Turbulent Fluid Motion

422 Pages·73.055 MB·English

by  Deissler, R

TURBULENT FLUID MOTION
Combustion: An International Series
Norman Chigier, Editor
Bayvel and 017.echowski, Liquid Atomization
Chen and Jaw, Fundamentals of Turbulence Modeling
Chigier, Combustion Measurements
Deissler, Turbulent Fluid Motion
Kuznetsov and Sabel'nikov, Turbulence and Combustion
Lefebvre, Atomization and Sprays
U, Applied Thermodynamics: Availability Method and Energy Conversion
Ubby, Introduction to Turbulence
Ray, Propulsion Combustion: Fuels to Emissions
TURBULENT FLUID MOTION

Robert G. Deissler
NASA Lewis Research Center
Cleveland, Ohio
USA Publishing Office: Taylor & Francis
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TURBULENT FLUID MOTION
© 1998 by Taylor & Francis Group, LLC

CRC Press is an imprint ofTaylor & Francis Group, an Informa business

All rights reserved. Printed in the United States ofAmerica. Except as permitted under the United

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To June and our family.

A CIP catalog record for this book is available from the British Ubrary.

@The paper in this publication meets the requirements of the ANSI Standard Z39.48-1984

(Permanence of Paper)

Ubrary of Congress cataloging-in-publication data is available from the publisher.

ISBN 1-56032-753-7 (cloth)

ISSN 1040-2756
CONTENTS

Preface ix

1 THE PHENOMENON OF FLUID TURBULENCE 3

1-1 What is Thrbulence? 3

1-2 Ubiquity of Thrbulence 4

1-3 Why Does Thrbulence Occur? 6

1-4 Closing Remarks 8

References 17

2 SCALARS, VECTORS, AND TENSORS 19

2-1 Introduction 19

2-2 Rotation of Coordinate Systems 19

2-3 Vectors (First-Order Tensors) 21

2-4 Second-Order Tensors 22

2-4-1 Definition and Simple Examples 22

2-4-2 Stress and the Quotient Law 23

2-4-3 Kronecker Delta, a Tensor 24

2-5 Third-and Higher-Order Tensors 25

2-5-1 Vorticity and the Alternating Tensor 25

2-5-2 A More General Quotient Law 26

2-6 Zero-Order Tensors and Contraction 27

2-7 Outer and Inner Products of Tensors of Higher Order 28

2-8 Subscripted Quantities That Are Not Tensors 28

2-9 Closing Remarks 29

References 29

V
vi CONTENfS
3
BASIC CONTINUUM EQUATIONS 31

3-1
Justification of the Use of a Continuum Approach for Thrbulence 31

3-2
Equation of Continuity (Conservation of Mass) 32

3-3
Navier-Stokes Equations (Conservation of Momentum) 34

3-3-1 Stress Tensor 34

3-3-2 Equations of Motion 38

3-3-3 Dimensionless Form of Constant-Property Auid-Aow Equations and

Dimensionless Correlation of Friction-Factor Data 41

3-4
Heat Transfer or Energy Equation (Conservation of Energy) 42

3-4-1 Dimensionless Form of Constant-Property Energy Equation and

Dimensionless Correlation of Heat-Transfer Data 44

3-5
Rule for Obtaining Additional Dimensionless Parameters as a System

Becomes More Complex 45

3-6
Closing Remarks 47

References 48

4
AVERAGES, REYNOLDS DECOMPOSmON, AND THE

CLOSURE PROBLEM 49

4-1
Average Values and Their Properties 49

4-1-1 Ecgodic Theory and the Randomness of Thrbulence 51

4-1-2 Remarks 51

4-1-3 Properties of Averaged Values 51

4-2
Equations in Terms of Mean and Auctuating Components 52

4-3
Averaged Equations 55

ss

4-3-1 Equations for Mean Aow and Mean Temperature
4-3-2 Simple Closures of the Equations for Mean Row and Temperature 57

