Herbert Oertel Editor

Prandtl-Essentials of Fluid Mechanics

Third Edition

With Contributions by P. Erhard, D. Etling, U. Müller, U. Riedel, K.R. Sreenivasan, J. Warnatz

Translated by Katherine Asfaw

With 536 Illustrations

Springer

Contents

Preface v

1. Introduction 1

2. Properties of Liquids and Gases 15 2.1 Properties of Liquids 15 2.2 State of Stress 16 2.3 Liquid Pressure 18 2.4 Properties of Gases 24 2.5 Gas Pressure 26 2.6 Interaction Between Gas Pressure and Liquid Pressure . . . . 29 2.7 Equilibrium in Other Force Fields 32 2.8 Surface Stress (Capillarity) 36 2.9 Problems 40

3. Kinematics of Fluid Flow 43 3.1 Methods of Representation 43 3.2 Acceleration of a Flow 47 3.3 Topology of a Flow 48 3.4 Problems 55

4. Dynamics of Fluid Flow 59 4.1 Dynamics of Inviscid Liquids 59

4.1.1 Continuity and the Bernoulli Equation 59 4.1.2 Consequences of the Bernoulli Equation 63 4.1.3 Pressure Measurement 71 4.1.4 Interfaces and Formation of Vortices 73 4.1.5 Potential Flow 76 4.1.6 Wing Lift and the Magnus Effect 88 4.1.7 Balance of Momentum for Steady Flows 91 4.1.8 Waves on a Free Liquid Surface 99 4.1.9 Problems 109

4.2 Dynamics of Viscous Liquids 114 4.2.1 Viscosity (Inner Friction), the Navier-Stokes Equation 114

V l l

viii Contents

4.2.2 Mechanical Similarity, Reynolds Number 118 4.2.3 Laminar Boundary Layers 119 4.2.4 Onset of Turbulence 122 4.2.5 Fully Developed Turbulence 132 4.2.6 Flow Separation and Vortex Formation 140 4.2.7 Secondary Flows 147 4.2.8 Flows with Prevailing Viscosity 149 4.2.9 Flows Through Pipes and Channels 156 4.2.10 Drag of Bodies in Liquids 161 4.2.11 Flows in Non-Newtonian Media 170 4.2.12 Problems 175

4.3 Dynamics of Gases 181 4.3.1 Pressure Propagation, Velocity of Sound 181 4.3.2 Steady Compressible Flows 185 4.3.3 Conservation of Energy 190 4.3.4 Theory of Normal Shock Waves 191 4.3.5 Flows past Corners, Free Jets 195 4.3.6 Flows with Small Perturbations 199 4.3.7 Flows past Airfoils 203 4.3.8 Problems 208

4.4 Aerodynamics 212 4.4.1 Bird Flight 213 4.4.2 Airfoils and Wings 215 4.4.3 Airfoil and Wing Theory 222 4.4.4 Aerodynamic Facilities 237 4.4.5 Transonic Aerodynamics, Swept Wings 238 4.4.6 Shock-Boundary-Layer Interaction 244 4.4.7 Flow Separation 250 4.4.8 Supersonic Aerodynamics, Delta Wings 252 4.4.9 Problems 259

5. Fundamental Equations of Fluid Mechanics 265 5.1 Continuity Equation 265 5.2 Navier-Stokes Equations 266

5.2.1 Laminar Flows 266 5.2.2 Reynolds Equations for Turbulent Flows 273

5.3 Energy Equation 278 5.3.1 Laminar Flows 278 5.3.2 Turbulent Flows 282

5.4 Fundamental Equations as Conservation Laws 284 5.4.1 Hierarchy of Fundamental Equations 284 5.4.2 Navier-Stokes Equations 287 5.4.3 Derived Model Equations 290 5.4.4 Reynolds Equations for Turbulent Flows 298 5.4.5 Turbulence Models 299

