Dimensional Analysis
J.C. Gibbings
Dimensional Analysis
123
Emeritus Reader J.C. Gibbings
University of Liverpool
Department of Engineering
Brownlow Hill
Liverpool
L69 3GH
UK
ISBN 978-1-84996-316-9
e-ISBN 978-1-84996-317-6
DOI 10.1007/978-1-84996-317-6
Springer London Dordrecht Heidelberg New York
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
© Springer-Verlag London Limited 2011
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the
Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to
the publishers.
The use of 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 laws and regulations and therefore free
for general use.
The publisher and the authors make no representation, express or implied, with regard to the accuracy
of the information contained in this book and cannot accept any legal responsibility or liability for any
errors or omissions that may be made.
Cover design: eStudioCalamar, Girona/Berlin
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Preface
. . . – a University, taken in its bare idea – has this object –. It
educates the intellect to reason well in all matters, to reach out
towards truth and to grasp it.
Cardinal J.H. Newman
Dimensional analysis combines great utility with a demanding intellectual rigour;
this is its delight. Together with the idea of similitude, it has a long and honourable
history. It was developed by the greatest of scientists and mathematicians; such included Newton, Fourier, D’Arcy Thompson, Vaschy, Rayleigh, Buckingham, Riabouchinsky, Einstein, Bridgman and Sedov. It is remarkable for its universality of
application, increasingly so in recent times.
This book is for engineers and scientists as a student’s learning volume, as a lecturer’s text, as a graduate’s reference handbook and, as it contains much that is new
from the author’s work, as a research publication. In serving students it is designed
to give instruction both to undergraduates and, in its more advanced applications, it
could form the basis of postgraduate courses. A selection of the very elementary elements would be suitable for school pupils of science. Since the definitive Franklin
Institute conference of 1971, there have been significant developments in setting
out the totality and ordering of the fundamental logic, in a resolution of outstanding problems basic to the analysis and in deriving a general and rigorous statement
of the theorem upon which the analysis rests, as well as further contributions. So
this volume is submitted as the first up-to-date comprehensive presentation of these
and other topics as well as of the various applications in a range of fields of study.
Another justification for this volume comes from the present author’s previous comment which was:1
The more elementary treatments of dimensional analysis too frequently can be faulted,
whilst specialised monographs often leave questions unanswered. Detailed scrutiny of many
expositions makes it difficult to advance a defence against the commonly made claim that
dimensional analysis is only effective because the correct answer has previously been otherwise obtained. When the position has been reached of publication of an unnecessarily
complicated derivation which in addition is not soundly based, arriving at an answer that is
incorrect, then the time has come to reconsider –
Introductory presentations that give the impression to students that dimensional
analysis is a trivially easy topic, mislead and do them a disservice. It is good that
1
J. Phys. A: Math. Gen., Vol. 13, pp. 75–89, 1980.
v
vi
Preface
scientists and engineers should reason in a manner that is logical and so without
ambiguity and lacunae. It is a quality that is necessary in the use of dimensional
analysis. Scientists practise logic when forming an analytical description of an observed phenomenon; mathematics for them is the language for doing this, prose
having too limited a function. Engineers also practise this logic, together with optimisation and balance in the process of design. Instilling well-ordered reasoning
is a prime function of education which distinguishes it from training. Dimensional
analysis produces relations between variables that are particularly useful when a
‘formal’ analysis is not available. However, alone it does not produce answers in the
sense of numerical values needed by the scientist and particularly by the engineer.
Attempts have been made to use it thus but, as discussed later, they can readily be
shown to fail. Its highly valuable strength is to be the support of experiment, through
checks on the validity of experimental design, in the ordering of the experimental
procedure, by the enabling of a synthesis of empirical data and in making feasible
some experimentation; in all these it is a very powerful tool. Not all forms of experiment allow the support of dimensional analysis. Where they do, not to employ it is
to make the experimental design much the clumsier and to make cumbersome and
limited the resulting output of data. Correctly used, the analysis does not give wrong
answers. The vital initial step to precision is the correct formulation of the physics
of the phenomenon being studied. This is stressed strongly here where guidance
comes from examples of errors and supported by exercises. Some of these could be
useful to tutors as topics for discussion. The ancient idea of the pupil learning at the
feet of the master remains an invaluable part of a university education. Resulting
from teaching experience, Chapter 1 presents an elementary introduction to explain
mathematical manoeuvring that can be novel in form to students and to bring them
to an easy initial understanding of both the method and the great power of the analysis. The full logic of the analysis in Chapters 2 and 3, is given in a book for the first
time. Other new matters include, in logical order, the following principal items:
1.
2.
3.
4.
5.
6.
7.
8.
A listing of the full logic of the analysis.
A listing of the basic primary quantities with original fundamental definitions.
A rigorous procedure for the incorporation of universal constants.
A general and rigorous proof of the pi-theorem.
A complete resolution of the Rayleigh–Riabouchinsky controversy.
An original criterion for a validity of experiment.
The careful justification of approximations to the full physics of a phenomenon.
The replacement of unjustified assumption in formal analysis.
Some matters such as the topic of partial modelling, would be of particular interest to engineers. Finally, attention is drawn to the wisdom of prior application of
dimensional analysis to experimental data to precede that of statistical analysis.
University of Liverpool,
Lent Term 2009
Acknowledgements
I am grateful for valuable correspondence with the late Dr. Ron Pankhurst and with
Prof. Nicholas Rott. I appreciate the kindness of Dr. R.J. Brook who provided a copy
of Dr. E.J. Miller’s paper on his milk mixing experiments. The statement of Kelvin’s
is given by permission of The Institution of Civil Engineers. The author is grateful
to Mr. Jeremy Benton for his selection of the three pictures related to and of the
Airbus 380 aeroplane and these are copyright Airbus and reproduced by permission
of Airbus UK. The picture of shell buckling in Figure 9.5 was kindly provided by
Prof. Norman Jones and this and the redrawn version of Darcy Thompson’s anatomical diagram in Figure 11.1 appear by permission of the Cambridge University Press.
The redrawn figure of Figure 9.9 showing the wave drag of a ship is reproduced
by permission of The Royal Institution of Naval Architects. The Figures 7.7, 7.12,
8.9–8.11 from the author’s papers are reproduced by permission of Elsevier Science
Publishers B.V. The pictures of the helicopter and ship interaction in Figure 9.6
are included by permission of Prof. G. Padfield. A few exercises are taken from the
author’s past examination papers and are reproduced by permission of the University
of Liverpool. Much assistance in computer preparation of the manuscript was given
by Mr. Roy Coates, by Mr. Michael Gibbings and by Mr. Kevin Rodgers whilst
invaluable support in preparation of all the diagrams was given by Mrs. Sandra
Collins. Originally I was encouraged by Prof. Sir John Horlock to review the subject
and then Prof. Norman Jones urged me to expand this aim into a book. I am grateful
for their kindly encouragement.
vii
Contents
1
An Elementary Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Purpose of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Units and Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Synthesis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Re-ordered Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Preliminary General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Fluid-mechanic Force on a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Benefits of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
3
3
5
6
10
10
13
16
19
23
2
Concepts, Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Summary of Basic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Definition of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Definition of Primary Physical Concepts . . . . . . . . . . . . . . . . . . .
2.4 The Definition of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The Definition of Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 The Definition of Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Summary of Primary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Constant Relative Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Dimensional Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Units-conversion factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Products of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Dimensional Equality in Functional Relations . . . . . . . . . . . . . . . . . . .
2.13 Limitation to Functional Operations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 The Complete Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Derived Concepts and Their Measure . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
27
28
29
30
32
32
32
33
33
34
35
36
36
37
39
ix
x
Contents
2.16 Dimensions of Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . . .
2.17 The Inclusion of Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . .
2.18 Formation of Dimensionless Groups from Units-conversion Factors
2.19 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
47
48
49
53
3
The Pi-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Outline Form of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Basic Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Generalised Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Linear Mass Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Non-linear Mass Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Impact of a Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Electromagnetic Field Energy . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Prior Proofs of the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 The Careful Choice of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 The Necessity for a Units-conversion Factor for Angle . . . . . . . . . . .
