Hermann Schlichting (Deceased) Klaus Gersten. Boundary- Layer Theory. Ninth Edition

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1 Hermann Schlichting (Deceased) Klaus Gersten Boundary- Layer Theory Ninth Edition

2 Boundary-Layer Theory

3 Hermann Schlichting (Deceased) Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes Ninth Edition 123

4 Hermann Schlichting (Deceased) Institute of Fluid Mechanics Technical University of Braunschweig Braunschweig Germany Klaus Gersten Institute of Thermodynamics and Fluid Mechanics Ruhr-University Bochum Bochum, Nordrhein-Westfalen Germany ISBN ISBN (ebook) DOI / Library of Congress Control Number: st edition: McGraw-Hill New York nd edition: Pergamon London th edition: McGraw-Hill New York th edition: McGraw-Hill New York th edition: McGraw-Hill New York th edition: Springer-Verlag Berlin Heidelberg th edition: Springer-Verlag Berlin Heidelberg 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 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 transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the 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 this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Typesetting: Katherine Mayes, Darmstadt and LE-TEX, Leipzig Cover design: de blik, Berlin Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg

5 Preface to the Ninth English Edition For this edition corrections have been carried out and additional important 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. Bochum, March 2016 Klaus Gersten

6 Preface to the Eighth English Edition According to the tradition of this book, a German edition has always been soon followed by the English translation. I am very grateful to Springer Verlag for undertaking this version and for securing a translator. My particular thanks go to Katherine Mayes for this excellent translation. In the course of the translation, some errors in the German edition were corrected and a number of additions carried out. In this connection I am very thankful to Prof. Dr. W. Schneider, Vienna, for several suggestions and improvements. I would like to thank Ursula Beitz again for her careful checking of the bibliography. I hope that the English edition will attain the same positive resonance as the ninth German edition. Bochum, May 1999 Klaus Gersten

7 Preface to the Ninth German Edition There is no doubt that Boundary Layer Theory by Hermann Schlichting is one of most important books within the sphere of fluid mechanics to appear in the last decade. Shortly before his death, Hermann Schlichting brought out the eighth edition which he revised together with his friend and former colleague 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 I spent at the institute of my highly respected teacher Hermann Schlichting, I had already been involved with earlier editions of the book and had revised some chapters. The burden was also eased by the fact that boundary layer theory in its widest sense has been my preferred direction of research for many years. It quickly became clear that a complete revision was necessary; indeed this was also known to Hermann Schlichting. In the preface to the eighth edition he wrote: Noting the systematic of our knowledge of today, it would have been desirable to fully revise this work; however such a process would have pushed back the appearance of this book by years. Compared to the eighth edition, the literature of the last 15 years had to be taken into account and recent developments, in turbulence models for example, had to be incorporated. In order to keep the size of the book tractable, some results those which no longer seem so important with today s computing potential had to be curtailed, or in some cases, left out altogether. Thus the necessity to completely rewrite the text emerged. The fundamental divisions within the book were retained; as before it consists of the four major sections: basic laws of the flows of viscous fluids, laminar boundary layers, the onset of turbulence, turbulent boundary layers. However a new fifth section on numerical methods in boundary layer theory has been added. The partition into chapters had to be somewhat modified in order to improve the style of presentation of the material. Because of the necessary restrictions on the material, the aim was to concentrate on boundary layer theory as the theory of high Reynolds number flows. Accordingly the chapter on creeping flows, that is flows at very small Reynolds numbers, was omitted.

8 X Preface It seemed natural to steer towards the style and level of presentation with the 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 complete overview. Consequently I am extremely grateful to two colleagues who supported me actively. Professor E. Krause wrote the new additional chapter on numerical methods in boundary layer theory, and Professor H. Oertel provided 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 Schäfer and Dr.-Ing. Detlev Vieth for a great many new sample calculations. Dr. Vieth also read the entire text discerningly. I am grateful to him for numerous improving suggestions. Renate Gölzenleuchtner deserves particular thanks for generating the figures which almost all had to be newly drawn up. I would like to thank Ursula Beitz particularly for her careful and exhaustive checking of the bibliography, while Marianne Ferdinand and Eckhard Schmidt were of first class assistance. It was by far impossible to adopt all citations, so that it may be necessary to revert to the eighth edition for specific references to earlier pieces of work. The printing firm of Jörg Steffenhagen is due particular praise for an extremely fruitful collaboration. My thanks also go to Springer Verlag for our 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

