Boundary-Layer Theory

Hermann Schlichting (Deceased)Klaus Gersten

Boundary-Layer Theory

Ninth Edition

123

With contributionsfrom Egon Krause and Herbert Oertel Jr.

Translated by Katherine Mayes

Hermann Schlichting (Deceased)Institute of Fluid MechanicsTechnical University of BraunschweigBraunschweigGermany

Klaus GerstenInstitute of Thermodynamics and FluidMechanics

Ruhr-University BochumBochum, Nordrhein-WestfalenGermany

ISBN 978-3-662-52917-1 ISBN 978-3-662-52919-5 (eBook)DOI 10.1007/978-3-662-52919-5

Library of Congress Control Number: 2016944848

1st edition: © McGraw-Hill New York 19552nd edition: © Pergamon London 19554th edition: © McGraw-Hill New York 19606th edition: © McGraw-Hill New York 19687th edition: © McGraw-Hill New York 19758th edition: © Springer-Verlag Berlin Heidelberg 20009th edition: © Springer-Verlag Berlin Heidelberg 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

Typesetting: Katherine Mayes, Darmstadt and LE-TEX, LeipzigCover design: de’blik, Berlin

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer-Verlag GmbH Berlin Heidelberg

Bochum, March 2016 Klaus Gersten

Preface to the Ninth English Edition

For this edition corrections have been carried out and additional impor-tant literature published in recent years have been included (66 additional references). Section 22.8 on plane turbulent wall jets has been completely rewritten. I am very thankful for valuable assistance to Prof. Dr. E. Krause, Prof. Dr. H. Oertel, Prof. Dr. W. Schneider, Prof. Dr. M. Breuer, Prof. Dr. H.-D. Papenfuß and last but not least Gertraude Odemar.

Preface to the Eighth English Edition

According to the tradition of this book, a German edition has always beensoon followed by the English translation. I am very grateful to Springer–Verlag for undertaking this version and for securing a translator. My partic-ular thanks go to Katherine Mayes for this excellent translation. In the courseof the translation, some errors in the German edition were corrected and anumber of additions carried out. In this connection I am very thankful toProf. Dr. W. Schneider, Vienna, for several suggestions and improvements. Iwould like to thank Ursula Beitz again for her careful checking of the biblio-graphy. I hope that the English edition will attain the same positive resonanceas the ninth German edition.

Bochum, May 1999 Klaus Gersten

Preface to the Ninth German Edition

There is no doubt that Boundary–Layer Theory by Hermann Schlichting isone of most important books within the sphere of fluid mechanics to appearin the last decade. Shortly before his death, Hermann Schlichting broughtout the eighth edition which he revised together with his friend and formercolleague Wilhelm Riegels.When this edition went out of print and a new edition was desired by the

publishers, I was very glad to take on the task. During the fifteen years Ispent at the institute of my highly respected teacher Hermann Schlichting, Ihad already been involved with earlier editions of the book and had revisedsome chapters. The burden was also eased by the fact that boundary–layertheory in its widest sense has been my preferred direction of research formany years.It quickly became clear that a complete revision was necessary; indeed

this was also known to Hermann Schlichting. In the preface to the eighthedition he wrote: “Noting the systematic of our knowledge of today, it wouldhave been desirable to fully revise this work; however such a process wouldhave pushed back the appearance of this book by years.” Compared to theeighth edition, the literature of the last 15 years had to be taken into accountand recent developments, in turbulence models for example, had to be incor-porated. In order to keep the size of the book tractable, some results – thosewhich no longer seem so important with today’s computing potential – hadto be curtailed, or in some cases, left out altogether.Thus the necessity to completely rewrite the text emerged. The funda-

mental divisions within the book were retained; as before it consists of thefour major sections: basic laws of the flows of viscous fluids, laminar bound-ary layers, the onset of turbulence, turbulent boundary layers. However anew fifth section on numerical methods in boundary–layer theory has beenadded.The partition into chapters had to be somewhat modified in order to

improve the style of presentation of the material. Because of the necessaryrestrictions on the material, the aim was to concentrate on boundary–layertheory as the theory of high Reynolds number flows. Accordingly the chap-ter on “creeping flows”, that is flows at very small Reynolds numbers, wasomitted.

It seemed natural to steer towards the style and level of presentation withthe same target audience as with Hermann Schlichting.The research area of boundary–layer theory is continuously growing, and

it has become so extensive that no single person can possess a completeoverview. Consequently I am extremely grateful to two colleagues who sup-ported me actively. Professor E. Krause wrote the new additional chapter onnumerical methods in boundary–layer theory, and Professor H. Oertel pro-vided the revision of the section on the onset of turbulence (stability theory).Further assistance was furnished from different sources. I am indebted to

Dr.-Ing. Peter Schafer and Dr.-Ing. Detlev Vieth for a great many new samplecalculations. Dr. Vieth also read the entire text discerningly. I am grateful tohim for numerous improving suggestions. Renate Golzenleuchtner deservesparticular thanks for generating the figures which almost all had to be newlydrawn up. I would like to thank Ursula Beitz particularly for her carefuland exhaustive checking of the bibliography, while Marianne Ferdinand andEckhard Schmidt were of first class assistance. It was by far impossible toadopt all citations, so that it may be necessary to revert to the eighth editionfor specific references to earlier pieces of work.The printing firm of Jorg Steffenhagen is due particular praise for an

extremely fruitful collaboration. My thanks also go to Springer–Verlag forour most agreeable work together.I hope we have been able to carry on the work of Hermann Schlichting as

he would have wished.