Contents ix

5.4.6 Multiphase Flows 317 5.4.7 Reactive Flows 329

5.5 Differential Equations of Perturbations 332 5.6 Problems 338

6. Instabilities and Turbulent Flows 345 6.1 Fundamentals of Turbulent Flows 345 6.2 Onset of Turbulence 346

6.2.1 Fluid-Mechanical Instabilities 347 6.2.2 Linear Stability Analysis 350 6.2.3 Transition to Turbulence 373

6.3 Developed Turbulence 378 6.3.1 The Notion of a Mixing Length 378 6.3.2 Turbulent Mixing 380 6.3.3 Energy Relations in Turbulent Flows 381

6.4 Classification of Turbulent Flows 384 6.4.1 Free Turbulence 385 6.4.2 Turbulence near Solid Boundaries 388 6.4.3 Rotating and Stratified Flows 391 6.4.4 Turbulence in Wind Tunnels 392 6.4.5 Two-Dimensional Turbulence 396 6.4.6 Structures and Statistics 399

6.5 Some New Developments in Turbulence 400 6.5.1 Decomposition into small and large scales 400 6.5.2 Lagrangian Investigations of Turbulence 406 6.5.3 Field-Theoretic Methods 407 6.5.4 Outlook 407

7. Convective Heat and Mass Transfer 409 7.1 Fundamentals of Heat and Mass Transfer 410

7.1.1 Free and Forced Convection 410 7.1.2 Heat Conduction and Convection 412 7.1.3 Diffusion and Convection 414

7.2 Free Convection 415 7.2.1 Rayleigh-Benard Convection 415 7.2.2 Convection at a Vertical Plate 426 7.2.3 Convection at a Horizontal Cylinder 432

7.3 Forced Convection 433 7.3.1 Pipe Flows 433 7.3.2 Boundary-Layer Flows 438 7.3.3 Bodies in Flows 443

7.4 Heat and Mass Exchange 444 7.4.1 Diffusion Convection 444 7.4.2 Mass Exchange at a Flat Plate 451

x Contents

8. Multiphase Flows 455 8.1 Fundamentals of Multiphase Flows 455

8.1.1 Definitions 456 8.1.2 Flow Patterns 459 8.1.3 Flow Pattern Maps 459

8.2 Flow Models 462 8.2.1 The One-Dimensional Two-Fluid Model 463 8.2.2 Mixing Models 466 8.2.3 The Drift-Flow Model 468 8.2.4 Bubbles and Drops 470 8.2.5 Spray Flows 475 8.2.6 Liquid-Solid Transport 479 8.2.7 Fluidization of Particle Beds 482

8.3 Pressure Loss and Volume Fraction in Hydraulic Components 484 8.3.1 Friction Loss in Horizontal Straight Pipes 485 8.3.2 Acceleration Losses 489

8.4 Propagation Velocity of Density Waves and Critical Mass Fluxes493 8.4.1 Density Waves 493 8.4.2 Critical Mass Fluxes 496 8.4.3 Cavitation 503

8.5 Instabilities in Two-Phase Flows 507 8.6 Turbulence in Dispersed Two-Phase Flows 513

8.6.1 General Aspects 513 8.6.2 The Mixing Length Concept 518 8.6.3 Transport Equation Models for Turbulence 520

9. Reactive Flows 523 9.1 Fundamentals of Reactive Flows 523

9.1.1 Rate Laws and Reaction Orders 525 9.1.2 Relation Between Forward and Reverse Reactions . . 526 9.1.3 Elementary Reactions and Reaction Molecularity . . . 527 9.1.4 Temperature Dependence of Rate Coefficients 531 9.1.5 Pressure Dependence of Rate Coefficients 532 9.1.6 Characteristics of Reaction Mechanisms 535