3.8 General Results from the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
55
57
57
59
61
61
62
63
64
66
67
72
74
76
77
81
4
The Development of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 The Case for the History of Dimensional Analysis . . . . . . . . . . . . . . .
4.2 The Onset of Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The Onset of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The Developing Use of the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 The Place of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 4.1 The Reynolds Pipe-Flow Experiment . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
83
83
84
84
85
90
90
92
5
The Choice of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Care in Choosing Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 The Number of Non-dimensional Groups . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Mass and Force Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Mass and Volume Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 Temperature and Quantity Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 Mass and Quantity Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.7 The Angle Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.8 Electrical Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.9 Use of Vectorial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents
xi
5.10 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6
Supplementation of Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1 Information from the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 The Bending of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4 Extrapolated Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Uncoupled Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Forced Convection of Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.6.1 Compressible-flow Energy Transfer . . . . . . . . . . . . . . . . . . . . . 124
6.6.2 Incompressible-flow Energy Transfer . . . . . . . . . . . . . . . . . . . 131
6.7 The Rayleigh–Riabouchinsky Problem . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8 Natural Thermal Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.9 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7
Systematic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1 The Benefits of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Reduction of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Further Reduction of Non-dimensional Groups . . . . . . . . . . . . . . . . . . 153
7.4 Alternate Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.5 Parameter Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.6 Range of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.7 Superfluous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.8 Missing Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.9 Influence of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.10 Measurement Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.11 Effectiveness of Experimental Variables . . . . . . . . . . . . . . . . . . . . . . . . 165
7.12 The Validity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.13 Synthesis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.14 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8
Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Analytical Results from Dimensional Analysis . . . . . . . . . . . . . . . . . . 179
8.2 Example I: Flow Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.3 The Complexity of Flow Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.4 The Physics of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5 The Turbulent-Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.6 Prandtl’s Mixing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.7 The Log-law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.8 Jet Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
xii
Contents
8.9 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.10 Example II: Particle Abrasion in Flows . . . . . . . . . . . . . . . . . . . . . . . . 191
8.11 The Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.12 The Wear Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.13 Classes of Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.14 Particle Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.15 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.16 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.17 Example III: Electrostatic Fluid Charging . . . . . . . . . . . . . . . . . . . . . . 199
8.18 The Physical Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.19 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.20 The Variables and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.21 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.22 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.23 Example IV: Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.24 The Kinetic Theory of Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.25 Mean-free Path Length in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.26 The Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.27 The Pressure and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.28 The Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.29 The Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.30 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.31 Electrical Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.32 The Einstein Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.33 Summarised Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.34 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
9
Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.1 The Application of Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.2 The Essence of Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.3 The Windmill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.4 The Oil-insulated Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.5 Collision Against a Spring Restraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9.6 Inapplicability of Hooke’s Law of Elasticity . . . . . . . . . . . . . . . . . . . . 227
9.7 Limitation to Elastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.8 Impossibility of Scale Structural Modelling . . . . . . . . . . . . . . . . . . . . . 231
9.9 Limitations to Partial Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.10 Full-scale Comparison Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.11 Non-effectiveness of a Single Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.12 Analytical Input Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.13 Partial Extrapolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.14 The Range Limitation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.15 The Distortion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Contents
xiii
9.16 Complexity of Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.17 Model Testing in Engineering Design . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.18 Assessment of the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10 Assessing Experimental Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10.1 Interpretation of Dimensionless Correlations . . . . . . . . . . . . . . . . . . . . 256
10.2 Interpretation of Experimental Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.3 Deduction of Physical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
10.4 Dimensional Analysis with Statistical Regression . . . . . . . . . . . . . . . . 264
10.5 A Mixing Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
10.6 The Regression Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.7 Statistical Analysis on the Non-dimensional Groups . . . . . . . . . . . . . 269
10.8 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
11 Similar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.1 The Concept of Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.2 Physical Significance of Non-dimensional Groups . . . . . . . . . . . . . . . 274
11.2.1 The Physical Significance of Reynolds Number . . . . . . . . . . . 275
11.2.2 The Physical Significance of Further Groups . . . . . . . . . . . . . 275
11.3 Numerical Value of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.4 The Use of Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.5 Similarity in Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.6 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A
Derivation of Dimensions of Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
A.1 Electro-magnetic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
A.2 Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
A.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
A.4 Illumination units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
A.5 Thermal Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
A.6 Mechanical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Chapter 1
An Elementary Introduction
It can be truly said that the very easiness of the process
of dimensional analysis tends somewhat to encourage
inexperienced investigators to plunge into the analysis
without sufficient preliminary study of the problem.
W.J. Duncan
Notation
a
a, b
A
A0
c
C
Cd
d
D
e
f
F
g
h
hp
H
kD
K1 , K2 , . . .
L
`
m
M0
n, N
p
P
q
Q
R
Re
Area; oscillation amplitude
Constants
Area
Units-conversion factor
Velocity of light
Electric capacitance
Drag coefficient
Diameter; distance
Diameter; drag force
Spring elasticity; elementary charge
Acceleration
Force
Gravity acceleration
Height
Planck constant
Power
Drag coefficient
Coefficients
Inductance
Length
Mass
Molecular molar mass
Rotational speed
Pressure
Propeller power
Velocity
Flow quantity rate
Gas constant; electric resistance
Reynolds number
J.C. Gibbings, Dimensional Analysis. © Springer 2011
1
2
1 An Elementary Introduction
s
t
T
u, v
V
x
y
f
Distance
Time
Temperature
Velocities
Flow velocity
Variable
Variable
Function indicator
"
˘
!
Permittivity
Viscosity
Non-dimensional product
Density
Potential difference
Frequency
“Is dimensionally equal”
A
C
F
H
L
M
n
T
˛
S
Electric current dimension
Luminous intensity dimension
Force dimension
Area dimension
Length dimension
Mass dimension
Quantity dimension
Time dimension
Plane angle dimension
Temperature dimension
Solid angle dimension
1.1 The Purpose of this Chapter
Reasons for this chapter have been set out in the Preface. Principally, it is aimed to
enthuse the reader for the widespread usefulness of the subject. Thus it is written for
those first coming to it and they include both the student and the research worker in
engineering and science who are still unaware of its scope and value.
Correspondingly, whilst the exposition in this chapter is elementary in nature,
the rest of this volume brings out the complexity of the topic that requires the care
needed in application of the full logic. This fine detail is well realised in the controversy between two such great masters as Lord Rayleigh and Professor Riabouchinsky which is now fully resolved here in Chapter 6 almost a century later.
1.3 Units-conversion Factors
3
1.2 Units and Dimensions
Dimensional analysis is a powerful means in the design, the ordering, the performance and the analysis of experiment and also the synthesis of the resulting data.
The great majority of experiment requires methods of measurement that use numerical scales from both defined units and dimensions. Rare exceptions to this are, for
example, botany and anatomy where classification can be in terms of graphical descriptions of shape and colour though even here some measure of size is commonly
used. In this book measurement is used as a basis of science and engineering and
hence of dimensional analysis.
In 1883 Kelvin stated this importance of measurement in an address to the Institution of Civil Engineers when he said [1]:
“In physical science a first essential step in the direction of learning any subject, is to find
principles of numerical reckoning, and methods for practicably measuring, some quality
connected with it. I often say that when you can measure what you are speaking about, and
express it in numbers, you know something about it; but when you cannot measure it, when
you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind:
it may be the beginning of knowledge, but you have scarcely, in your thought, advanced to
the stage of science, whatever the matter may be.”
The first step in the present logic requires consideration of a basic feature of
measurement in science. There are two matters in measurement; one is that of the
dimensions used, the other of the units used. To illustrate: if a pupil states that one
cow approximately equals ten sheep then this statement is faulty on two counts. Two
possible correct statements are that the mass of ten sheep roughly equals the mass
of one cow or alternatively the cost of ten sheep approximately equals the cost of
one cow. Now there are equality of dimensions. In these cases the dimensions are of
mass or of cost respectively. But these statements have to be further qualified. The
first must refer to the same unit which could be the kilogramme; the second could
be the unit of the United States dollar. The first part of these statements gets the
equality of dimensions right; the second does so for the units.