Bochum, October 1996 Klaus Gersten

X Preface

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XX I

Part I. Fundamentals of Viscous Flows

1. Some Features of Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Real and Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Laminar and Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Asymptotic Behaviour at Large Reynolds Numbers . . . . . . . . . 141.6 Comparison of Measurements

Using the Inviscid Limiting Solution . . . . . . . . . . . . . . . . . . . . . . 141.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2. Fundamentals of Boundary–Layer Theory . . . . . . . . . . . . . . . . . 292.1 Boundary–Layer Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Laminar Boundary Layer on a Flat Plate at Zero Incidence . . 302.3 Turbulent Boundary Layer on a Flat Plate at Zero Incidence . 332.4 Fully Developed Turbulent Flow in a Pipe . . . . . . . . . . . . . . . . . 362.5 Boundary Layer on an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.6 Separation of the Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . 392.7 Overview of the Following Material . . . . . . . . . . . . . . . . . . . . . . . 48

3. Field Equations for Flows of Newtonian Fluids . . . . . . . . . . . 513.1 Description of Flow Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 General Stress State of Deformable Bodies . . . . . . . . . . . . . . . . . 533.5 General State of Deformation of Flowing Fluids . . . . . . . . . . . . 573.6 Relation Between Stresses and Rate of Deformation . . . . . . . . . 623.7 Stokes Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.8 Bulk Viscosity and Thermodynamic Pressure . . . . . . . . . . . . . . 663.9 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

XII Contents

3.10 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.11 Equations of Motion

for Arbitrary Coordinate Systems (Summary) . . . . . . . . . . . . . . 733.12 Equations of Motion

for Cartesian Coordinates in Index Notation . . . . . . . . . . . . . . . 763.13 Equations of Motion in Different Coordinate Systems . . . . . . . 79

4. General Properties of the Equations of Motion . . . . . . . . . . . 834.1 Similarity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Similarity Laws for Flow with Buoyancy Forces

(Mixed Forced and Natural Convection) . . . . . . . . . . . . . . . . . . . 864.3 Similarity Laws for Natural Convection . . . . . . . . . . . . . . . . . . . 904.4 Vorticity Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.5 Limit of Very Small Reynolds Numbers . . . . . . . . . . . . . . . . . . . 934.6 Limit of Very Large Reynolds Numbers . . . . . . . . . . . . . . . . . . . 944.7 Mathematical Example of the Limit Re→∞ . . . . . . . . . . . . . . 964.8 Non–Uniqueness of Solutions of the Navier–Stokes Equations . 99

5. Exact Solutions of the Navier–Stokes Equations . . . . . . . . . . 1015.1 Steady Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Couette–Poiseuille Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1.2 Jeffery–Hamel Flows

(Fully Developed Nozzle and Diffuser Flows) . . . . . . . . . 1045.1.3 Plane Stagnation–Point Flow . . . . . . . . . . . . . . . . . . . . . . 1105.1.4 Flow Past a Parabolic Body . . . . . . . . . . . . . . . . . . . . . . . 1155.1.5 Flow Past a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . 115

5.2 Steady Axisymmetric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2.1 Circular Pipe Flow (Hagen–Poiseuille Flow) . . . . . . . . . 1165.2.2 Flow Between Two Concentric Rotating Cylinders . . . . 1175.2.3 Axisymmetric Stagnation–Point Flow . . . . . . . . . . . . . . . 1185.2.4 Flow at a Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.5 Axisymmetric Free Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Unsteady Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3.1 Flow at a Wall Suddenly Set into Motion

(First Stokes Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3.2 Flow at an Oscillating Wall

(Second Stokes Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.3.3 Start–up of Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3.4 Unsteady Asymptotic Suction . . . . . . . . . . . . . . . . . . . . . . 1315.3.5 Unsteady Plane Stagnation–Point Flow . . . . . . . . . . . . . 1315.3.6 Oscillating Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.4 Unsteady Axisymmetric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.4.1 Vortex Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.4.2 Unsteady Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Contents XIII

Part II. Laminar Boundary Layers

6. Boundary–Layer Equations in Plane Flow;Plate Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.1 Setting up the Boundary–Layer Equations . . . . . . . . . . . . . . . . . 1456.2 Wall Friction, Separation and Displacement . . . . . . . . . . . . . . . . 1506.3 Dimensional Representation

of the Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 1526.4 Friction Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.5 Plate Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7. General Properties and Exact Solutionsof the Boundary–Layer Equationsfor Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.1 Compatibility Condition at the Wall . . . . . . . . . . . . . . . . . . . . . . 1667.2 Similar Solutions of the Boundary–Layer Equations . . . . . . . . . 167

