundary-Layer Separation 706

14.3 Turbulence 710 14.3.1 Laminar, Transitional and Turbulent Flow 710 14.3.2 General Properties of Turbulence 715 14.3.3 Reynolds Averaging 718

CONTENTS xiii

14.3.4 Reynolds-Averaged Equations 719 14.3.5 Turbulent Boundary-Layer Structure 721 14.3.6 Prandtl's Mixing-Length Hypothesis 727 14.3.7 Modern Turbulence Theories 730

14.4 Computational Fluid Dynamics 734 14.4.1 Parabolic Marching Methods 734 14.4.2 Stretched Finite-Difference Grids 734 14.4.3 Blottner's Variable-Grid Method 738 14.4.4 Predicting Separation on a Rankine Oval 740 Chapter Summary 744 Problems 746

15 Viscous and 2-D Compressible Flow 767 15.1 The Energy Principle 768

15.1.1 Integral Form 768 15.1.2 Constitutive Relation for the Heat-Flux Vector 769 15.1.3 Differential Form 770 15.1.4 Entropy Generation Revisited 774 15.1.5 Surface Boundary Conditions 774

15.2 Fanno Flow 777 15.2.1 Fanno-Flow Equations 777 15.2.2 Solution of the Fanno-Flow Equations 780 15.2.3 Subsonic Inlet 781 15.2.4 Supersonic Inlet 783 15.2.5 Mollier Diagram 785

15.3 Rayleigh Flow 787 15.3.1 Rayleigh-Flow Equations 787 15.3.2 Solution of the Rayleigh-Flow Equations 788 15.3.3 Subsonic Inlet With Heating 788 15.3.4 Supersonic Inlet With Heating 790 15.3.5 Mollier Diagram 791

15.4 Oblique Shock Waves 793 15.4.1 Oblique-Shock Relations 794 15.4.2 Attached and Detached Shocks 800 15.4.3 Shock Reflecting from a Solid Boundary 803

15.5 Prandtl-Meyer Expansion 804 15.5.1 Prandtl-Meyer Function 805 15.5.2 Shock Reflecting from a Solid Boundary Revisited 807 15.5.3 Supersonic Flow Past an Airfoil 808

15.6 Compressible Boundary Layers 811 15.6.1 Laminar Flow 812 15.6.2 Turbulent Flow 813 15.6.3 Compressible Law of the Wall 816

15.7 Computational Fluid Dynamics 818 15.7.1 Conservative Differencing 818 15.7.2 Numerical Dissipation and Dispersion 820 15.7.3 Propagating Shocks and Expansions 822 Chapter Summary 823 Problems 826

XIV

A Fluid Properties 841 A.l Perfect-Gas Properties 841 A.2 Pressure 841 A.3 Density 842 A.4 Compressibility and Speed of Sound 843 A.5 Surface Tension 843 A.6 Viscosity 844 A.7 Vapor Pressure 846

В Compressible Flow Tables 847 B. 1 Isentropic Flow and Normal-Shock Relations 847 B.2 Prandtl-Meyer Function and Mach Angle 853 B.3 Fanno Flow 855 B.4 Rayleigh Flow 858

С Vector Differential Operator V 861 C.l Definition of V 861 C.2 Gradient, Divergence and Curl 861 C.3 Directional Derivative 862 C.4 Operations and Identities 862

D Equations of Motion in Various Coordinate Systems 863 D.l Rectangular Cartesian Coordinates 864 D.2 Cylindrical Polar Coordinates 865 D.3 Spherical Coordinates 866 D.4 Rotating Coordinate System 867 D.5 Natural (Streamline) Coordinates 868

E Companion Software 871

Answers to Selected Problems 873

Bibliography 875

Index 881