The fundamental basis of this is that it is quite meaningless to add quantities
having different measures. Addition of physical quantities is only meaningful when
both the dimensions and the units are identical. There is no useful meaning in adding
a length to a force; equally, nor is there in adding acres directly to hectares. It follows
that an equality is under the same restrictions. This principle, though simple, is the
foundation of the development of dimensional analysis. It is the first stage in the
logic of this subject: it is the primary statement as being an acceptable affirmation
from it being self evident. Thus it forms the basic premiss for the present work.
1.3 Units-conversion Factors
The use of units and of dimensions is now introduced through a simple example.
The definition of measure of an area, denoted by a, is the arithmetic sum of unit
4
1 An Elementary Introduction
squares. It follows that the area of a square is given by the square of the side length,
`, or a D `2 . But in general it is necessary to write;
a D A0 `2 :
(1.1)
This nomenclature is explained by three cases, viz.:
a)
when a is measured in acres (a) and ` is measured in yards (yd), then:
(i)
(ii)
(iii)
A0 has the dimensions of [area length2 ];
A0 has the units of acres per square yard: this is written as; a yd2 ;
A0 has the numerical value of 1/4840.
To illustrate, the square of side 500 yd has an area of;
1
500 500 D 51:7 a
4840
b)
where a is measured in hectares (ha) and ` is measured in metres (m) then:
(i)
(ii)
(iii)
A0 has dimensions of area length2 ;
A0 has units of ha m2
A0 has the numerical value of 1/10 000.
Now, the square of side 200 m has an area of;
1
200 200 D 4 ha
10 000
c)
when a is measured in square metres, denoted here by m2 and ` is measured in
m, then:
(i)
(ii)
(iii)
A0 has no dimensions;
A0 has no units;
A0 has the numerical value of 1 0.
This time, the square of side 200 m has an area of;
200 200 D 40 000 m2
Comparing these three cases, it is seen in the third one, that by using the same units
for area and for length – in this example the use of metres – the factor A0 has no
dimensions and no units: it can be excluded from the calculating process. Then only
one and not two units are needed for this problem of the measure of the concept of
area. The factor A0 is called a units-conversion factor [2]. This feature, that removal
of a unit results in the removal of a units-conversion factor, is a general one in
dimensional analysis.
Just as the amount 2 54 is the value of the units-conversion factor to change
inches into centimetres, so the quantity A0 is called a units-conversion factor and
represents a number to change the units of length squared into those of area. Each
1.4 Dimensional System
5
of the three above procedures are used. When (c) is adopted, then, because A0 has
a value of unity and is dimensionless, its existence can go unrecognised; but in
principle it is present.
From these three examples it is seen that, by writing Equation 1.1, then this
inclusion of the units-conversion factor, A0 , makes this equation consistent for the
dimensions and valid for all sets of units. In the next chapter it will be shown that
such consistency and validity is obtained for all the basic physical equations.
This discussion introduces also the idea of the combination of units and dimensions. For case (a) (i) above, the dimensions of area being denoted by the symbol H
and those of length by L, then the dimensions of A0 are H=L2 . In a similar context
Jeffreys pointed out that the dimensions of ` and of A0 could be specified so that
those of a would follow: the choice is arbitrary [3]. With case (c)(i) the dimensions
of area are L2 and so those of A0 are L2 =L2 D 1: that is it is dimensionless. This
introduces the concept of cancellation of dimensions.
Another elementary example of this cancellation is that in which velocity multiplied by time equals length. Denoting the dimension of time by the symbol T, then
in terms of the dimensions this is:
velocity time.L=T/ T D L
Here the symbol indicates ‘is dimensionally equal to’.
1.4 Dimensional System
Maxwell introduced the symbolism to denote dimensions that sets that for mass as
M, for length as L, and so on [4].
The full set of dimensions in the Systeme International d’Unites together with
the symbols of dimensions is now listed in Table 1.1.
Table 1.1
Quantity
Unit
Dimension
Symbol
Length
Area
Time
Force
Mass
Electric current
Temperature
Luminous intensity
Quantity
Plane angle
Solid angle
metre; m
metre2 ; m2
second; s
Newton; N
kilogram; kg
ampere; A
Kelvin; K
candela; cd
mole; mol
radian; rad
steradian; sr
L
H
T
F
M
A
C
n
˛
S
6
1 An Elementary Introduction
Table 1.2
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
Quantity
Dimensions
Dimension symbols
Velocity
Acceleration
Velocity gradient
Density
Force
Pressure, stress
Viscosity
Work
Power
Electric potential
Length/time
Velocity/time
Velocity/length
Mass/volume
Mass acceleration
Force/area
Stress/velocity gradient
Force length
Work/time
Power/current
L T1
L T2
T1
ML3
MLT2
ML1 T2
ML1 T1
ML2 T2
ML2 T3
ML2 A1 T3
Table 1.3
(1)
ıx x
(2)
dy
dx
(3)
dn y
dx n
(4)
D Ltıx!0 ıy
ıx
y
x
ıy
y
D Ltıx!0 .ıx/
n xn
R
P
ydx D
yıx yx
This listing is justified later. The dimensions of physical quantities can then be
developed from these basic ones.1 Some examples are now listed in Table 1.2.
In succeeding chapters this derivation of dimensions is amplified, but it should
be noted here that items (i) to (iv), (vi) and (viii) to (x) come from the definitions
of these physical properties. Items (v) and (vii) come from a physical law, they are
respectively Newton’s law of motion and that of the Newton–Navier statement of
viscous shear [5].
Dimensions in the calculus are indicated by the examples set out in Table 1.3
where again the symbol means ‘is dimensionally equal to’.
1.5 Synthesis of Experimental Data
The power of dimensional analysis in ordering and synthesising experimental data
is now shown through examples of the results of four experiments.
Experiment (a)
Figure 1.1 shows the setting of a thin circular disc perpendicular to an oncoming
uniform stream of Newtonian fluid. The airstream is classified by the velocity, V ,
1
We scientists are not very bright when we adopt the basic unit of mass having the prefix kilo- and
when in a decimal system a time scale goes by the factors 60, 60 and 24.
1.5 Synthesis of Experimental Data
7
Figure 1.1 Sketch of a plate set normal to a uniform stream
the density of the fluid, and the viscosity, . The size of the disc is represented by
a length, `, that could be taken, for example, as the diameter, d . As a result of the
flow a drag force, D, is exerted upon the disc.
Over limited ranges – which can only be quantified later in this discussion – if
the stream velocity alone was varied in this experiment then it would be found that
D / V 2 . This result would plot as sketched by the full line in Figure 1.2. Repeating
the experiment but now by varying only the plate size, `, would give D / `2 .
This would give the series of points on the vertical line shown in Figure 1.2. One
is not immediately justified in concluding that both these results combine to give
D / `2 V 2 for this implies a complete family of straight lines on Figure 1.2. To get
the full result the second experiment has to be repeated for a range of values of V .
For example, if only five data points are obtained on each line, as now shown dotted
in Figure 1.2, then for five lines, the number of data points required is 52 D 25. More
than five might be judged necessary for each line [6]. Repeating the experiment by
individually varying the density, and then the viscosity, , gives in turn, D / and D ¤ f ./. Extending the previous illustration, in total the number of data
points required is 54 D 625.
Figure 1.2 Sketch of the
experimental values of the
drag as a function of the
square of the velocity
8
1 An Elementary Introduction
It is worth pausing here to counter an erroneous conclusion on the physics of
this phenomenon. To say that D ¤ f ./ is not to imply that the viscosity has no
influence upon the drag. A well known analytical result shows that in any fictitious
flow that has no viscous shear stress, the drag and indeed also any force including
a lift and a thrust, must be zero [6–8]. What has happened in the present example is
that the numerical value of the viscosity is of no influence, which is something quite
different. Viscosity causes the flow to separate from the sharp edge of the disc and
this is independent, as is the rest of the flow pattern in the region of the plate, of the
numerical value of the viscosity: then so also is the drag.