7.2.1 Derivation of the Ordinary Differential Equation . . . . . 167A Boundary Layers with Outer Flow . . . . . . . . . . . . . . . 169B Boundary Layers Without Outer Flow . . . . . . . . . . . . 172

7.2.2 Wedge Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.2.3 Flow in a Convergent Channel . . . . . . . . . . . . . . . . . . . . . 1747.2.4 Mixing Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.2.5 Moving Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.2.6 Free Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.2.7 Wall Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.3 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.3.1 Gortler Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.3.2 v. Mises Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.3.3 Crocco Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.4 Series Expansion of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 1847.4.1 Blasius Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.4.2 Gortler Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.5 Asymptotic Behaviour of Solutions Downstream . . . . . . . . . . . . 1877.5.1 Wake Behind Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.5.2 Boundary Layer at a Moving Wall . . . . . . . . . . . . . . . . . . 190

7.6 Integral Relations of the Boundary Layer . . . . . . . . . . . . . . . . . . 1917.6.1 Momentum–Integral Equation . . . . . . . . . . . . . . . . . . . . . 1917.6.2 Energy–Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1927.6.3 Moment–of–Momentum Integral Equations . . . . . . . . . . 194

8. Approximate Methods for Solving the Boundary–LayerEquations for Steady Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . 1958.1 Integral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

XIV Contents

8.2 Stratford’s Separation Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.3 Comparison of the Approximate Solutions

with Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.3.1 Retarded Stagnation–Point Flow . . . . . . . . . . . . . . . . . . . 2028.3.2 Divergent Channel (Diffuser) . . . . . . . . . . . . . . . . . . . . . . 2048.3.3 Circular Cylinder Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.3.4 Symmetric Flow past a Joukowsky Airfoil . . . . . . . . . . . 207

9. Thermal Boundary Layers

to the Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.1 Boundary–Layer Equations for the Temperature Field . . . . . . . 2099.2 Forced Convection for Constant Properties . . . . . . . . . . . . . . . . 2119.3 Effect of the Prandtl Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2159.4 Similar Solutions of the Thermal Boundary Layer . . . . . . . . . . 2189.5 Integral Methods for Computing the Heat Transfer . . . . . . . . . 2239.6 Effect of Dissipation;

Distribution of the Adiabatic Wall Temperature . . . . . . . . . . . . 226

10. Thermal Boundary Layerswith Coupling of the Velocity Fieldto the Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.2 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.3 Boundary Layers with Moderate Wall Heat Transfer

(Without Gravitational Effects) . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.3.1 Perturbation Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.3.2 Property Ratio Method (Temperature Ratio Method) . 23710.3.3 Reference Temperature Method . . . . . . . . . . . . . . . . . . . . 240

10.4 Compressible Boundary Layers(Without Gravitational Effects) . . . . . . . . . . . . . . . . . . . . . . . . . . 24110.4.1 Physical Property Relations . . . . . . . . . . . . . . . . . . . . . . . 24110.4.2 Simple Solutions of the Energy Equation . . . . . . . . . . . . 24410.4.3 Transformations of the Boundary–Layer Equations . . . 24610.4.4 Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24910.4.5 Integral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25810.4.6 Boundary Layers in Hypersonic Flows . . . . . . . . . . . . . . . 263

10.5 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.5.1 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 26510.5.2 Transformation of the Boundary–Layer Equations . . . . 27010.5.3 Limit of Large Prandtl Numbers (Tw = const) . . . . . . . 27110.5.4 Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27310.5.5 General Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27710.5.6 Variable Physical Properties . . . . . . . . . . . . . . . . . . . . . . . 27810.5.7 Effect of Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

without Coupling of the Velocity Field

Contents XV

10.6 Indirect Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28110.7 Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

11. Boundary–Layer Control (Suction/Blowing) . . . . . . . . . . . . . . 29111.1 Different Kinds of Boundary–Layer Control . . . . . . . . . . . . . . . . 29111.2 Continuous Suction and Blowing . . . . . . . . . . . . . . . . . . . . . . . . . 295

11.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.2.2 Massive Suction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29711.2.3 Massive Blowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.2.4 Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30211.2.5 General Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

1. Plate Flow with Uniform Suction or Blowing . . . . . . 3072. Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

11.2.6 Natural Convection with Blowing and Suction . . . . . . . 31011.3 Binary Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

11.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.3.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31211.3.3 Analogy Between Heat and Mass Transfer . . . . . . . . . . . 31611.3.4 Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

12. Axisymmetric and Three–Dimensional Boundary Layers . . 32112.1 Axisymmetric Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12.1.1 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 32112.1.2 Mangler Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32312.1.3 Boundary Layers

on Non–Rotating Bodies of Revolution . . . . . . . . . . . . . . 32412.1.4 Boundary Layers on Rotating Bodies of Revolution . . . 32712.1.5 Free Jets and Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