9 Contents Introduction... XXI Part I. Fundamentals of Viscous Flows 1. Some Features of Viscous Flows Real and Ideal Fluids Viscosity Reynolds Number Laminar and Turbulent Flows Asymptotic Behaviour at Large Reynolds Numbers Comparison of Measurements Using the Inviscid Limiting Solution Summary Fundamentals of Boundary Layer Theory Boundary Layer Concept Laminar Boundary Layer on a Flat Plate at Zero Incidence Turbulent Boundary Layer on a Flat Plate at Zero Incidence Fully Developed Turbulent Flow in a Pipe Boundary Layer on an Airfoil Separation of the Boundary Layer Overview of the Following Material Field Equations for Flows of Newtonian Fluids Description of Flow Fields Continuity Equation Momentum Equation General Stress State of Deformable Bodies General State of Deformation of Flowing Fluids Relation Between Stresses and Rate of Deformation Stokes Hypothesis Bulk Viscosity and Thermodynamic Pressure Navier Stokes Equations... 68

10 XII Contents 3.10 Energy Equation Equations of Motion for Arbitrary Coordinate Systems (Summary) Equations of Motion for Cartesian Coordinates in Index Notation Equations of Motion in Different Coordinate Systems General Properties of the Equations of Motion Similarity Laws Similarity Laws for Flow with Buoyancy Forces (Mixed Forced and Natural Convection) Similarity Laws for Natural Convection Vorticity Transport Equation Limit of Very Small Reynolds Numbers Limit of Very Large Reynolds Numbers Mathematical Example of the Limit Re Non Uniqueness of Solutions of the Navier Stokes Equations Exact Solutions of the Navier Stokes Equations Steady Plane Flows Couette Poiseuille Flows Jeffery Hamel Flows (Fully Developed Nozzle and Diffuser Flows) Plane Stagnation Point Flow Flow Past a Parabolic Body Flow Past a Circular Cylinder Steady Axisymmetric Flows Circular Pipe Flow (Hagen Poiseuille Flow) Flow Between Two Concentric Rotating Cylinders Axisymmetric Stagnation Point Flow Flow at a Rotating Disk Axisymmetric Free Jet Unsteady Plane Flows Flow at a Wall Suddenly Set into Motion (First Stokes Problem) Flow at an Oscillating Wall (Second Stokes Problem) Start up of Couette Flow Unsteady Asymptotic Suction Unsteady Plane Stagnation Point Flow Oscillating Channel Flow Unsteady Axisymmetric Flows Vortex Decay Unsteady Pipe Flow Summary

11 Contents XIII Part II. Laminar Boundary Layers 6. Boundary Layer Equations in Plane Flow; Plate Boundary Layer Setting up the Boundary Layer Equations Wall Friction, Separation and Displacement Dimensional Representation of the Boundary Layer Equations Friction Drag Plate Boundary Layer General Properties and Exact Solutions of the Boundary Layer Equations for Plane Flows Compatibility Condition at the Wall Similar Solutions of the Boundary Layer Equations Derivation of the Ordinary Differential Equation A Boundary Layers with Outer Flow B Boundary Layers Without Outer Flow Wedge Flows Flow in a Convergent Channel Mixing Layer Moving Plate Free Jet Wall Jet Coordinate Transformation Görtler Transformation v. Mises Transformation Crocco Transformation Series Expansion of the Solutions Blasius Series Görtler Series Asymptotic Behaviour of Solutions Downstream Wake Behind Bodies Boundary Layer at a Moving Wall Integral Relations of the Boundary Layer Momentum Integral Equation Energy Integral Equation Moment of Momentum Integral Equations Approximate Methods for Solving the Boundary Layer Equations for Steady Plane Flows Integral Methods

12 XIV Contents 8.2 Stratford s Separation Criterion Comparison of the Approximate Solutions with Exact Solutions Retarded Stagnation Point Flow Divergent Channel (Diffuser) Circular Cylinder Flow Symmetric Flow past a Joukowsky Airfoil Thermal Boundary Layers without Coupling of the Velocity Field to the Temperature Field Boundary Layer Equations for the Temperature Field Forced Convection for Constant Properties Effect of the Prandtl Number Similar Solutions of the Thermal Boundary Layer Integral Methods for Computing the Heat Transfer Effect of Dissipation; Distribution of the Adiabatic Wall Temperature Thermal Boundary Layers with Coupling of the Velocity Field to the Temperature Field Remark Boundary Layer Equations Boundary Layers with Moderate Wall Heat Transfer (Without Gravitational Effects) Perturbation Calculation Property Ratio Method (Temperature Ratio Method) Reference Temperature Method Compressible Boundary Layers (Without Gravitational Effects) Physical Property Relations Simple Solutions of the Energy Equation Transformations of the Boundary Layer Equations Similar Solutions Integral Methods Boundary Layers in Hypersonic Flows Natural Convection Boundary Layer Equations Transformation of the Boundary Layer Equations Limit of Large Prandtl Numbers (T w =const) Similar Solutions General Solutions Variable Physical Properties Effect of Dissipation