A similar result is found, for examples, in the case of flow past buildings, bridges
and other so-called bluff bodies such as some road vehicles [9].
The foregoing experiments will not ensure that all physical quantities that might
influence the value of D have been taken into account. A full consideration requires
an understanding of the physics of the phenomenon. In the present one, recourse
to the equations of motion shows the presence of forces from pressure and viscous
stresses that are balanced by inertia terms which contain and V . Forces involve the
size of the system, `, whilst the viscous stresses derive from . This more careful
reasoning gives preliminary assurance that no important parameter has been omitted. Later discussion shows how this conclusion has to be amplified in one important
respect.
The final result of this particular experiment is that, over some limited range,
D / V 2 `2 , or:
D D K1 V 2 `2 :
(1.2)
Checking the dimensions of the quantities on both sides of Equation 1.2, through
reference to Table 1.2, gives the dimensions of D as:
D MLT2 :
Also,
2
V 2 `2 ML3 LT1 L2 D MLT2
A comparison with Equation 1.2 shows that, as both terms above have identical
dimensions, then for this equation to be meaningful by having an equality of dimensions, the coefficient K1 must be dimensionless; it is a pure number. This is
written as:
K1 1
1.5 Synthesis of Experimental Data
9
Experiment b)
The second experiment now described is one on a small sphere, of diameter, `, also
set in a uniform stream. Following through the arguments and the test as previously
for the disc, and again over limited ranges, the result obtained from a series of experiments would be:
D / V `
or,
D D K2 V ` :
(1.3)
Reference again to the prior tabulation of dimensions gives
V ` ML1 T1 LT1 L D MLT2 :
Comparison with the dimensions of D from above shows that K2 also is a nondimensional number.
Experiment c)
A third experiment would be one on a small and thin flat plate of streamwise length,
`, and of unit width, set tangentially to the oncoming flow. Again over limited ranges
the result would be found to be:
D D K3 1=2 V 3=2 1=2 `3=2 :
(1.4)
A check as before, left as an exercise, would show that again K3 is a dimensionless
constant.
Experiment d)
The final test would be one on the flow through a length of smooth pipe of varying
length, `, and diameter whilst holding the ratio of length to diameter constant. Again
over limited ranges the result obtained would be,
D D K4 V `
with, as for experiment (b), K4 being a dimensionless constant.
10
1 An Elementary Introduction
1.6 Comparison of Results
The results of all four experiments are collected in Table 1.4.
Table 1.4
(1)
a) Disc
D D K1 V 2 `2
(2)
D=V `
`
K1 V
b) Sphere
D D K2 V `
K2
c) Plate
D D K3 1=2 V 3=2 1=2 `3=2
K3
d) Pipe
D D K4 V `
K4
V `
1=2
(3)
D=V 2 `2
(4)
CD
K1
2K1
K2
K3
K4
V `
V `
V `
1
1=2
1
2K2 =Re
2K3 =Re1=2
2K4 =Re
Column 1 of Table 1.4 lists the foregoing results. If it was wished to compare
the drags of these four objects of different shape under like conditions this could
hardly be done by inspection of this column except for items (b) and (d). This latter
is only possible because the forms of the two equations are identical and so the
relative values of the non-dimensional constants, K2 and K4 give the required direct
comparison. However, this comparison is now a much simpler one in not requiring
separate and individual specification of the values of , V and of `.
Column 2 lists values of D=.V `/ as obtained from Equations 1.2–1.4. Three
most useful results are now revealed.
First, inspection shows that comparison of resistances is now possible because
this would be obtained from a graph of D=.V `/ against the quantity .V `/=.
This graph for these four particular objects is shown in Figure 1.3; note that logarithmic scales are used to give the four straight lines at the appropriate slope for
each. The general comparison of drag is clearly seen in this greatly condensed presentation of all the proposed four sets.
Secondly, following the previous suggestion of sets of five data points this now
shows a remarkable condensation of the 2500 data points down to only those needed
to determine four curves on a single graph: possibly twenty points.
And thirdly, the limits, determined experimentally, over which the data is valid
are now seen by the end marks on each curve and so are now quantified in a very
general form as values of only the combination .V `/=, with corresponding limiting values of D=.V `/ and not as separate limits to each of the four variables in
this product group.
1.7 Re-ordered Functions
It follows from the prior discussion of Equations 1.2 and 1.3 that the groups
D=.V 2 `2 / and D=.V `/ are each non-dimensional. Comparison of columns 1
1.7 Re-ordered Functions
11
and 2 of Table 1.4 reveals that so also must be .V `/=. Such groupings by multiplication and division of variables forming products which are then non-dimensional
are, following Buckingham, called pi-groups and given the symbol ˘ [10].
Defining as follows:
˘1 D=.V 2 `2 / ;
˘2 D=.V `/ ;
˘3 .V `/=
then column 2 says that
˘2 D f .˘3 /
(1.5)
for all the four shapes tested, the form of the functions being different between the
shapes.
The practice of multiplying and dividing non-dimensional groups of variables
whilst retaining a functional relationship will be used here. Teaching experience
shows that this mathematical operation can initially be puzzling. Explained simply,
it is noted that the existence of a known real function y D f1 .x/ implies that values of y can be plotted against those of x even if the function is multi-valued: for
every value of x the value of y is known and vice versa. So for every value of x
any combination of y and x can be calculated so that this combination can be plotted against any other one.; for example, y=x D f2 .x 2 y/. The changed suffix on f
merely indicates a fresh form of the function; a different shape graph. This is illustrated in Figures 1.4(a) and (b). The point ‘A’ in Figure 1.4(a) transforms into the
point ‘A’ in Figure 1.4(b). In the same way, other forms of presentation of the comparison in column 2 are possible. For example, and bearing in mind Equation 1.5,
Figure 1.3 Experimental results for the four different shapes
12
1 An Elementary Introduction
Figure 1.4 Illustration of
a transformation of variables
˘1 D ˘2 =˘3 D f Œ˘3 and this gives the set of relations in column 3. Now comparison can be made from a graph of values of D=V 2 `2 plotted against those of
.V `/=.
In the study of fluid motion it is practice to define a drag coefficient, CD by;2
. 1
CD D
V 2 `2
2
whilst the non-dimensional number .V `/= is called the Reynolds number and
given the symbol Re . Making these changes gives the final column 4 and the graphs
in these terms are shown in Figure 1.5.
This final synthesis of all the data of at least 625 data points representing just
one object, into just one curve, requiring a mere five data points, is only one of the
remarkable powers of dimensional analysis. It will be noted that the reproduction
in Figure 1.5 of the lines in Figure 1.3 are indicated by the markers on extended
curves that cover much greater ranges than shown in the latter figure and which are
no longer straight lines over the whole length of each. That such general curves can
be demonstrated to exist for each shape of these objects will be shown later as yet
another power of dimensional analysis.
In the older European literature, which is still used for data, kD is defined without the factor of
1=2 so that kD D CD =2.
2
1.8 Preliminary General Analysis
13
Figure 1.5 Experimental results for drag coefficient against Reynolds number for the four different
shapes
1.8 Preliminary General Analysis
Equation 1.5 is a particular example of a general result that can be obtained by use
of what is known as the pi-theorem. The principles of this theorem have been given
elsewhere as a generalised proof [11]. It will be illustrated here initially by just
a simple example, because that proof and the operation of the theorem are as one.
Suppose a rigid solid of mass, m, is suspended vertically from a linear elastic
spring which is attached to a rigid support, the whole being within a perfect vacuum.
This is illustrated in Figure 1.6. The container enclosing the vacuum is given a single
jerk to set the solid oscillating vertically with a frequency of !.
The physical laws involved are:
a)
b)
Hooke’s law of linear elastic springs.
Newton’s law of motion.