12.2 Three–Dimensional Boundary Layers . . . . . . . . . . . . . . . . . . . . . . 33512.2.1 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 33512.2.2 Boundary Layer at a Cylinder . . . . . . . . . . . . . . . . . . . . . 34112.2.3 Boundary Layer at a Yawing Cylinder . . . . . . . . . . . . . . 34212.2.4 Three–Dimensional Stagnation Point . . . . . . . . . . . . . . . . 34412.2.5 Boundary Layers in Symmetry Planes . . . . . . . . . . . . . . . 34512.2.6 General Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

13. Unsteady Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34913.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

13.1.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34913.1.2 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 35013.1.3 Similar and Semi–Similar Solutions . . . . . . . . . . . . . . . . . 35113.1.4 Solutions for Small Times (High Frequencies) . . . . . . . . 35213.1.5 Separation of Unsteady Boundary Layers . . . . . . . . . . . . 35313.1.6 Integral Relations and Integral Methods . . . . . . . . . . . . . 354

13.2 Unsteady Motion of Bodies in a Fluid at Rest . . . . . . . . . . . . . . 355

XVI Contents

13.2.1 Start–Up Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35513.2.2 Oscillation of Bodies in a Fluid at Rest . . . . . . . . . . . . . 362

13.3 Unsteady Boundary Layers in a Steady Basic Flow . . . . . . . . . 36513.3.1 Periodic Outer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36513.3.2 Steady Flow with a Weak Periodic Perturbation . . . . . . 36713.3.3 Transition Between Two Slightly Different Steady

Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36913.4 Compressible Unsteady Boundary Layers . . . . . . . . . . . . . . . . . . 370

13.4.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37013.4.2 Boundary Layer Behind a Moving

Normal Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37113.4.3 Flat Plate at Zero Incidence with Variable Free Stream

Velocity and Wall Temperature . . . . . . . . . . . . . . . . . . . . 373

14. Extensions to the Prandtl Boundary–Layer Theory . . . . . . . 37714.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37714.2 Higher Order Boundary–Layer Theory . . . . . . . . . . . . . . . . . . . . 37914.3 Hypersonic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38914.4 Triple–Deck Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39214.5 Marginal Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40314.6 Massive Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Part III. Laminar–Turbulent Transition

15. Onset of Turbulence (Stability Theory) . . . . . . . . . . . . . . . . . . . 41515.1 Some Experimental Results

on the Laminar–Turbulent Transition . . . . . . . . . . . . . . . . . . . . . 41515.1.1 Transition in the Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . 41515.1.2 Transition in the Boundary Layer . . . . . . . . . . . . . . . . . . 419

15.2 Fundamentals of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . 42415.2.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42415.2.2 Fundamentals of Primary Stability Theory . . . . . . . . . . 42515.2.3 Orr–Sommerfeld Equation . . . . . . . . . . . . . . . . . . . . . . . . . 42715.2.4 Curve of Neutral Stability

and the Indifference Reynolds Number . . . . . . . . . . . . . . 434a Plate Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 436b Effect of Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . 445c Effect of Suction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457d Effect of Wall Heat Transfer . . . . . . . . . . . . . . . . . . . . . 460e Effect of Compressibility . . . . . . . . . . . . . . . . . . . . . . . . 463f Effect of Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . 467g Further Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

15.3 Instability of the Boundary Layerfor Three–Dimensional Perturbations . . . . . . . . . . . . . . . . . . . . . 473

Contents XVII

15.3.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.3.2 Fundamentals of Secondary Stability Theory . . . . . . . . . 47615.3.3 Boundary Layers at Curved Walls . . . . . . . . . . . . . . . . . . 48115.3.4 Boundary Layer at a Rotating Disk . . . . . . . . . . . . . . . . . 48515.3.5 Three–Dimensional Boundary Layers . . . . . . . . . . . . . . . 487

15.4 Local Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Part IV. Turbulent Boundary Layers

16. Fundamentals of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . 49916.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49916.2 Mean Motion and Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . .16.3 Basic Equations for the Mean Motion of Turbulent Flows . . . . 504

16.3.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50416.3.2 Momentum Equations (Reynolds Equations) . . . . . . . . . 50516.3.3 Equation for the Kinetic Energy

of the Turbulent Fluctuations (k-Equation) . . . . . . . . . . 50716.3.4 Thermal Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . 510

16.4 Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16.5 Description of the Turbulent Fluctuations . . . . . . . . . . . . . . . . . 512

16.5.1 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51216.5.2 Spectra and Eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51316.5.3 Turbulence of the Outer Flow . . . . . . . . . . . . . . . . . . . . . . 51516.5.4 Edges of Turbulent Regions and Intermittence . . . . . . . 515

16.6 Boundary–Layer Equations for Plane Flows . . . . . . . . . . . . . . . . 516

17. Internal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51917.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

17.1.1 Two–Layer Structure of the Velocity Fieldand the Logarithmic Overlap Law . . . . . . . . . . . . . . . . . . 519

17.1.2 Universal Laws of the Wall . . . . . . . . . . . . . . . . . . . . . . . . 52417.1.3 Friction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53617.1.4 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53817.1.5 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

