Incompressible Flow

RONALD L. PANTON is the J. H. Herring Centennial Professor Emeritus in the Department of Mechanical Engineering at The University of Texas at Austin.

Preface xi

Preface to the Third Edition xiii

Preface to the Second Edition xv

Preface to the First Edition xvii

1 Continuum Mechanics 1

1.1 Continuum Assumption 3

1.2 Fundamental Concepts, Definitions, and Laws 3

1.3 Space and Time 5

1.4 Density, Velocity, and Internal Energy 7

1.5 Interface between Phases 10

1.6 Conclusions 12

Problems 13

2 Thermodynamics 15

2.1 Systems, Properties, and Processes 15

2.2 Independent Variables 16

2.3 Temperature and Entropy 16

2.4 Fundamental Equations of Thermodynamics 18

2.5 Euler’s Equation for Homogenous Functions 19

2.6 Gibbs–Duhem Equation 20

2.7 Intensive Forms of Basic Equations 20

2.8 Dimensions of Temperature and Entropy 21

2.9 Working Equations 21

2.10 Ideal Gas 22

2.11 Incompressible Substance 25

2.12 Compressible Liquids 26

2.13 Conclusions 26

Problems 26

3 Vector Calculus and Index Notation 28

3.1 Index Notation Rules and Coordinate Rotation 29

3.2 Definition of Vectors and Tensors 32

3.3 Special Symbols and Isotropic Tensors 33

3.4 Direction Cosines and the Laws of Cosines 34

3.5 Algebra with Vectors 35

3.6 Symmetric and Antisymmetric Tensors 37

3.7 Algebra with Tensors 38

3.8 Vector Cross-Product 41

*3.9 Alternative Definitions of Vectors 42

*3.10 Principal Axes and Values 44

3.11 Derivative Operations on Vector Fields 45

3.12 Integral Formulas of Gauss and Stokes 48

3.13 Leibnitz’s Theorem 51

3.14 Conclusions 52

Problems 53

4 Kinematics of Local Fluid Motion 54

4.1 Lagrangian Viewpoint 54

4.2 Eulerian Viewpoint 57

4.3 Substantial Derivative 59

4.4 Decomposition of Motion 60

4.5 Elementary Motions in a Linear Shear Flow 64

*4.6 Proof of Vorticity Characteristics 66

*4.7 Rate-of-Strain Characteristics 68

4.8 Rate of Expansion 69

*4.9 Streamline Coordinates 70

4.10 Conclusions 72

Problems 72

5 Basic Laws 74

5.1 Continuity Equation 74

5.2 Momentum Equation 78

5.3 Surface Forces 79

*5.4 Stress Tensor Derivation 79

5.5 Interpretation of the Stress Tensor Components 81

5.6 Pressure and Viscous Stress Tensor 83

5.7 Differential Momentum Equation 84

*5.8 Moment of Momentum, Angular Momentum, and Symmetry of Tij 89

5.9 Energy Equation 90

5.10 Mechanical and Thermal Energy Equations 92

5.11 Energy Equation with Temperature as the Dependent Variable 94

*5.12 Second Law of Thermodynamics 94

5.13 Integral Form of the Continuity Equation 95

5.14 Integral Form of the Momentum Equation 97

*5.15 Momentum Equation for a Deformable Particle of Variable Mass 100

*5.16 Integral Form of the Energy Equation 103

5.17 Integral Mechanical Energy Equation 104

5.18 Jump Equations at Interfaces 106

5.19 Conclusions 108

Problems 108

6 Newtonian Fluids and the Navier–Stokes Equations 111

6.1 Newton’s Viscosity Law 111

6.2 Molecular Model of Viscous Effects 114

6.3 Non-Newtonian Liquids 118

*6.4 Wall Boundary Conditions; The No-Slip Condition 120

6.5 Fourier’s Heat Conduction Law 123

6.6 Navier–Stokes Equations 125

6.7 Conclusions 125

Problems 126

7 Some Incompressible Flow Patterns 127

7.1 Pressure-Driven Flow in a Slot 127

7.2 Mechanical Energy, Head Loss, and Bernoulli Equation 132

7.3 Plane Couette Flow 136

7.4 Pressure-Driven Flow in a Slot with a Moving Wall 138

7.5 Double Falling Film on a Wall 139

7.6 Outer Solution for Rotary Viscous Coupling 142

7.7 The Rayleigh Problem 143

7.8 Conclusions 148

Problems 148

8 Dimensional Analysis 150

8.1 Measurement, Dimensions, and Scale Change Ratios 150