For (a) the force exerted by a linear spring can be represented by the spring elasticity, e, which measures the force per unit deflexion. For (b) the acceleration can be
represented by the frequency, !, and the amplitude of the oscillation, a. The force
from the spring is accounted for under (a). Preliminary inspection recognises that
there is also a weight force acting on the mass. However this force merely causes
a fixed deflexion of the spring; obtaining vibration requires an oscillating force and
this comes from the spring alone. There is no aerodynamic force because a perfect
14
1 An Elementary Introduction
Figure 1.6 Sketch of the
arrangement for a spring
mounted mass
vacuum is specified. Finally, the mass of the spring is taken as negligible compared
with that of the suspended solid.
Setting out the dimensions gives the following lines (a) and (b):
(a) — Variables
!
m
a
e
(b) — Dimensions
1
T
M
L
M
T2
It has just been specified that:
! D f1 Œm; a; e
(1.6)
! 2 D f2 Œm; a; e
(1.7)
so that:
the subscripts on the function indicators emphasising the difference of these two
functions. Then, noting line (b) above, Equation 1.7 is rewritten as:
!2
e D f2 Œm; a; e :
e
(1.8)
Setting out the dimensions now gives:
(c) — Variables
(d) — Dimensions
!2
e
1
M
m
a
e
M
L
M
T2
By dividing ! 2 by e, a quantity has been obtained for which the dimension in T has
been cancelled out.
It is specified that Equation 1.8 must be such that the dimensions of both sides
of this equation are identical. Considering first the dimension of T then, observing
line (d) in conjunction with Equation 1.8 shows that now only the variable e contains
the dimension T. This means that this equation must be of the form of:
!2
e D ef3 Œm; a
e
(1.9)
1.8 Preliminary General Analysis
15
for the dimension in T to balance; there is no other permissible form. The variable
e then cancels out leaving Equation 1.9 as:
!2
D f3 Œm; a :
e
(1.10)
Going back to lines (a) and (b) above, it is seen that by cancelling and thus eliminating the dimension in T, the variable, e, on its own, is now eliminated. It is combined
as effectively the single variable of .! 2 =e/.
Now considering dimensional equality in M and noting line (d) above, Equation 1.10 is rearranged to the form:
! 2m 1
D f4 Œm; a :
e m
(1.11)
Again setting out the dimensions gives:
(e) — Variables
!2 m
e
(f) — Dimensions
1
1
m
1
M
a
L
where the unit dimension means that the quantity is non-dimensional. Now Equation 1.11 has to take the form of:
!2m 1
1
D f5 Œa
e m
m
(1.12)
for the dimension in M to balance. Thus the variable m cancels so that:
!2m
D f5 Œa :
e
(1.13)
Now there is the situation that it is not possible to balance the dimension in L from
the variable a. It is simply concluded that a is not a relevant variable in this problem
so that finally:
!2m
D constant
e
(1.14)
a result that is most useful. A detailed formal analysis would confirm that the variable a does not arise because the spring is taken as being linearly elastic. For a nonlinear spring the amplitude does become a relevant variable as will be shown later.
Summarising the steps in the above lines of analysis gives the convenient layout
of Compact Solution 1.1.
This can be condensed further, for convenient working, into the form of Compact
Solution 1.2 [6]:
This demonstration is an illustration of the pi-theorem and its application. This
theorem with various ramifications is considered in a general form in Chapter 3.
16
1 An Elementary Introduction
Compact Solution 1.1
(a) — Variables
!
m
a
e
(b) — Dimensions
1
T
M
L
M
T2
(c) — Variables
!2
e
m
a
e
(d) — Dimensions
1
M
M
L
M
T2
(e) — Variables
!2 m
e
1
m
a
(f) — Dimensions
1
1
L
M
Compact Solution 1.2
!
Variable
Dimensions
m
M
1
T
a
L
e
M
T2
Variables
!2
e
–
Dimensions
1
M
–
Variables
!2 m
e
–
–
Dimensions
1
–
–
1.9 Fluid-mechanic Force on a Body
The derivation of the general result for the force on a body in a uniform incompressible flow, as illustrated for example, in Figure 1.5 is now given.
First, for the force, F , it is specified that:
F D f Œ; V; `; :
(1.15)
As for the previous example, setting out the variables and dimensions gives:
(a) — Variables
F
V
`
(b) — Dimensions
ML
T2
M
L3
L
T
L
M
LT
Then Equation 1.15 can be rewritten as:
F
D f ; V; `; :
(1.16)
1.9 Fluid-mechanic Force on a Body
17
Figure 1.7 Sketch to illustrate acceptable rearrangement of variables
It is seen that now both F and have been divided and multiplied by . Setting out
the variables and dimensions shows why this has been done as follows:
F
V
`
L4
T2
M
L3
L
T
L
L2
T
The dimension of M now occurs in only the variable . For fixed values of V and
` Equations 1.15 and 1.16 can be illustrated by Figure 1.7. This can be transformed
into the diagram of Figure 1.8: for example, numerical values associated with point
‘A’ in Figure 1.7 enable point ‘B’ to be calculated in Figure 1.8. This then illustrates
why Equation 1.16 can now be transformed into:
F
D f ; V; `;
:
(1.17)
Figure 1.8 Sketch to illustrate rearrangement of
variables
18
1 An Elementary Introduction
Compact Solution 1.3
F
V
`
ML
T2
M
L3
L
T
F
–
L4
T2
–
L2
T
F
V 2
–
–
V
L2
–
–
L
F
V 2 `2
–
–
–
V `
1
–
–
–
1
M
LT
Following the previous discussion and noting the dimensions of the variables given
above, then this equation must be of the form of:
F
D f V; `;
or, the variable now being cancelled gives:
F
D f V; `;
:
As before, this procedure can be continued and is now tabulated in full in Compact
Solution 1.3.
This layout becomes the operational one for this solution of the pi-theorem. It
gives the final result of:
F
Df
V 2 `2
V `
which is rewritten as:
F
Df
V 2 `2
V `
:
(1.18)
This is the general form illustrated for the drag in Figure 1.5. Thus dimensional
analysis can both prove the existence of those curves and show the general nondimensional nature of them.
This example illustrates two important matters. It shows how two or more groups
of variables can be rearranged to cancel a single dimension whilst retaining a functional relationship. It is not acceptable merely to operate on variables to cancel
dimensions. It is the essence of operating the pi-theorem of dimensional analysis
1.10 Benefits of Dimensional Analysis
19
that a functional relationship is retained at all steps in a procedure. This important
matter is returned to in Chapter 3.
1.10 Benefits of Dimensional Analysis
The principal benefits of dimensional analysis to experiment now start to be revealed. In the first example of elastic vibration an associated experiment would not
require separate measurement of the effects of the variables m, e and a. It is only
necessary to set up an experiment in which m and e are fixed at a single value each,
then to read the corresponding single value of ! so that the value of the constant in
Equation 1.14 is determined. The value of ! is then known for all values of m and
e without having to vary these two independent variables at all in an experiment:
there is a great saving in experimental measurement.
Returning to the examples illustrated in Table 1.4 and Figure 1.3, it is now seen
from Equation 1.18 that each curve can be obtained by varying in the experiment
only one of , V , , and ` and measuring the corresponding values of D. Clearly, the
variable chosen can now readily be the one that is most convenient experimentally.
This reveals the quite outstanding benefit to simplification of experiment that results.
To illustrate the saving in experimental effort consider the oscillation experiment.
From Equation 1.6, with the variables of !, m and e, just varying all these would
require the following:
a)
b)
c)
d)
three variables require three readings for each experimental point;
say five points as a minimum are required for determination of each curve;
graphically, this would require a family of curves on a single graph;
therefore the number of readings would be:
3:52 D 75 readings
Again, to illustrate the fluid drag experiment we have:
a)
b)
c)
there are five variables so each point requiring five readings;
graphically, this would require a family of a family of a family of curves;
so the number of readings would be:
5:54 D 3125 readings :
instead of the thirteen readings to determine one of the lines of Figure 1.3; that
is, of three fixed variables and two varied ones.