4-3-3 One-Point Correlation Equations 91

4-3-4 1\vo-Point Correlation Equations 98

4-4
Closing Remarks 104

References 106

s
FOURIER ANALYSIS, SPEcrRAL FORM OF THE CONTINUUM

EQUATIONS, AND HOMOGENEOUS TURBULENCE 109

5-1
Fourier Analysis of the Two-Point Averaged Continuum Equations 110

5-1-1 Analysis of1\vo-Point Averaged Quantities 110

5-1-2 Analysis of the 1\vo-Point Correlation Equations 112

5-2
Fourier Analysis of the Unaveraged (Instantaneous) Continuum Equations 116

5-2-1 Analysis of Instantaneous Quantities 116

5-2-2 Analysis of Instantaneous Continuum Equations 118

5-3
Homogeneous Turbulence without Mean Velocity or Temperature

(Scalar) Gradients 122

5-3-1 Basic Equations 122

5-3-2 IUustrative Solutions of the Basic Equations 131

5-4
Homogeneous Turbulence and Heat Transfer with Uniform Mean-Velocity

or -Temperature Gradients 220
CO!IITENTS vii
5-4-1 Basic Equations 220
5-4-2 Cases for Which Mean Gradients Are Large or the Thrbu1ence Is Weak 222
5-4-3 Uniformly and Steadily Sheared Homogeneous Thrbulence If Triple
Correlations May Be Important 355
5-5 Closing Remarks 365
References 365
6 11JRBULENCE, NONLINEAR DYNAMICS, AND
DETERMINISTIC CHAOS 373
6-1 Low-Order Nonlinear System 374
6-2 Basic Equations and a Long-Term Thrbulent Solution with Steady Forcing 376
6-3 Some Computer Animations of a Thrbulent Flow 382
6-4 Some Thrbulent and Nonturbulent Navier-Stokes Flows 383
6-4-1 Time Series 385
6-4-2 Phase Portraits 386
6-4-3 Poincare Sections 392
6-4-4 Liapunov Exponent 394
6-4-5 Ergodic Theory Interpretations 398
6-4-6 Power Spectra 399
6-4-7 Dimensions of the Attractors 400
6-5 Closing Remarks 402
References 403
AFTER WORD 405
INDEX 407

PREFACE

Researchers have been active in serious studies of turbulence for more than a century.
Today, as it was a century ago, turbulence is ubiquitous. Although it is still an active
field of research, there is no general deductive theory of strong turbulence.
The literature on turbulence is now far too voluminous for anything like a full pre­
sentation to be given in a moderately-sized volume. Rather, it is attempted here to give a
coherent account of one line of development. Part of this has been given in abbreviated
form in Chapter 7 of Handbook ofTurbulence, Volume 1 (Plenum, 1977). In particular,
the scope of the work, which was somewhat limited by our inability to solve the fun­
damental nonlinear equations, has been considerably increased by numerical solutions.
Moreover, applications of dynamic systems theory in conjunction with numerical solu­
tions have resulted in, among other things, a sharper characterization of turbulence and
a deliniation of routes to turbulence.
The present work is based on a series of six NASA Technical Memoranda by the
writer. Throughout the book the emphasis is on understanding the physical processes
in turbulent flow. This is done to a large extent by obtaining and interpreting analytical
or numerical solutions of the equations of fluid motion. No attempt is made to either
emphasize or avoid the use of mathematical analysis. Because most of the material is
given in some detail, the student or research worker with a modest knowledge of fluid
mechanics should not find the text particularly hard to follow. Some familiarity with
Cartesian-tensor notation and Fourier analysis may be helpful, although background
material in those subjects is given.
Although turbulence, as it occurs, is more often strong than weak, it appears that
much can be learned about its nature by considering weak or moderately weak turbulence,
as often is done here. In general, the same processes occur in moderately weak turbulence
as occur at much higher Reynolds numbers; the differences are quantitative rather than
qualitative. The crux of the matter therefore might be accessible through low- and
moderate-Reynolds-number studies.
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