Fluid Mechanics F Durst Fluid Mechanics An Introduction to the Theory of Fluid Flows With 347 Figures and 13 Tables 123 Prof Dr Dr h.c Franz Durst FMP Technology GmbH Am Weichselgarten 34 91058 Erlangen Germany f.durst@fmp-technology.com English Translation: Ingeborg Arnold Fliederstrasse 40 66119 Saarbrücken Germany arnoldwi@t-online.de ISBN: 978-3-540-71342-5 e-ISBN: 978-3-540-71343-2 Library of Congress Control Number: 2007937409 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, 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 Cover design: eStudio Calamar S.L., F Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper springer.com This book is dedicated to my wife Heidi and my sons Bodo Andr´e and Heiko Brian and their families Preface of German Edition Some readers familiar with fluid mechanics who come across this book may ask themselves why another textbook on the basics of fluid mechanics has been written, in view of the fact that the market in this field seems to be more than saturated The author is quite conscious of this situation, but he thinks all the same that this book is justified because it covers areas of fluid mechanics which have not yet been discussed in existing texts, or only to some extent, in the way treated here When looking at the textbooks available on the market that give an introduction into fluid mechanics, one realizes that there is hardly a text among them that makes use of the entire mathematical knowledge of students and that specifically shows the relationship between the knowledge obtained in lectures on the basics of engineering mechanics or physics and modern fluid mechanics There has been no effort either to activate this knowledge for educational purposes in fluid mechanics This book therefore attempts to show specifically the existing relationships between the above fields, and moreover to explain them in a way that is understandable to everybody and making it clear that the motions of fluid elements can be described by the same laws as the movements of solid bodies in engineering mechanics or physics The tensor representation is used for describing the basic equations, showing the advantages that this offers The present book on fluid mechanics makes an attempt to give an introductory structured representation of this special subject, which goes far beyond the potential-theory considerations and the employment of the Bernoulli equation, that often overburden the representations in fluid mechanics textbooks The time when potential theory and energy considerations, based on the Bernoulli equation, had to be the center of the fluid mechanical education of students is gone The development of modern measuring and computation techniques, that took place in the last quarter of the 20th century, up to the application level, makes detailed fluid-flow investigations possible nowadays, and for this aim students have to be educated vii viii Preface Using the basic education obtained in mathematics and physics, the present book strives at an introduction into fluid mechanics in such a way that each chapter is suited to provide the material for a one-week or two-week lectures, depending on the educational and knowledge level of the students The structure of the book helps students, who want to familiarize themselves with fluid mechanics, to recognize the material which they should study in addition to the lectures to become acquainted, chapter by chapter, with the entire field of fluid mechanics Moreover, the present text is also suited to study fluid mechanics on one’s own Each chapter is an introduction into a subfield of fluid mechanics Having acquired the substance of one chapter, it is easier to read more profound books on the same subfield, or to pursue advanced education by reading conference and journal publications In the description of the basic and most important fluid characteristic for fluid mechanics, the viscosity, much emphasis is given so that its physical cause is understood clearly The molecular-caused momentum transport, leading to the τij -terms in the basic fluid mechanical equations, is dealt with analogously to the molecular-dependent heat conduction and mass diffusion in fluids Explaining viscosity by internal “fluid friction” is physically wrong and is therefore not dealt with in this form in the book This text is meant to contribute so that readers familiarizing themselves with fluid mechanics gain quick access to this special subject through physically correctly presented fluid flows The present book is based on the lectures given by the author at the University of Erlangen-Nă urnberg as an introduction into uid mechanics Many students have contributed greatly to the compilation of this book by referring to unclarified points in the lecture manuscripts I should like to express my thanks for that I am also very grateful to the staff of the Fluid Mechanics Chair who supported me in the compilation and final proof-reading of the book and without whom the finalization of the book would not have been possible My sincere thanks go to Dr.-Ing C Bartels, Dipl.-Ing A Schneider, Dipl.