8.2 Physical Variables and Functions 153

8.3 Pi Theorem and Its Applications 155

8.4 Pump or Blower Analysis: Use of Extra Assumptions 159

8.5 Number of Primary Dimensions 163

*8.6 Proof of Bridgman’s Equation 165

*8.7 Proof of the Pi Theorem 167

8.8 Dynamic Similarity and Scaling Laws 170

8.9 Similarity with Geometric Distortion 171

8.10 Nondimensional Formulation of Physical Problems 174

8.11 Conclusions 179

Problems 180

9 Compressible Flow 182

9.1 Compressible Couette Flow: Adiabatic Wall 182

9.2 Flow with Power Law Transport Properties 186

9.3 Inviscid Compressible Waves: Speed of Sound 187

9.4 Steady Compressible Flow 194

9.5 Conclusions 197

Problems 197

10 Incompressible Flow 198

10.1 Characterization 198

10.2 Incompressible Flow as Low-Mach-Number Flow with Adiabatic Walls 199

10.3 Nondimensional Problem Statement 201

10.4 Characteristics of Incompressible Flow 205

10.5 Splitting the Pressure into Kinetic and Hydrostatic Parts 207

*10.6 Mathematical Aspects of the Limit Process M2 → 0 210

*10.7 Invariance of Incompressible Flow Equations under Unsteady Motion 211

*10.8 Low-Mach-Number Flows with Constant-Temperature Walls 213

*10.9 Energy Equation Paradox 216

10.10 Conclusions 218

Problems 219

11 Some Solutions of the Navier–Stokes Equations 220

11.1 Pressure-Driven Flow in Tubes of Various Cross Sections: Elliptical Tube 221

11.2 Flow in a Rectangular Tube 224

11.3 Asymptotic Suction Flow 227

11.4 Stokes’s Oscillating Plate 228

11.5 Wall under an Oscillating Free Stream 231

*11.6 Transient for a Stokes Oscillating Plate 234

11.7 Flow in a Slot with a Steady and Oscillating Pressure Gradient 236

11.8 Decay of an Ideal Line Vortex (Oseen Vortex) 241

11.9 Plane Stagnation Point Flow (Hiemenz Flow) 245

11.10 Burgers Vortex 251

11.11 Composite Solution for the Rotary Viscous Coupling 253

11.12 Von K´arm´an Viscous Pump 257

11.13 Conclusions 262

Problems 263

12 Streamfunctions and the Velocity Potential 266

12.1 Streamlines 266

12.2 Streamfunction for Plane Flows 269

12.3 Flow in a Slot with Porous Walls 272

*12.4 Streamlines and Streamsurfaces for a Three-Dimensional Flow 274

*12.5 Vector Potential and the E2 Operator 277

12.6 Stokes’s Streamfunction for Axisymmetric Flow 282

12.7 Velocity Potential and the Unsteady Bernoulli Equation 283

12.8 Flow Caused by a Sphere with Variable Radius 284

12.9 Conclusions 286

Problems 287

13 Vorticity Dynamics 289

13.1 Vorticity 289

13.2 Kinematic Results Concerning Vorticity 290

13.3 Vorticity Equation 292

13.4 Vorticity Diffusion 293

13.5 Vorticity Intensification by Straining Vortex Lines 295

13.6 Production of Vorticity at Walls 296

13.7 Typical Vorticity Distributions 300

13.8 Development of Vorticity Distributions 300

13.9 Helmholtz’s Laws for Inviscid Flow 306

13.10 Kelvin’s Theorem 307

13.11 Vortex Definitions 308

13.12 Inviscid Motion of Point Vortices 310

13.13 Circular Line Vortex 312

13.14 Fraenkel–Norbury Vortex Rings 314

13.15 Hill’s Spherical Vortex 314

13.16 Breaking and Reconnection of Vortex Lines 317

13.17 Vortex Breakdown 317

13.18 Conclusions 323

Problems 324

14 Flows at Moderate Reynolds Numbers 326

14.1 Some Unusual Flow Patterns 327

14.2 Entrance Flows 330

14.3 Entrance Flow into a Cascade of Plates: Computer Solution by the Streamfunction–Vorticity Method 331

14.4 Entrance Flow into a Cascade of Plates: Pressure Solution 341

14.5 Entrance Flow into a Cascade of Plates: Results 342

14.6 Flow Around a Circular Cylinder 346

14.7 Jeffrey–Hamel Flow in a Wedge 362

14.8 Limiting Case for Re → 0; Stokes Flow 367