17.2 Fully Developed Internal Flows (A = const) . . . . . . . . . . . . . . . . 54317.2.1 Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54317.2.2 Couette–Poiseuille Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 54417.2.3 Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

17.3 Slender–Channel Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

18. Turbulent Boundary Layers

to the Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55718.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

without Coupling of the Velocity Field

501

511

XVIII Contents

18.1.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55718.1.2 Algebraic Turbulence Models . . . . . . . . . . . . . . . . . . . . . . 55918.1.3 Turbulent Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . 56018.1.4 Two–Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56218.1.5 Reynolds Stress Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 56518.1.6 Heat Transfer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56818.1.7 Low–Reynolds–Number Models . . . . . . . . . . . . . . . . . . . . 57018.1.8 Large–Eddy Simulation

and Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . 57118.2 Attached Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

18.2.1 Layered Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57218.2.2 Boundary–Layer Equations Using

the Defect Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57418.2.3 Friction Law and Characterisitic Quantities

of the Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57718.2.4 Equilibrium Boundary Layers . . . . . . . . . . . . . . . . . . . . . . 58018.2.5 Boundary Layer on a Plate at Zero Incidence . . . . . . . . 582

18.3 Boundary Layers with Separation . . . . . . . . . . . . . . . . . . . . . . . . . 58918.3.1 Stratford Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58918.3.2 Quasi–Equilibrium Boundary Layers . . . . . . . . . . . . . . . . 591

18.4 Computation of Boundary Layers Using Integral Methods . . . 59418.4.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59418.4.2 Inverse Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

18.5 Computation of Boundary Layers Using Field Methods . . . . . . 59818.5.1 Attached Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 59818.5.2 Boundary Layers with Separation . . . . . . . . . . . . . . . . . . 60118.5.3 Low–Reynolds–Number Turbulence Models . . . . . . . . . . 60318.5.4 Additional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

18.6 Computation of Thermal Boundary Layers . . . . . . . . . . . . . . . . 60718.6.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60718.6.2 Computation of Thermal Boundary Layers

Using Field Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

19. Turbulent Boundary Layerswith Coupling of the Velocity Fieldto the Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61119.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

19.1.1 Time Averaging for Variable Density . . . . . . . . . . . . . . . . 61119.1.2 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 613

19.2 Compressible Turbulent Boundary Layers . . . . . . . . . . . . . . . . . 61719.2.1 Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61719.2.2 Overlap Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61919.2.3 Skin–Friction Coefficient and Nusselt Number . . . . . . . . 62119.2.4 Integral Methods for Adiabatic Walls . . . . . . . . . . . . . . . 623

Contents XIX

19.2.5 Field Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62519.2.6 Shock–Boundary–Layer Interaction . . . . . . . . . . . . . . . . . 625

19.3 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

20. Axisymmetric and Three–DimensionalTurbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63120.1 Axisymmetric Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 631

20.1.1 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 63120.1.2 Boundary Layers without Body Rotation . . . . . . . . . . . 63220.1.3 Boundary Layers with Body Rotation . . . . . . . . . . . . . . . 635

20.2 Three–Dimensional Boundary Layers . . . . . . . . . . . . . . . . . . . . . . 63720.2.1 Boundary–Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 63720.2.2 Computation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64120.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

21. Unsteady Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . 64521.1 Averaging and Boundary–Layer Equations . . . . . . . . . . . . . . . . . 64521.2 Computation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64821.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

22. Turbulent Free Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65322.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65322.2 Equations for Plane Free Shear Layers . . . . . . . . . . . . . . . . . . . . 65522.3 Plane Free Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

22.3.1 Global Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65922.3.2 Far Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66022.3.3 Near Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66522.3.4 Wall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

22.4 Mixing Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66722.5 Plane Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66922.6 Axisymmetric Free Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 671

22.6.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67122.6.2 Free Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67222.6.3 Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

22.7 Buoyant Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67522.7.1 Plane Buoyant Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67522.7.2 Axisymmetric Buoyant Jet . . . . . . . . . . . . . . . . . . . . . . . . 676

22.8 Plane Wall Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

.

XX Contents

Part V. Numerical Methods in Boundary–Layer Theory

23. Numerical Integration of the Boundary–Layer Equations . 68323.1 Laminar Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

23.1.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68323.1.2 Note on Boundary–Layer Transformations . . . . . . . . . . . 68423.1.3 Explicit and Implicit Discretisation . . . . . . . . . . . . . . . . . 68523.1.4 Solution of the Implicit Difference Equations . . . . . . . . . 68923.1.5 Integration of the Continuity Equation . . . . . . . . . . . . . . 69123.1.6 Boundary–Layer Edge and Wall Shear Stress . . . . . . . . 69123.1.7 Integration of the Transformed Boundary–Layer

Equations Using the Box Scheme . . . . . . . . . . . . . . . . . . . 69223.2 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

