Prandtl-essentials of fluid mechanics

Applied Mathematical Sciences

Editors

S.S. Antman J.E. Marsden L. Sirovich

Advisors

J.K. Hale P. Holmes J. Keener

J. Keller B.J. Matkowsky A. Mielke

C.S. Peskin K.R. Sreenivasan

Volume 158

For further volumes:

http://www.springer.com/series/34

Herbert Oertel

Editor

Prandtl–Essentials of Fluid

Mechanics

Third Edition

With Contributions by

Translated by Katherine Asfaw

With 536 Illustrations

K.R. Sreenivasan, J. Warnatz

P. Erhard, D. Etling, U. Müller, U. Riedel,

Dir. MPI für Strömungsforschung, † 1953

© Springer Science+Business Media, LLC 2010

subject to proprietary rights.

Printed on acid-free paper

permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they

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software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

are not identified as such, is not to be taken as an expression of opinion as to whether or not they are

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Library of Congress Control Number: 2009938172

ISSN 0066-5452

ISBN 978-1-4419-1563-4 e-ISBN 978-1-4419-1564-1

Herbert Oertel

Kaiserstr. 12

Germany

Institute for FFluid Mechanics

University of Karlsruhe

D-76131 Karlsruhe

University of Göttingen,

Editors:

S.S. Antman

and

Institute for Physical Science

and Technology

University of Maryland

College Park

USA

[email protected]

J.E. Marsden

Control and Dynamical

Systems, 107-81

California Institute

Pasadena, CA 91125

USA

[email protected]

L. Sirovich

Mathematics

Biomathematical Sciences

New York, NY 10029-6574

USA

[email protected]

Ludwig Prandtl

Laboratory of Applied

MD 20742-4015

Department of Mathematics

of Technology

Department of

DOI 10.1007/978-1-4419-1564-1

Mount Sinai School of Medicine

Originally published in the German language by Vieweg+Teubner, 65189 Wiesbaden, Germany

© Vieweg+Teubner |GWV Fachverlage GmbH, Wiesbaden 2008

th revised and enlarged edition”

All rights reserved. This work may not be translated or copied in whole or in part without the written

as “Oertel: Prandtl – Führer durch die Strömungslehre. 12

Preface

Ludwig Prandtl, with his fundamental contributions to hydrodynamics, aero￾dynamics, and gas dynamics, greatly influenced the development of fluid me￾chanics as a whole, and it was his pioneering research in the first half of the

last century that founded modern fluid mechanics. His book F¨uhrer durch

die Str¨omungslehre, which appeared in 1942, originated from previous publi￾cations in 1913, Lehre von der Fl¨ussigkeit und Gasbewegung, and 1931, Abriß

der Str¨omungslehre. The title F¨uhrer durch die Str¨omungslehre, or Essentials

of Fluid Mechanics, is an indication of Prandtl’s intentions to guide the reader

on a carefully thought-out path through the different areas of fluid mechan￾ics. On his way, the author advances intuitively to the core of the physical

problem, without extensive mathematical derivations. The description of the

fundamental physical phenomena and concepts of fluid mechanics that are

needed to derive the simplified models has priority over a formal treatment

of the methods. This is in keeping with the spirit of Prandtl’s research work.

The first edition of Prandtl’s F¨uhrer durch die Str¨omungslehre was the

only book on fluid mechanics of its time and, even today, counts as one of

the most important books in this area. After Prandtl’s death, his students

Klaus Oswatitsch and Karl Wieghardt undertook to continue his work, and to

add new findings in fluid mechanics in the same clear manner of presentation.

When the ninth edition went out of print and a new edition was desired

by the publishers, we were glad to take on the task. The first four chapters of

this book keep to the path marked out by Prandtl in the first edition, in 1942.

The original historical text has been linguistically revised, and leads, after the

Introduction, to chapters on Properties of Liquids and Gases, Kinematics of

Flow, and Dynamics of Fluid Flow. These chapters are taught to science and

engineering students in introductory courses on fluid mechanics even today.

We have retained much of Prandtl’s original material in these chapters, but

added a section on the Topology of a Flow in Chapter 3 on Flows of Non￾Newtonian Media and Aerodynamics in Chapter 4. Chapter 5 on Fundamental

Equations of Fluid Mechanics enlarges the material in the original, and forms

the basis for the treatment of different branches of fluid mechanics that appear

in subsequent chapters.

The major difference from previous editions lies in the treatment of addi￾tional topics of fluid mechanics. The field of fluid mechanics is continuously

v

Preface

growing, and has now become so extensive that a selection had to be made.

I am greatly indebted to my colleagues K.R. Sreenivasan, U. M¨uller, J. War￾natz, U. Riedel, D. Etling, and P. Erhard, who revised individual chapters in

their own research areas, keeping Prandtl’s purpose in mind and presenting

the latest developments of the last seventy years in Chapters 6 to 12. Some of

these chapters can be found in some form in Prandtl’s book, but have under￾gone substantial revisions; others are entirely new. The original chapters on

Wing Aerodynamics, Heat Transfer, Stratified Flows, Turbulent Flows, Mul￾tiphase Flows, Flows in the Atmosphere and the Ocean, and Turbomachinery

have been revised, while the chapters on Instabilities and Turbulent Flows,

Flows with Chemical Reactions, Microflows and Biofluid Mechanics are new.

References to the literature in the individual chapters have intentionally been

kept to those few necessary for comprehension and completion. The extensive

historical citations may be found by referring to previous editions.