In general the number of readings for n variables becomes:
nr .n1/
where r is the number of points needed to determine a single curve.
20
1 An Elementary Introduction
So as well as this economy of experiment, there is an equally valuable synthesis
of the resulting experimental data.
Exercises
1.1
Using the system of dimensions of Table 1.1, derive those for the following:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
1.2
1.3
1.4
1.5
elastic strain energy;
Young’s modulus;
Poisson’s elastic strain ratio;
work;
heat;
thermal internal energy;
coefficient of specific heat;
thermal conductivity;
electrical voltage;
electrical field strength;
electrical charge;
electrical permittivity;
electrical conductivity;
diffusion coefficient;
ionic concentration;
ionic mobility;
work function;
Avogadro constant;
Planck constant;
Faraday constant;
Boltzman constant;
luminance;
illumination.
If instead of adopting a dimension of mass, M, one of force, F, was used,
what would be the dimensions of the quantities in Table 1.2? The quantities
in that Table have been listed in a logical order of derivation. Set out a similar
order for this change of dimension.
Using the force, length and time or FLT system of dimensions derive those
for the items of Exercise 1.1.
Show that the Reynolds number is non-dimensional.
Check the dimensions of Equation 1.4.
1.10 Benefits of Dimensional Analysis
1.6
21
Show that the following equations which govern the linear motion of a particle satisfy the equality of dimensions:
v D uCft;
s D ut C .1=2/f t 2 ;
v 2 D u2 C 2f s :
1.7
Then rearrange each equation so that all the terms are non-dimensional and
show that in each of these three equations three variables are reduced to two
non-dimensional groups.
Show that the Bernoulli equation, that is:
p
C gh C .1=2/q 2 D constant :
1.8
1.9
1.10
1.11
1.12
satisfies the equality of dimensions.
From the three pi groups of Section 1.6 derive one that does not contain V
and check that it is non-dimensional.
Consider the physics of the example of Section 1.8 and show how that phenomenon might be extended to the case of the oscillation of a large elastic
structure.
Consider the physics of the example of Section 1.8 and show how that phenomenon might be extended to the case of the time of swing of a pendulum.
The power, P given by a ship propeller of a diameter d rotating at n radians
per second in water of a density of is quoted as being given by:
P D constant n3 d 5 :
What are the dimensions of the constant?
The discharge rate of a water pump, Q, in m3 per second is a function of the
rotational speed n and the pump diameter d . This is commonly expressed by
Q
D a numerical constant :
nd 3
1.13
1.14
Check the dimensions of this equation.
The force between the two plates of a condenser can be expressed by:
F
A
Df
:
2
"
d2
where the force is F , the potential difference is , the permittivity is ", the
plate area is A and the distance between the plates is d . Check the dimensions of this equation.
An electrical circuit having a resistance, R, a capacitance, C , and an inductance, L, can be made to oscillate at a frequency of !. This can be repre-
22
1 An Elementary Introduction
sented by:
!L
Df
R
1.15
CR2
L
:
Check the dimensions of the two terms.
The equation of state of an ideal gas is given by:
p
RT
D
;
M0
1.16
1.17
where p is the pressure, is the density, T is the absolute temperature, M0
is the molecular molar mass and R is the universal gas constant. Determine
the dimensions of R.
The equation of state of a real gas is given approximately by the equation:
a2
M0
pC 2
b D RT :
M0
Determine the dimensions of a and of b and show that this equation then has
a balance of dimensions.
The “fine-structure constant”, a, which indicates the strength of the electromagnetic force, is given by:
aD
e2
;
2"0 chp
where,
e
"0
c
hp
1.18
is the elementary charge;
is the permittivity of free space;
is the velocity of light;
is Planck’s constant.
Show that a is dimensionless.
Accepting that the power, H , developed by a windmill is a function of the
airspeed, V , the air density, , the rotational speed, n, and the diameter, d ,
show that:
H
V
D
f
:
n3 d 5
nd
An experiment in a wind tunnel enables a graph of results to be plotted as
a single line in the form of this equation. However this is not convenient
for design studies that have to determine a design diameter. How could the
results be re-plotted so that the diameter could be derived directly?
References
23
References
1. W. Thomson (Sir; Lord Kelvin). Electrical units of measurement. The practical applications
of electricity; a series of lectures, The Institution of Civil Engineers, pp. 149–174, 1884.
2. J.C. Gibbings. On dimensional analysis, J. Phys. A: Math. Gen., Vol. 13, pp. 75–89, 1980.
3. H. Jeffreys. Units and dimensions, Philos. Mag., Vol. 34, pp. 837–840, (see p. 839, Ll. 22–25),
1943.
4. J.C. Maxwell. On the mathematical classification of physical quantities, Proc. Lond. Math.
Soc., Vol. 3, Pt. 34, pp. 224, March 1871.
5. S. Goldstein (Ed.) Modern developments in fluid dynamics, Dover, New York, pp. 3–4, 676–
680, 1965.
6. J.C. Gibbings (Ed.) The systematic experiment, Cambridge, Ch. 9, 1986.
7. N.A.V. Piercy. Aerodynamics, English Univ. Press, London, Art. 118, 1937.
8. J.C. Gibbings. Thermomechanics, Arts. 14.6, 14.7, Pergamon, Oxford, 1970.
9. J.C. Gibbings. Some recent developments in the mechanics of fluids, Phys. Bull., Vol. 20,
pp. 460–465, Inst. Phys., London, Nov. 1969. [See also Vol. 21, p. 135]
10. E. Buckingham. On physically similar systems: illustration of the use of dimensional equations, Phys. Rev., Vol. 4, pp. 345–376, 1914.
11. J.C. Gibbings. A logic of dimensional analysis, J. Phys. A: Math. Gen. Vol. 15, pp. 1991–2002,
1982.
Chapter 2
Concepts, Dimensions and Units
The results of this investigation have both a practical and a
philosophical aspect.
Osborne Reynolds, 1883
Notation
a
A; B
Am
A0
c
e
E, Ex , Ey
Er , Ek , El
f
F
Fa
g0
G0
hp
H
I
J
k, k1 , k2
kB
kf , ks
K
`
L
m
me
M0
N
Na
p
Area, acceleration
Quantities
Unit atomic mass
Units-conversion-factor for area
Velocity of light, particle velocity
Electron charge
Electrical field strength
Energy; radiation, kinetic, luminance
Frequency, acceleration
Force
Faraday constant
Inertia constant; motion
Gravitational constant
Planck constant
Magnetic field strength
Electrical current
Mechanical equivalent of heat
Multipliers
Boltzmann constant
Scaling factors
Constant
Length
Luminance
Mass, electron mass
Electron mass
Molecular ‘weight’
Sound level
Avogadro number
Pressure
J.C. Gibbings, Dimensional Analysis. © Springer 2011
25
26
2 Concepts, Dimensions and Units
q
Q
r
R
Ry
s
t
T
u
v, V
w
x; y
z
Electrical charge
Heat, quantity
Radius
Gas constant
Rydberg number
Distance
Time
Temperature
Velocity
Velocity
Ci
Unit measures
x C iy
˛
"0
0
Angle
Permittivity; vacuum
Field direction
Luminous flux
Permeability; vacuum
Density
Potential function
Flux function
Angle
!
Units-conversion factors:
c
Velocity of light
g0
Mass
G0
Gravity
hp
Radiation; Planck constant
J
Heat
kB
Temperature; Boltzmann constant
P0
Luminous flux
R
Particle-quantity
ˇ0
Plane angle
0
Solid angle
"0
Electrical charge
0
Magnetic field
Dimensions notation:
A
Electric current
C
Luminous intensity
E
Field
F
Force
L
Length
M
Mass
2.1 Summary of Basic Logic
n
H
T
˛
27
Quantity
Area
Time
Plane angle
Temperature
Solid angle
2.1 Summary of Basic Logic
In the previous chapter stress was laid upon the necessity in dimensional analysis
that reasoning be quite logical. Thus as a first starting point, such discussion should
commence from a basic premiss or premisses that have a sound basis. As a bonus,
the following discussion forms an exercise in this important ability. The aim here
is to develop the logic of dimensional analysis from very first principles so that its
basis is a sound one. A pioneering approach to this reasoning was by Bridgman [1].