23.2.1 Method of Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . 69523.2.2 Low–Reynolds–Number Turbulence Models . . . . . . . . . . 700

23.3 Unsteady Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70123.4 Steady Three–Dimensional Boundary Layers . . . . . . . . . . . . . . . 703

List of Frequently Used Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

References and Index of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Introduction

Short historical review

At the end of the 19th century, fluid mechanics had split into two differentdirections which hardly had anything more in common. On one side was thescience of theoretical hydrodynamics, emanating from Euler’s equations ofmotion and which had been developed to great perfection. However this hadvery little practical importance, since the results of this so–called classicalhydrodynamics were in glaring contradiction to everyday experience. Thiswas particularly true in the very important case of pressure loss in tubes andchannels, as well as that of the drag experienced by a body moved through afluid. For this reason, engineers, on the other side, confronted by the practicalproblems of fluid mechanics, developed their own strongly empirical science,hydraulics. This relied upon a large amount of experimental data and differedgreatly from theoretical hydrodynamics in both methods and goals.It is the great achievement of Ludwig Prandtl which, at the beginning of

this century, set forth the way in which these two diverging directions of fluidmechanics could be unified. He achieved a high degree of correlation betweentheory and experiment, which, in the first half of this century, has led tounimagined successes in modern fluid mechanics. It was already known thenthat the great discrepancy between the results in classical hydrodynamicsand reality was, in many cases, due to neglecting the viscosity effects in thetheory. Now the complete equations of motion of viscous flows (the NavierStokes equations) had been known for some time. However, due to the greatmathematical difficulty of these equations, no approach had been found to themathematical treatment of viscous flows (except in a few special cases). Fortechnically important fluids such as water and air, the viscosity is very small,and thus the resulting viscous forces are small compared to the remainingforces (gravitational force, pressure force). For this reason it took a long timeto see why the viscous forces ignored in the classical theory should have animportant effect on the motion of the flow.In his lecture on “Uber Flussigkeitbewegung bei sehr kleiner Reibung”

(On Fluid Motion with Very Small Friction) at the Heidelberg mathemati-cal congress in 1904, L. Prandtl (1904) showed how a theoretical treatmentcould be used on viscous flows in cases of great practical importance. Usingtheoretical considerations together with some simple experiments, Prandtl

XXII Introduction

showed that the flow past a body can be divided into two regions: a very thinlayer close to the body (boundary layer) where the viscosity is important,and the remaining region outside this layer where the viscosity can be ne-glected. With the help of this concept, not only was a physically convincingexplanation of the importance of the viscosity in the drag problem given,but simultaneously, by hugely reducing the mathematical difficulty, a pathwas set for the theoretical treatment of viscous flows. Prandtl supported histheoretical work by some very simple experiments in a small, self–built waterchannel, and in doing this reinitiated the lost connection between theory andpractice. The theory of the Prandtl boundary layer or the frictional layer hasproved to be exceptionally useful and has given considerable stimulation toresearch into fluid mechanics since the beginning of this century. Under theinfluence of a thriving flight technology, the new theory developed quicklyand soon became, along with other important advances – airfoil theory andgas dynamics – a keystone of modern fluid mechanics.One of the most important applications of boundary–layer theory is the

calculation of the friction drag of bodies in a flow, e.g. the drag of a flatplate at zero incidence, the friction drag of a ship, an airfoil, the body of anairplane, or a turbine blade. One particular property of the boundary layeris that, under certain conditions, a reverse flow can occur directly at thewall. A separation of the boundary layer from the body and the formationof large or small eddies at the back of the body can then occur. This resultsin a great change in the pressure distribution at the back of the body, lead-ing to the form or pressure drag of the body. This can also be calculatedusing boundary–layer theory. Boundary–layer theory answers the importantquestion of what shape a body must have in order to avoid this detrimentalseparation. It is not only in flow past a body where separation can occur, butalso in flow through a duct. In this way boundary–layer theory can be usedto describe the flow through blade cascades in compressors and turbines, aswell as through diffusers and nozzles. The processes involved in maximumlift of an airfoil, where separation is also important, can only be understoodusing boundary–layer theory. The boundary layer is also important for heattransfer between a body and the fluid around it.Initially boundary–layer theory was developed mainly for the laminar flow

of an incompressible fluid, where Stokes law of friction could be used as anansatz for the viscous forces. This area was later researched in very manypieces of work, so that today it can be considered to be fully understood.Later the theory was extended to the practically important turbulent in-compressible boundary–layer flows. Around 1890, O. Reynolds (1894) hadalready introduced the fundamentally important concept of apparent turbu-lent stresses, but this did not yet permit the theoretical treatment of tur-bulent flows. The introduction of the concept of the Prandtl mixing length,cf. L. Prandtl (1925), contributed considerable advances and, together withsystematic experiments, allowed turbulent flows to be treated theoretically