9.2 Laminar Reactive Flows 540 9.2.1 Structure of Premixed Flames 540 9.2.2 Flame Velocity of Premixed Flames 542 9.2.3 Sensitivity Analysis 543 9.2.4 Nonpremixed Counterflow Flames 544 9.2.5 Nonpremixed Jet Flames 547 9.2.6 Nonpremixed Flames with Fast Chemistry 548 9.2.7 Exhaust Gas Cleaning with Plasma Sources 550 9.2.8 Flows in Etching Reactors 551 9.2.9 Heterogeneous Catalysis 553

9.3 Turbulent Reactive Flows 555

Contents xi

9.3.1 Overview and Concepts 555 9.3.2 Direct Numerical Simulation 556 9.3.3 Mean Reaction Rates 558 9.3.4 Eddy-Break-Up Models 563 9.3.5 Turbulent Nonpremixed Flames 563 9.3.6 Turbulent Premixed Flames 575

9.4 Hypersonic Flows 581 9.4.1 Physical-Chemical Phenomena in Re-Entry Flight . . 581 9.4.2 Chemical Nonequilibrium 583 9.4.3 Thermal Nonequilibrium 585 9.4.4 Surface Reactions on Re-entry Vehicles 588

10.Flows in the Atmosphere and in the Ocean 593 10.1 Fundamentals of Flows in the Atmosphere and in the Ocean . 593

10.1.1 Introduction 593 10.1.2 Fundamental Equations in Rotating Systems 593 10.1.3 Geostrophic Flow 597 10.1.4 Vorticity 599 10.1.5 Ekman Layer 602 10.1.6 Prandtl Layer 605

10.2 Flows in the Atmosphere 607 10.2.1 Thermal Wind Systems 607 10.2.2 Thermal Convection 611 10.2.3 Gravity Waves 613 10.2.4 Vortices 616 10.2.5 Global Atmospheric Circulation 621

10.3 Flows in the Ocean 623 10.3.1 Wind-Driven Flows 624 10.3.2 Water Waves 626

10.4 Application to Atmospheric and Oceanic Flows 629 10.4.1 Weather Forecast 629 10.4.2 Greenhouse Effect and Climate Prediction 631 10.4.3 Ozone Hole 635

l l .Microflows 639 11.1 Fundamentals of Microflows 639

11.1.1 Application of Microflows 639 11.1.2 Fluid Models 641 11.1.3 Microflows of Gases 643 11.1.4 Microflows of Liquids 645

11.2 Molecular Models 647 11.2.1 Fundamentals of Molecular Models 647 11.2.2 Monte-Carlo-Simulation 650 11.2.3 Molecular Dynamic Simulation 653

11.3 Continuum Models 655

xii Contents

11.3.1 Similarity Discussion 655 11.3.2 Modifications of Boundary Conditions 657 11.3.3 Electrokinetic Effects 661 11.3.4 Wetting and Thin Films 670

11.4 Experiments 678 11.4.1 Pressure Drop 679 11.4.2 Laminar-Turbulent Transition 681 11.4.3 Heat Transfer 682

12.Biofluid Mechanics 685 12.1 Fundamentals of Biofluid Mechanics 685

12.1.1 Biofluid Mechanics of Animals 687 12.1.2 Biofluid Mechanics of Humans 690 12.1.3 Blood Rheology 697

12.2 Swimming and Flight 700 12.2.1 Motion of Protozoa 700 12.2.2 Swimming of Fish 703 12.2.3 Flow Control 705 12.2.4 Bird Flight 707

12.3 Human Heart Flow 712 12.3.1 Physiology and Anatomy of the Heart 713 12.3.2 Structure of the Heart 715 12.3.3 Excitation Physiology of the Heart 719 12.3.4 Flow in the Heart 722 12.3.5 Cardiac Valves 731

12.4 Flow in Blood Vessels 734 12.4.1 Unsteady Pipe Flow 738 12.4.2 Unsteady Arterial Flow 742 12.4.3 Arterial Branchings 745 12.4.4 Microcirculation 749

Selected Bibliography 753

Index 785