Essentials of Fluid Mechanics is targeted to science and engineering stu￾dents who, having had some basic exposure to fluid mechanics, wish to attain

an overview of the different branches of fluid mechanics. The presentation

postpones the use of vectors and eschews the use integral theorems in order

to preserve the accessibility to this audience. For more general and compact

mathematical derivations we refer to the references. In order to give students

the possibility of checking their learning of the subject matter, Chapters 2

to 5 are supplemented with problems. The book will also give the expert in

research or industry valuable stimulation in the treatment and solution of

fluid-mechanical problems.

We hope that we have been able, with the treatment of the different

branches of fluid mechanics, to carry on the work of Ludwig Prandtl as he

would have wished. Chapters 1–5, 7, and 12 were written by H. Oertel, Chap￾ter 6 by K.R. Sreenivasan and H. Oertel, Chapter 8 by U. M¨uller, Chapter

9 by J. Warnatz and U. Riedel, Chapter 10 by D. Etling, and Chapter 11 by

P. Erhard. Thanks are due to those colleagues whose numerous suggestions

have been included in the text.

I thank Katherine Aswaf for the translation and typesetting of the English

manuscript and K. Fritsch-Kirchner for the completion of the text files. The

extremely fruitful collaboration with Springer-Verlag also merits particular

praise.

Karlsruhe, July 2009 Herbert Oertel

vi

Preface

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

vii

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

viii Contents

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–B´enard 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

Contents ix

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

x Contents

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

11.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

Contents xi

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

xii Contents

1. Introduction

The development of modern fluid mechanics is closely connected to the name

of its founder, Ludwig Prandtl. In 1904 it was his famous article on fluid

motion with very small friction that introduced boundary-layer theory. His

article on airfoil theory, published the following decade, formed the basis

for the calculation of friction drag, heat transfer, and flow separation. He

introduced fundamental ideas on the modeling of turbulent flows with the

Prandtl mixing length for turbulent momentum exchange. His work on gas

dynamics, such as the Prandtl–Glauert correction for compressible flows, the

theory of shock waves and expansion waves, as well as the first photographs

of supersonic flows in nozzles, reshaped this research area. He applied the

methods of fluid mechanics to meteorology, and was also pioneering in his

contributions to problems of elasticity, plasticity, and rheology.

Prandtl was particularly successful in bringing together theory and ex￾periment, with the experiments serving to verify his theoretical ideas. It was

this that gave Prandtl’s experiments their importance and precision. His fa￾mous experiment with the tripwire, through which he discovered the turbu￾lent boundary layer and the effect of turbulence on flow separation, is one

example. The tripwire was not merely inspiration, but rather was the result

of consideration of discrepancies in Eiffel’s drag measurements on spheres.

Two experiments with different tripwire positions were enough to establish

the generation of turbulence and its effect on the flow separation. For his

experiments Prandtl developed wind tunnels and measuring apparatus, such

as the G¨ottingen wind tunnel and the Prandtl stagnation tube. His scientific

results often seem to be intuitive, with the mathematical derivation present

only to serve the physical understanding, although it then does indeed deliver

the decisive result and the simplified physical model. According to Werner

Heisenberg, Prandtl was able to “see” the solutions of differential equations

without calculating them.

Selected individual examples aim to introduce the reader to the path to

understanding of fluid mechanics prepared by Prandtl and to the contents and

modeling in each chapter. As an example of the dynamics of flows (Chapter

4), the different regimes in the flow past a vehicle, an incompressible flow,

and in the flow past an automobile, a compressible flow, are described.

H. Oertel (ed.), Prandtl-Essentials of Fluid Mechanics, 1

Applied Mathematical Sciences 158, DOI 10.1007/978-1-4419-1564-1_1,

© Springer Science+Business Media, LLC 2010

2 1. Introduction

In flow past a vehicle, we differentiate between the free flow over the

surface and the flow between the vehicle moving with velocity U∞ and the

street which is at rest. At the stagnation point, where the pressure is at its

maximum, the flow divides, and is accelerated along the hood and past the

spoiler along the base of the vehicle. This leads to a pressure drop and to a

negative downward pressure on the street, as shown in Figure 1.1. The flow

again slows down at the windshield, and is decelerated downstream along

the roof and the trunk. This leads to a pressure increase with a positive lift,

while the negative downward pressure on the street along the lower side of

the vehicle remains.

Viscous flow (Section 4.2) on the upper and lower sides of the vehicle

is restricted to the boundary-layer flow, which becomes the viscous wake

at the back edge of the vehicle. In the wind tunnel experiment the flow is

made visible with smoke, and this shows that downstream from the back of

the automobile, a backflow region forms. This is seen in the figure as the

black region. Outside the boundary layer and the wake, the flow is essentially

inviscid (Section 4.1).

In order to be able to understand the different flow regimes, and therefore

to establish a basis for the aerodynamic design of a motor vehicle, Prandtl

worked out the carefully prepared path (Chapters 2 to 4) from the properties

of liquids and gases, to kinematics, and to the dynamics of inviscid and viscous

flows. By following this path, too, the reader will successively gain physical

understanding of this first flow example.

The second flow example considers compressible flow past a wing with a

shock wave (Sections 4.3 and 4.4.5). The free flow toward the wing has the

Sichtbarmachung im Nachlauf

boundary layer

wake

inviscid flow

wake flow visualization

Fig. 1.1. Flow past a vehicle