Introduction XXIII

with the help of boundary–layer theory. Even today a rational theory of fullydeveloped turbulent flows remains to be found. Thanks to the great increaseof velocities in flight technology, boundary layers in compressible flows weresubsequently also thoroughly examined. As well as the boundary layer inthe velocity field, a thermal boundary layer also forms; this is of great im-portance for the heat transfer between the flow and the body. Because ofinternal friction (dissipation) at high Mach numbers, the body surface heatsup greatly. This causes many problems, particularly in flight technology andsatellite flights (“thermal barrier”).The transition from laminar to turbulent flow, important for all of fluid

mechanics, was first examined in pipe flow at the end of the last century byO. Reynolds (1883). Using the flow about a sphere, in 1914 Prandtl was ableto show experimentally that the boundary layer also can be both laminaror turbulent and that the process of separation and thus the drag problemare controlled by this laminar–turbulent transition, cf. L. Prandtl (1914)The theoretical investigations into this transition assume Reynolds’ idea ofthe instability of the laminar flow. This was treated by Prandtl in 1921.After some futile attempts, W. Tollmien (1929) and H. Schlichting (1933)were able to theoretically calculate the indifference Reynolds number for theflat plate at zero incidence. However it took more than ten years before thetheory could be confirmed by careful experiments by H.L. Dryden (1946–1948) and his coworkers. The effect of other parameters on the transition(pressure gradient, suction, Mach number, heat transfer) were clarified usingthe stability theory of the boundary layer. This theory has found importantapplication with, among other things, airfoils with very low drag (laminarairfoils).An important characteristic of modern research into fluid mechanics in

general and more specifically into the branch of boundary–layer theory is theclose connection between theory and experiment. The most crucial advanceshave been achieved through a few fundamental experiments together withtheoretical considerations. Many years ago, A. Betz (1949) produced a re-view of the development of boundary–layer theory, with particular emphasison the mutual fructification of theory and experiment. Research into bound-ary layers, inspired by Prandtl from 1904, were, in the first 20 years up untilPrandtl’s Wilbur Wright memorial lecture at the Royal Aeronautical Soci-ety in London, (L. Prandtl (1927)) almost exclusively confined to Prandtl’sinstitute in Gottingen. It is only since 1930 that other researchers have beeninvolved in the further expansion of boundary–layer theory, initially in Eng-land and the USA. Today boundary–layer theory has spread over the wholeworld; together with other branches it forms one of the most important pillarsof fluid mechanics.In the mid–fifties, mathematical methods into singular perturbation the-

ory were being systematically developed, cf. S. Kaplun (1954), S. Kaplun;P.A. Lagerstrom (1957), M. Van Dyke (1964b), also W. Schneider (1978).

XXIV Introduction

It became clear that the boundary–layer theory heuristically developed byPrandtl was a classic example of the solution of a singular perturbationproblem. Thus boundary–layer theory is a rational asymptotic theory ofthe solution of the Navier–Stokes equations for high Reynolds numbers, cf.K. Gersten (2000). This opened the possibility of a systematic developmentto higher order boundary–layer theory, cf. M. Van Dyke (1969), K. Gersten(1972), K. Stewartson (1974), K. Gersten; J.F. Gross (1976), V.V. Sychevet al. (1998), I.J. Sobey (2000). The asymptotic methods which were firstdeveloped for laminar flows were then, at the start of the seventies, carriedover to turbulent flows, cf. K.S. Yajnik (1970), G.L. Mellor (1972). Reviewsof asymptotic theory of turbulent flows are to be found in K. Gersten (1987),K. Gersten (1989c), A. Kluwick (1989a), as well as W. Schneider (1991).K. Gersten; H. Herwig (1992) have presented a systematic application ofasymptotic methods (regular and singular perturbation methods) to the the-ory of viscous flows. The book by P.A. Libby (1998) also gives preferentialtreatment to asymptotic methods. Most of the characteristics of the asymp-totic theory for high Reynolds-number flows can already be found in Prandtl’swork, cf. K. Gersten (2000).In turbulence modelling, the mixing length hypothesis developed by

L. Prandtl (1925) led to an algebraic turbulence model. Twenty years later,L. Prandtl (1945) showed how transport equations for turbulent quantitiessuch as the kinetic energy of the random motion, the dissipation and theReynolds shear stress could be applied to improve to turbulence models. Cal-culation methods for turbulent boundary layers with highly refined turbu-lence models have been developed by, for example, P. Bradshaw et al. (1967),W.P. Jones; B.E. Launder (1973), K. Hanjalic; B.E. Launder (1976), as wellas J.C. Rotta (1973). Overviews on turbulence modelling are presented byW.C. Reynolds (1976) and V.C. Patel et al. (1985). In two extremely notewor-thy events at Stanford University in the years 1968 and 1980/81, the existingmethods for calculating boundary layers were compared and examined inspecially chosen experiments; see the reports by S.J. Kline et al. (1968) andS.J. Kline et al. (1981). A review on Reynolds number effects in wall-boundedturbulent flows by M. Gad-el-Hak; P.R. Bandyopadhyay (1994) is also worthmentioning.The following tendency is emerging from the rapid developments in the

area of supercomputing: the future will consist more of direct numerical solu-tions of the Navier–Stokes equations without any simplifications, and also ofthe computation of turbulent flows using direct numerical simulation (DNS),i.e. without using a turbulence model or by modelling only high frequencyturbulent fluctuations (“large–eddy simulations”), cf. D.R. Chapman (1979).However numerical methods in computing flows at high Reynolds numbersonly become efficient if the particular layered structure of the flow, as givenby the asymptotic theory, is taken into account, as occurs if a suitable grid