13 Contents XV 10.6 Indirect Natural Convection Mixed Convection Boundary Layer Control (Suction/Blowing) Different Kinds of Boundary Layer Control Continuous Suction and Blowing Fundamentals Massive Suction Massive Blowing Similar Solutions General Solutions Plate Flow with Uniform Suction or Blowing Airfoil Natural Convection with Blowing and Suction Binary Boundary Layers Overview Basic Equations Analogy Between Heat and Mass Transfer Similar Solutions Axisymmetric and Three Dimensional Boundary Layers Axisymmetric Boundary Layers Boundary Layer Equations Mangler Transformation Boundary Layers on Non Rotating Bodies of Revolution Boundary Layers on Rotating Bodies of Revolution Free Jets and Wakes Three Dimensional Boundary Layers Boundary Layer Equations Boundary Layer at a Cylinder Boundary Layer at a Yawing Cylinder Three Dimensional Stagnation Point Boundary Layers in Symmetry Planes General Configurations Unsteady Boundary Layers Fundamentals Remark Boundary Layer Equations Similar and Semi Similar Solutions Solutions for Small Times (High Frequencies) Separation of Unsteady Boundary Layers Integral Relations and Integral Methods Unsteady Motion of Bodies in a Fluid at Rest

14 XVI Contents Start Up Processes Oscillation of Bodies in a Fluid at Rest Unsteady Boundary Layers in a Steady Basic Flow Periodic Outer Flow Steady Flow with a Weak Periodic Perturbation Transition Between Two Slightly Different Steady Boundary Layers Compressible Unsteady Boundary Layers Remark Boundary Layer Behind a Moving Normal Shock Wave Flat Plate at Zero Incidence with Variable Free Stream Velocity and Wall Temperature Extensions to the Prandtl Boundary Layer Theory Remark Higher Order Boundary Layer Theory Hypersonic Interaction Triple Deck Theory Marginal Separation Massive Separation Part III. Laminar Turbulent Transition 15. Onset of Turbulence (Stability Theory) Some Experimental Results on the Laminar Turbulent Transition Transition in the Pipe Flow Transition in the Boundary Layer Fundamentals of Stability Theory Remark Fundamentals of Primary Stability Theory Orr Sommerfeld Equation Curve of Neutral Stability and the Indifference Reynolds Number a Plate Boundary Layer b Effect of Pressure Gradient c Effect of Suction d Effect of Wall Heat Transfer e Effect of Compressibility f Effect of Wall Roughness g Further Effects Instability of the Boundary Layer for Three Dimensional Perturbations

15 Contents XVII Remark Fundamentals of Secondary Stability Theory Boundary Layers at Curved Walls Boundary Layer at a Rotating Disk Three Dimensional Boundary Layers Local Perturbations Part IV. Turbulent Boundary Layers 16. Fundamentals of Turbulent Flows Remark Mean Motion and Fluctuations Basic Equations for the Mean Motion of Turbulent Flows Continuity Equation Momentum Equations (Reynolds Equations) Equation for the Kinetic Energy of the Turbulent Fluctuations (k-equation) Thermal Energy Equation Closure Problem Description of the Turbulent Fluctuations Correlations Spectra and Eddies Turbulence of the Outer Flow Edges of Turbulent Regions and Intermittence Boundary Layer Equations for Plane Flows Internal Flows Couette Flow Two Layer Structure of the Velocity Field and the Logarithmic Overlap Law Universal Laws of the Wall Friction Law Turbulence Models Heat Transfer Fully Developed Internal Flows (A = const) Channel Flow Couette Poiseuille Flows Pipe Flow Slender Channel Theory Turbulent Boundary Layers without Coupling of the Velocity Field to the Temperature Field Turbulence Models

Boundary-Layer Theory

Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22