14.9 Limiting Case for Re→−∞ 368

14.10 Conclusions 372

Problems 372

15 Asymptotic Analysis Methods 374

15.1 Oscillation of a Gas Bubble in a Liquid 374

15.2 Order Symbols, Gauge Functions, and Asymptotic Expansions 377

15.3 Inviscid Flow over a Wavy Wall 380

15.4 Nonuniform Expansions: Friedrich’s Problem 384

15.5 Matching Process: Van Dyke’s Rule 386

15.6 Composite Expansions 391

15.7 Characteristics of Overlap Regions and Common Parts 393

15.8 Composite Expansions and Data Analysis 399

15.9 Lagerstrom’s Problems 403

15.10 Conclusions 406

Problems 407

16 Characteristics of High-Reynolds-Number Flows 409

16.1 Physical Motivation 409

16.2 Inviscid Main Flows: Euler Equations 411

16.3 Pressure Changes in Steady Flows: Bernoulli Equations 414

16.4 Boundary Layers 418

16.5 Conclusions 428

Problems 428

17 Kinematic Decomposition of Flow Fields 429

*17.1 General Approach 429

*17.2 Helmholtz’s Decomposition; Biot–Savart Law 430

*17.3 Line Vortex and Vortex Sheet 431

*17.4 Complex Lamellar Decomposition 434

*17.5 Conclusions 437

*Problems 437

18 Ideal Flows in a Plane 438

18.1 Problem Formulation for Plane Ideal Flows 439

18.2 Simple Plane Flows 442

18.3 Line Source and Line Vortex 445

18.4 Flow over a Nose or a Cliff 447

18.5 Doublets 453

18.6 Cylinder in a Stream 456

18.7 Cylinder with Circulation in a Uniform Stream 457

18.8 Lift and Drag on Two-Dimensional Shapes 460

18.9 Magnus Effect 462

18.10 Conformal Transformations 464

18.11 Joukowski Transformation: Airfoil Geometry 468

18.12 Kutta Condition 473

18.13 Flow over a Joukowski Airfoil: Airfoil Lift 475

18.14 Numerical Method for Airfoils 482

18.15 Actual Airfoils 484

*18.16 Schwarz–Christoffel Transformation 487

*18.17 Diffuser or Contraction Flow 489

*18.18 Gravity Waves in Liquids 494

18.19 Conclusions 499

Problems 499

19 Three-Dimensional Ideal Flows 502

19.1 General Equations and Characteristics of Three-Dimensional Ideal Flows 502

19.2 Swirling Flow Turned into an Annulus 504

19.3 Flow over a Weir 505

19.4 Point Source 507

19.5 Rankine Nose Shape 508

19.6 Experiments on the Nose Drag of Slender Shapes 510

19.7 Flow from a Doublet 513

19.8 Flow over a Sphere 515

19.9 Work to Move a Body in a Still Fluid 516

19.10 Wake Drag of Bodies 518

*19.11 Induced Drag: Drag due to Lift 519

*19.12 Lifting Line Theory 524

19.13 Winglets 525

*19.14 Added Mass of Accelerating Bodies 526

19.15 Conclusions 531

Problems 531

20 Boundary Layers 533

20.1 Blasius Flow over a Flat Plate 533

20.2 Displacement Thickness 538

20.3 Von K´arm´an Momentum Integral 540

20.4 Von K´arm´an–Pohlhausen Approximate Method 541

20.5 Falkner–Skan Similarity Solutions 543

20.6 Arbitrary Two-Dimensinoal Layers: Crank–Nicolson Difference Method 547

*20.7 Vertical Velocity 556

20.8 Joukowski Airfoil Boundary Layer 558

20.9 Boundary Layer on a Bridge Piling 563

20.10 Boundary Layers Beginning at Infinity 564

20.11 Plane Boundary Layer Separation 570

20.12 Axisymmteric Boundary Layers 573

20.13 Jets 576

20.14 Far Wake of Nonlifting Bodies 579

20.15 Free Shear Layers 582

20.16 Unsteady and Erupting Boundary Layers 584

*20.17 Entrance Flow into a Cascade, Parabolized Navier–Stokes Equations 587

*20.18 Three-Dimensional Boundary Layers 589

*20.19 Boundary Layer with a Constant Transverse Pressure Gradient 593

*20.20 Howarth’s Stagnation Point 598

*20.21 Three-Dimensional Separation Patterns 600

20.22 Conclusions 603

Problems 605

21 Flow at Low Reynolds Numbers 607

21.1 General Relations for Re → 0: Stokes’s Equations 607

21.2 Global Equations for Stokes Flow 611