It might be thought that those matters now to be discussed seem, at first sight, to be
trivial: often in this subject, especially in the early stages of elementary teaching,
matters are taken as granted. Such neglect of the full logical process can lead to
error in more advanced applications; examples appear later in this text: here the
foundations are laid firmly.
Dimensional analysis is founded upon basic principles of science. Not doing this
has in the past resulted in difficulties with and dissensions over this subject. As
Jeffreys pointed out during an extended controversy [2], confusion even has come
from semantic misunderstandings. Other problems arise from the use of definitions
that are lacking in rigour; one such is the use of mass both in a momentum context
and also in a thermal internal-energy one as a unit of amount of substance [3]. Again,
as discussed later in Chapter 6, a difficulty has arisen from whether a coefficient of
specific heat should be mass or volume based.
Here the fine details and practices of measurement in science are not considered. But the fundamental principles of measurement have to be discussed because
only after that can matters, which are basic in dimensional analysis, be dealt with
adequately.
A complete order of the logic of dimensional analysis is now set out. Each item
is discussed in turn in this and the succeeding chapter, each step in the argument
being referred back to this list. The order of the logic follows the successive stages
summarised in the following list.
(i)
(ii)
(iii)
(iv)
The full definition of a scientific concept is in two distinct and consecutive
parts: first, the nature of a concept is defined; then the definition of the
measure of that concept follows from the first.
The idea of a ‘primary’ physical concept is defined.
The unit reference measure is defined.
The measure of a ‘primary’ concept is defined as addition of unit measures.
28
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
(xvii)
(xviii)
2 Concepts, Dimensions and Units
The measure by addition of unit measures, together with a specified origin
of zero measure, results in a linear scale of measure.
The existence of linear scales together with common origins results in the
constancy of relative magnitude.
There is a principle of the meaningful addition of numerical values, this
requiring both dimensional and unit equality.
Dimensions are assignable to products of variables so that dimensions in
these products can be multiplied and cancelled.
The multiplication and cancellation of dimensions is required for Item (vii)
so that variables can be combined only as products.
The constancy of relative magnitude is retained for products of concepts.
Dimensional and unit equality is required by Item (vii) for the addition of
products of variables.
There are limitations in dimensional analysis to the permissible functional
operations on equations.
The complete equation is a consequence of item (xi).
Derived concepts are in the form of products of concepts with unitsconversion factors and are defined through either an arbitrary defining relation or an observed physical law.
The arbitrary choice of a dimension with its unit measure for a derived
concept requires the introduction of a units-conversion factor to retain dimensional and unit equality.
The foregoing of an arbitrary measure with its unit measure results in
the removal from a dimensional equation of the corresponding unitsconversion factor together with the corresponding dimension.
The appearance of units-conversion factors, in a functional statement between variables that govern a phenomenon, is directly related to the inclusion of the corresponding defining relations in the analytical model.
Within an equation, variables can be grouped in products, the limit to this
grouping occurring when the products become non-dimensional.
These steps in the logic have been set out before [4]. This particular ordering and
totality of the logic is now to be clarified and amplified step by step.
2.2 The Definition of Concepts
A difficulty that often arises in philosophical writings [5,6], occurs because the careful distinction, necessary in science, between the nature of a physical concept and
the definition of its measure is not made. One good example, described later, comes
from the various attempts by philosophers to define the concept of time where the
nature of time is not separated from the measurement of it. The nature of a concept
has to be defined before its measurement can be considered.
There is a difference in approach between the ‘philosopher’ on the one hand
and the scientist and the engineer on the other because the latter are faced with the
2.3 The Definition of Primary Physical Concepts
29
reality of having to obtain numerical measures and so having to derive a rigorous
definition of the nature of the concept on which to build a rigorous definition of how
to measure values of it. The importance of this is made clear in the quotation from
Kelvin given in Chapter 1, Section 1.2. Fourier thought of a physical quantity first
being defined as being a concept and then as having a numerical measure [7]. This
approach is used here.
2.3 The Definition of Primary Physical Concepts
In science, concern is with observation of material things forming a system, as that
term is defined in a generalised thermodynamic sense [8,9], and in the way that they
behave during a process. Quantification of the state of a system is by quantification
of its properties and these properties derive from concepts. In taking the first step
of defining a concept it is recognised philosophically that all words are defined in
terms of other words so that there is no absolute of verbal definition [10].
Starting from the foundation of absolute basic ideas, consider first the concept
of extension. A physicist calls this ‘length‘ whereas the ‘philosopher’ uses the word
‘length’ to denote the numerical measure. We can appreciate position by our senses;
we can observe the absence of sameness in position between two positions. This
difference is defined here as the concept of extension. Before a definition of its
numerical measure can be formed it is necessary to define the straight line. This is
left as the Exercise 2.1.
The definition of the numerical measure must then be a consequence of these two
definitions; it is measured by counting a set of standard lengths along the straight
line joining the two end positions; here this is called the length. Basically, this must
be along a straight line to achieve a unique value. Analytically, using the infinitesimal calculus, the distance along a curve can be calculated as distinct from being
measured.
This example illustrates why it first is necessary to define a concept and then
to follow this by a separate definition of its measure. This logical approach is the
reverse of that proposed by Taylor [11] who wrote that ‘it is important that the
concept – as determined from the rules for its measurement –’. The precision of
measurement is controlled by the size of the smallest standard length used in the
measurement; in the language of physics, by the degree of division of the standard
metre. This specifies a limiting tolerance on the accuracy.
The definition of extension is described as a ‘primary’ one because it is quite
independent of the definition of any other measurable quantity. Extension is a primary concept. These definitions must contain the requirements of the secondary
definition of length measurement. That is why, in this definition of extension, one
has to recognise that objects must be placed to terminate an extension; the present
‘absence of sameness in position’ must be in relation to objects occupying the two
positions. There can be three positions, A, B, and C successively along a straight
line as sketched in Figure 2.1. Because the measure of length is specified here by
30
2 Concepts, Dimensions and Units
simple addition, it follows that this measure is by a linear scale as illustrated in
Figure 2.1. Then the length AC is the sum of those of AB and BC. If it was by a logarithmic scale this would not be so. The measurement of length is a difference and
this is analogous to the First Law of Thermodynamics in which internal energy is
measured as a difference [12]. Any origin of measurement is then arbitrary.
Then a ‘position’ in empty space is an invaluable analytical abstraction as is
‘infinity’ or a ‘point’. Physical measurement is a matter of comparison, and the
observer having made the measurement is not concerned, remembering tolerances
and mistakes, that another would record otherwise.
In contrast, the definition of angle is a derived one in that it requires the prior
definitions of extension and then of the straight line. This is also left as Exercise 2.2.
The importance in dimensional analysis of linear and hence additive scales of
measure will be used later and will now be repeated for other acceptable concepts.
2.4 The Definition of Time
Amongst ‘philosophers’ difficulty still exists over the definition of time. It seems
that some philosophers use motion and time as synonyms; to a physicist motion
is the relation between space and time measured, for example, as a velocity. Plotinus [5] criticised Aristotle’s definition as being circular in that it relied on the definition of motion which in turn relied on the definition of time. This difficulty is
still discussed in modern times [6]. An approach analogous to that just given for
extension and length is now presented.
We can appreciate an accumulation of experience. Then we can appreciate a difference in a total of experience. This difference is now defined as time [13]. It is
not then a definition linked with motion and is another ‘primary’ concept because it
stands alone.
