Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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Fluid Mechanics

F. Durst

Fluid Mechanics

of Fluid Flows

An Introduction to the Theory

123

With 347 Figures and 13 Tables

FMP Technology GmbH

Am Weichselgarten 34

91058 Erlangen

Germany

ISBN: 978-3-540-71342-5 e-ISBN: 978-3-540-71343-2

 2008 Springer-Verlag Berlin Heidelberg

liable to prosecution under the German Copyright Law.

and regulations and therefore free for general use.

Printed on acid-free paper

987654321

springer.com

Library of Congress Control Number: 2007937409

Cover design:

eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain

f.durst@fmp-technology.com

Germany

Fliederstrasse 40

Ingeborg Arnold

Prof. Dr. Dr. h.c. Franz Durst

66119 Saarbrücken

arnoldwi@t-online.de

English Translation:

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

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

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 intro-

duction 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 ed-

ucational 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 introduc-

tory 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 text-

books. 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 physi-

cal cause is understood clearly. The molecular-caused momentum transport,

leading to the τ

-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-N¨urnberg as an introduction into ﬂuid 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. Gl¨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, Franz Durst

February 2006

Preface of English Edition

Fluid mechanics is a still growing subject, due to its wide application in engi-

neering, 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 ba-

sic 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 ap-

plied, 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 treat-

ment 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.

x Preface

On the above basis, the present book provides, in a systematic manner,

introductions to important “subﬁelds of ﬂuid mechanics”, such as wave mo-

tions, 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 publi-

cations 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 “Grundla-

gen der Str¨omungsmechanik: Eine Einf¨uhrung in die Theorie der Str¨omungen

von Fluiden”. The translation was carried out with the support of Ms. Inge

Arnold of Saarbr¨ucken, Germany. Her eﬀorts 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-N¨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 sup-

port, 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

1 Introduction, Importance and Development

of Fluid Mechanics ....................................... 1

1.1 FluidFlowsandtheir Signiﬁcance....................... 1

1.2 Sub-Domains of Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 HistoricalDevelopments ............................... 9

References ................................................. 14

2 Mathematical Basics ...................................... 15

2.1 IntroductionandDeﬁnitions............................ 15

2.2 Tensorsof ZeroOrder(Scalars)......................... 16

2.3 Tensorsof FirstOrder(Vectors) ........................ 17

2.4 TensorsofSecond Order ............................... 21

2.5 FieldVariablesandMathematicalOperations ............ 23

2.6 Substantial Quantities and Substantial Derivative . . . . . . . . . 26

2.7 Gradient, Divergence, Rotation and Laplace Operators . . . . 27

2.8 Line,Surface andVolume Integrals...................... 29

2.9 IntegralLawsofStokesandGauss ...................... 31

2.10 Diﬀerential Operators in Curvilinear Orthogonal

Coordinates.......................................... 32

2.11 Complex Numbers .................................... 36

2.11.1 Axiomatic Introduction to Complex Numbers . . . . . . 37

2.11.2 Graphical Representation of Complex Numbers . . . . . 38

2.11.3 TheGauss Complex NumberPlane ............... 39

2.11.4 TrigonometricRepresentation .................... 39

2.11.5 StereographicProjection......................... 41

2.11.6 ElementaryFunction ............................ 42

References ................................................. 47

3PhysicalBasics........................................... 49

3.1 Solidsand Fluids ..................................... 49

3.2 Molecular Properties and Quantities of Continuum

Mechanics ........................................... 51