21.3 Streamfunction for Plane and Axisymmetric Flows 613

21.4 Local Flows, Moffatt Vortices 616

21.5 Plane Internal Flows 623

21.6 Flows between Rotating Cylinders 628

21.7 Flows in Tubes, Nozzles, Orifices, and Cones 631

21.8 Sphere in a Uniform Stream 636

21.9 Composite Expansion for Flow over a Sphere 641

21.10 Stokes Flow near a Circular Cylinder 642

*21.11 Axisymmetric Particles 644

*21.12 Oseen’s Equations 646

*21.13 Interference Effects 647

21.14 Conclusions 648

Problems 649

22 Lubrication Approximation 650

22.1 Basic Characteristics: Channel Flow 650

22.2 Flow in a Channel with a Porous Wall 653

22.3 Reynolds Equation for Bearing Theory 655

22.4 Slipper Pad Bearing 657

22.5 Squeeze-Film Lubrication: Viscous Adhesion 659

22.6 Journal Bearing 660

22.7 Hele-Shaw Flow 664

22.8 Conclusions 667

Problems 668

23 Surface Tension Effects 669

23.1 Interface Concepts and Laws 669

23.2 Statics: Plane Interfaces 676

23.3 Statics: Cylindrical Interfaces 679

23.4 Statics: Attached Bubbles and Drops 681

23.5 Constant-Tension Flows: Bubble in an Infinite Stream 683

23.6 Constant-Tension Flows: Capillary Waves 686

23.7 Moving Contact Lines 688

23.8 Constant-Tension Flows: Coating Flows 691

23.9 Marangoni Flows 695

23.10 Conclusions 703

Problems 705

24 Introduction to Microflows 706

24.1 Molecules 706

24.2 Continuum Description 708

24.3 Compressible Flow in Long Channels 709

24.4 Simple Solutions with Slip 712

24.5 Gases 715

24.6 Couette Flow in Gases 719

24.7 Poiseuille Flow in Gases 722

24.8 Gas Flow over a Sphere 726

24.9 Liquid Flows in Tubes and Channels 728

24.10 Liquid Flows near Walls; Slip Boundaries 730

24.11 Conclusions 735

25 Stability and Transition 737

25.1 Linear Stability and Normal Modes as Perturbations 738

25.2 Kelvin–Helmholtz Inviscid Shear Layer Instability 739

25.3 Stability Problems for Nearly Parallel Viscous Flows 744

25.4 Orr–Sommerfeld Equation 746

25.5 Invsicid Stability of Nearly Parallel Flows 747

25.6 Viscous Stability of Nearly Parallel Flows 749

25.7 Experiments on Blasius Boundary Layers 752

25.8 Transition, Secondary, Instability, and Bypass 756

25.9 Spatially Developing Open Flows 759

25.10 Transition in Free Shear Flows 759

25.11 Poiseuille and Plane Couette Flows 761

25.12 Inviscid Instability of Flows with Curved Streamlines 763

25.13 Taylor Instability of Couette Flow 765

25.14 Stability of Regions of Concentrated Vorticity 767

25.15 Other Instabilities: Taylor, Curved, Pipe, Capillary Jets, and G¨ortler 769

25.16 Conclusions 771

26 Turbulent Flows 772

26.1 Types of Turbulent Flows 772

26.2 Characteristics of Turbulent Flows 773

26.3 Reynolds Decomposition 776

26.4 Reynolds Stress 777

*26.5 Correlation of Fluctuations 780

*26.6 Mean and Turbulent Kinetic Energy 782

*26.7 Energy Cascade: Kolmogorov Scales and Taylor Microscale 784

26.8 Wall Turbulence: Channel Flow Analysis 789

26.9 Channel and Pipe Flow Experiments 797

26.10 Boundary Layers 800

26.11 Wall Turbulence: Fluctuations 804

26.12 Turbulent Structures 811

26.13 Free Turbulence: Plane Shear Layers 817

26.14 Free Turbulence: Turbulent Jet 822

26.15 Bifurcating and Blooming Jets 824

26.16 Conclusions 825

A Properties of Fluids 827

B Differential Operations in Cylindrical and Spherical Coordinates 828

C Basic Equations in Rectangular, Cylindrical, and Spherical Coordinates 833

D Streamfunction Relations in Rectangular, Cylindrical, and Spherical Coordinates 838

E Matlab R Stagnation Point Solver 842

F Matlab R Program for Cascade Entrance 844

G Matlab R Boundary Layer Program 847

References 851

Index 869