-Ing M Glă uck for their intense reading of the book I owe special thanks to Mrs I.V Paulus, as without her help the final form of the book would not have come about Erlangen, February 2006 Franz Durst Preface of English Edition Fluid mechanics is a still growing subject, due to its wide application in engineering, science and medicine This wide interest makes it necessary to have a book available that provides an overall introduction into the subject and covers, at the same time, many of the phenomena that fluid flows show for different boundary conditions The present book has been written with this aim in mind It gives an overview of fluid flows that occur in our natural and technical environment The mathematical and physical background is provided as a sound basis to treat fluid flows Tensor notation is used, and it is explained as being the best way to express the basic laws that govern fluid motions, i.e the continuity, the momentum and the energy equations These equations are derived in the book in a generally applicable manner, taking basic kinematics knowledge of fluid motion into account Particular attention is given to the derivations of the molecular transport terms for momentum and heat In this way, the generally formulated momentum equations are turned into the well-known Navier–Stokes equations These equations are then applied, in a relatively systematic manner, to provide introductions into fields such as hydro- and aerostatics, the theory of similarity and the treatment of engineering flow problems, using the integral form of the basic equations Potential flows are treated in an introductory way and so are wave motions that occur in fluid flows The fundamentals of gas dynamics are covered, and the treatment of steady and unsteady viscous flows is described Low and high Reynolds number flows are treated when they are laminar, but their transition to turbulence is also covered Particular attention is given to flows that are turbulent, due to their importance in many technical applications Their statistical treatment receives particular attention, and an introduction into the basics of turbulence modeling is provided Together with the treatment of numerical methods, the present book provides the reader with a good foundation to understand the wide field of modern fluid mechanics In the final sections, the treatment of flows with heat transfer is touched upon, and an introduction into fluid-flow measuring techniques is given ix x Preface On the above basis, the present book provides, in a systematic manner, introductions to important “subfields of fluid mechanics”, such as wave motions, gas dynamics, viscous laminar flows, turbulence, heat transfer, etc After readers have familiarized themselves with these subjects, they will find it easy to read more advanced and specialized books on each of the treated specialized fields They will also be prepared to read the vast number of publications available in the literature, documenting the high activity in fluid-flow research that is still taking place these days Hence the present book is a good introduction into fluid mechanics as a whole, rather than into one of its many subfields The present book is a translation of a German edition entitled Grundlagen der Stră omungsmechanik: Eine Einfă uhrung in die Theorie der Stră omungen von Fluiden The translation was carried out with the support of Ms Inge Arnold of Saarbră ucken, Germany Her eorts to publish this book are greatly appreciated The final proof-reading was carried out by Mr Phil Weston of Folkestone in England The author is grateful to Mr Nishanth Dongari and Mr Dominik Haspel for all their efforts in finalizing the book Very supportive help was received in proof-reading different chapters of the book Especially, the author would like to thank Dr.-Ing Michael Breuer, Dr Stefan Becker and Prof Ashutosh Sharma for reading particular chapters The finalization of the book was supported by Susanne Braun and Johanna Grasser Many students at the University of Erlangen-Nă urnberg made useful suggestions for corrections and improvements and contributed in this way to the completion of the English version of this book Last but not least, many thanks need to be given to Ms Isolina Paulus and Mr Franz Kaschak Without their support, the present book would have not been finalized The author hopes that all these efforts were worthwhile, yielding a book that will find its way into teaching advanced fluid mechanics in engineering and natural science courses at universities March 2008 Franz Durst Contents Introduction, Importance and Development of Fluid Mechanics 1.1 Fluid Flows and their Significance 1.2 Sub-Domains of Fluid Mechanics 1.3 Historical Developments References 14 Mathematical Basics 2.1 Introduction and Definitions 2.2 Tensors of Zero Order (Scalars) 2.3 Tensors of First Order (Vectors) 2.4 Tensors of Second Order 2.5 Field Variables and Mathematical Operations 2.6 Substantial Quantities and Substantial Derivative 2.7 Gradient, Divergence, Rotation and Laplace Operators 2.8 Line, Surface and Volume Integrals 2.9 Integral Laws of Stokes and Gauss 2.10 Differential Operators in Curvilinear Orthogonal Coordinates 2.11 Complex Numbers 2.11.1 Axiomatic Introduction to Complex Numbers 2.11.2 Graphical Representation of Complex Numbers 2.11.3 The Gauss Complex Number Plane 2.11.4 Trigonometric Representation 2.11.5 Stereographic Projection 2.11.6 Elementary Function References 15 15 16 17 21 23 26 27 29 31 32 36 37 38 39 39 41 42 47 Physical Basics 49 3.1 Solids and Fluids 49 3.2 Molecular Properties and Quantities of Continuum Mechanics 51 xi .. .Fluid Mechanics F Durst Fluid Mechanics An Introduction to the Theory of Fluid Flows With 347 Figures and 13 Tables 123 Prof Dr Dr h.c Franz Durst FMP Technology GmbH... Chapter Introduction, Importance and Development of Fluid Mechanics 1.1 Fluid Flows and their Significance Flows occur in all fields of our natural and technical environment and anyone perceiving their... sees that the sixteenth and seventeenth centuries were marked by the development of the understanding of important basics of fluid mechanics In the course of the development of mechanics, the basic