Introduction XXV

is used for computation. Boundary–layer theory will therefore retain its fun-damental place in the calculation of high Reynolds number flows.The first summary of boundary–layer theory is to be found in two short

articles byW. Tollmien (1931) in the Handbuch der Experimentalphysik. Someyears later Prandtl’s comprehensive contribution appeared in AerodynamicTheory, edited by W.F. Durand, L. Prandtl (1935). In the six decades sincethen, the extent of this research area has become extraordinarily large. cf.H. Schlichting (1960) and also I. Tani (1977), A.D. Young (1989), K. Gersten(1989a), A. Kluwick (1998) and T. Cebeci; J. Cousteix (2005). According toa review by H.L. Dryden (1955), about 100 articles appeared in the year 1955,and 45 years later this number has grown to about 800 per year.

In spite of these developments concerning the numerical solutions of the full Navier-Stokes equations it can also be noticed that in recent years the extensions of boundary layer theory received increasing attention. Several textbooks have been published recently.

The Interactive Boundary Layer Theory by I.J. Sobey (2000) and the Asymptotic Theory of Separated Flows by V.V. Sychev et al. (1998) refer to laminar flows. The books by E.H. Hirschel et al. (2014), T. Cebeci (1999, 2004), M. Hallbäck et al. (Eds.) (1996) and T.K. Sengupta (2012) are

totic methods for high Reynolds number flows will be discussed in the appropriate Chapters 14 and 18.

About the year 2000 the numerical methods in fluid mechanics had reached a standard that made it possible to solve the full Navier-Stokes equations. So-called Navier-Stokes/RANS methods (Reynolds Averaged Navier-Stokes) were in wide use at universities, research establishments and industry. Reynolds-stress models came into use, which in principle take into account non-isotropy of turbulence. Turbulence flow separation still remains to be a major issue. Maybe hybrid RANS-LES methods (Large-Eddy Simulation, LES) are the ultimate industrial methods for the simu-lation of flow fields past realistic vehicle configurations, see E.H. Hirschel et al. (2014).

concentrated on turbulent and transitional flows. The state of research canalso be found in H. Steinrück (Ed.) (2010), A. Kluwick (Ed.) (1998) and G.E.A. Meier et al. (Eds.) (2006). These new developments of the asymp-

Abstract

The nonlinear partial differential equations describing all general flow fields arecalled Navier-Stokes equations. In non-dimensional form these equations dependon the Reynolds number, which determines the effect of the viscosity in the flow.The higher the Reynolds number, the lower are the viscosity effects on the flow.

The boundary-layer theory is the asymptotic theory of the Navier-Stokes equa-tions for high Reynolds numbers. This theory has been developed by LudwigPrandtl (1904). Although this theory is now more than 110 years old, it is nowadaysstill being applied in industry and research, because many important fields of fluidmechanics (i.e. aeronautics, ship hydrodynamics, automobile aerodynamics) referto flows at high Reynolds numbers.

The book has 23 Chapters and is divided into 5 Parts:

I Fundamentals of Viscous FlowsII Laminar Boundary LayersIII Laminar-Turbulent TransitionIV Turbulent Boundary LayersV Numerical Methods in Boundary Layer Theory

The boundary-layer theory is a perturbation method, because it starts from thelimiting solution for infinitely large Reynolds number (solution for inviscid fluid)and is then perturbed by viscosity effects. Since the inviscid flow solution does notsatisfy the no-slip condition at the wall, boundary-layer theory is called a singularperturbation method. For high Reynolds numbers the entire flow field consists oftwo different regions. In the larger of these two regions the flow is inviscid. Thesecond region is a very narrow layer close to the wall called the boundary layer.The complete solution can be found by the method of matched asymptoticexpansions as long as the boundary layer keeps attached to the wall.

When the boundary layer leads to separation, the limiting invisced solution (asstarting solution) is generally not known a priori. In this case a dimensionless lengthparameter (for instance: a step height compared to a flat plate) must be taken intothe analysis. The two dimensionless parameters (geometrical parameter andReynolds number) lead to a three-layer structure of the resulting flow field called

triple deck theory or asymptotic interaction theory. Within this theory boundarylayers with separation and reattachment can be calculated without any singularities(Chapter 14.4). Where separation occurs in a flow, the limiting solution can often bechosen to be that flow where the geometry is changed just so that there is noseparation (marginal separation: laminar: Chapter 14.5; turbulent: Chapter 18.5.2).

XXVIII Abstract