For amplification, as illustrated in Figure 2.2, consider that experimenter ‘A’
recognises a ‘point’ in time by observing one or more events, indicated by ‘ab ’
before that time and ‘aa ’ after it. Experimenter ‘B’ recognises the same point in
time by observing ‘b’ which contains some or all of ‘a’ so that some events either
Figure 2.1 The linear scale
of extension with an arbitrary
origin
2.4 The Definition of Time
31
side of the ‘point’ are common to both. Then ‘A’ and ‘B’ can agree on this ‘point’ in
time. Then experimenter ‘C’ recognises the same ‘point’ in time by observing ‘c’.
If ‘c’ contains some or all of ‘b’ so that again the events either side of the ‘point’
are common to ‘B’, then ‘B’ and ‘C’ agree on the location of the ‘point’ in time;
‘c’ may contain none of ‘a’ but still ‘C’ and ‘A’ can agree. Then another later point
in time gives a time difference. Again there is an analogy here with the statement of
the Zeroth Law of Thermodynamics.
The present definition overcomes further philosophical difficulties concerning
the extent of ‘past’, ‘present’ and ‘future’ [14]. Saint Augustine posed the problem,
asking ‘– the present, which we found was the only one of the three divisions of time
(past, present, future) that could possibly be said to be long –’ and ‘– the present
– has no duration’. If here ‘present’ is regarded as synonymous with a prescribed
‘point’ in time it becomes the useful analytical abstraction that a ‘point’ in extension
is.
Thus the present definition is analytically precise unlike Bergson’s philosophical
comment that ‘present is that which is acting and, for which Russell pointed out, led
to a circular definition [15].
As before, the numerical measure of time then follows; in physics it is measured
by counting events such as the oscillation of the current across a crystal. Here there
appears an uncertainty. We have to assume that these events are identical measures
of time. Defining unit times we are forced, by the passage of time in one direction,
to make this assumption. We cannot go backwards in time to check. However, there
is some evidence for this constancy in that highly accurate measurements showing
that the rotation of the earth is slowing are supported by observations of total solar
Figure 2.2 Illustration of the definition of time
32
2 Concepts, Dimensions and Units
eclipses going back to 700 BC [16]. A further support for this constancy of the unit
of time is given later.
This specification of measurement results again in a linear scale of measure.
2.5 The Definition of Force
The concept of force can be defined as being sensed by the loading experienced
from the gravitational attraction upon an object held in one’s hand. One can sense
no such attraction when the object is removed from the hand.
It follows from this definition that, unlike those for extension and time, force is
an absolute quantity and not a difference, the zero force being absolutely defined.
It further follows that force can be measured by defining a unit force as the gravitational attraction upon a reference object in a reference location under reference
conditions. Then using a balance, another force can be measured by the summing
of a number of unit balancing forces. Thus the measure of force defined in this way
provides a linear scale.
As before, this definition of force is independent of other concepts and so is
a primary concept.
2.6 The Definition of Quantity
The concept of particle-quantity is defined as the number of elementary entities in
a system, such as the number of molecules. The unit of measurement of quantity can
be a single entity or a specified number of them. This definition makes the origin of
the scale to be absolutely defined and also provides a linear scale of measurement.
This again lays down a definition and a unit of measure that are independent of all
others and so particle-quantity is a primary concept. However, this concept stands
alone from the previous three in that it can hardly be humanly sensed.
2.7 Summary of Primary Concepts
Here the definitions of length, time, force and particle-quantity are each quite independent of any other definition of a concept. The first three can also be comprehended by the human senses. The measures of the first two are defined in terms of
differences whilst the latter two each have an absolute zero. These four concepts are
here called ‘primary’ concepts. A unit measure for each of these concepts also can
be defined quite independently of each other. The measure of each of these ‘primary’
concepts is then specified as being by addition of these unit quantities so resulting in
linear scales. The definition of the measure of these ‘primary’ concepts is separate
2.9 Dimensional Equality
33
from, though consequent upon, the definition of the concept. In this approach we
follow Bacon who wrote “And it is a grand error to assert that sense is the measure
of things” [17].
This independence of the measurement was advanced by Esnault-Pelterie [18]
as the definition of a primary concept, rather than as here where the initial definition of each concept is independent of all others. Here again we follow Bacon
who wrote “For information begins with the senses. But our whole work ends in
Practice –” [19]. The definition by the idea of independence of measurement does
not seem to accord with the choice by Esnault-Pelterie of mass, rather than force as
used here, as the third primary concept.
This discussion amplifies logical points numbers (i) to (iv).
2.8 Constant Relative Magnitude
The measure of primary concepts by addition of unit measures results in a linear
scale of measurement with a scale zero at zero amount. Then if x is the size of the
unit measure, for example 1 mm, then a length, `, can be written ` D kx where k is
an integer number. A power of ` has a numerical value of `a D k a x a . The ratio of
these powers of two lengths, `1 and `2 , is:
a
`a1
k1 x a
k1
D
;
a D
`2
k2 x
k2
which is independent of x and so, as Bridgman pointed out [1], the ratio of two measures of the same concept is independent of the size chosen for the ‘unit measure’;
as he put it, there is a constancy of relative magnitude.
The existence of linear scales with a common scale point of zero is important.
For example, the ratio of two temperatures are the same whether measured in the
Celsius or in the Réamur scales of temperature but not between either and the Kelvin
scale.
From the definition of the measure of a primary concept, a multiple of the smallest distinguishable amount is chosen as a convenient unit amount. In principle, all
other amounts are then taken as the arithmetical count of the content of smallest
amounts.
The logical steps numbers (v) and (vi) have now been considered.
2.9 Dimensional Equality
The discussion in Chapter 1 on dimension and unit equality is now amplified. By
writing:
ADB
34
2 Concepts, Dimensions and Units
four different equalities are commonly implied as follows:
a)
b)
c)
The measurement scale of A is identical to that of B. The unit measure must be
identical as must be the origin of the scale.
In units that are common to both, the numerical value of A equals that of B.
The dimensions of A are identical to those of B. Here that is written as:
AB
d)
The vectorial direction of A is identical to that of B. Usually, a special symbolism is used to indicate vectorial equality. This can be
ADB
or
AE D BE :
The symbols for dimensions as introduced by Maxwell [20] have been described in
Chapter 1. Also, the full set of dimensions in the Système International d’Unités (SI
units) together with the symbols of dimensions used here were listed in Table 1.1.
The third, fourth, fifth and ninth four entries in that table are for the primary
concepts just discussed. All the concepts with their units, excluding those of area
and of force, are those of the SI system.
It is incorrect to identify the use of the dimension equality symbolism with a numerical equality. It is as different from the latter as is the directional equality of two
vectors.
2.10 Units-conversion factors
As discussed in Chapter 1, to add two quantities having different unit measures
requires the introduction of a units-conversion factor. Problems have arisen in the
literature over the necessity and choice of introduction of these factors in dimensional analysis. Their use requires care and so will now be considered in some detail. Some of them, like J in the First Law of Thermodynamics and g0 in Newton’s
law have fallen into disuse largely as a result of the adoption of metric measures. In
those countries that still use the old Imperial system of units the use of these two
units-conversion factors still occasionally remain in the literature.
Other factors are more troublesome. The chemist uses two, both quite different,
neither with a generally allotted symbol, and both of the same numerical value; that
is of 103 . One arises from the measurement of volume in litres, the other from the
use of the mole based on the gramme in conjunction with the SI system of units.1
1
Delightful examples of units abound. One large chemical firm has used a heat transfer coefficient
in kg cal per sq.ft. per ı F per h. The ’X’ Co. Ltd. used its own ’X’ litre based on the volume
of a chemical it produces, whilst a lecturer once quoted a specific impulse for ion thrusters for
space vehicles in W per lb. Also conversion of units is sometimes taken very seriously: on
an archway leading into a British government department a traffic notice once read ’Maximum
width 7’0" (2.1336 M). By Order’. A serious omission arose in the National Aeronautics and Space
2.11 Products of Concepts
35
The use of units-conversion factors was introduced in Chapter 1, Section 1.2
through the example of the relation for the area of a square. This was:
a D A0 `2 :
(2.1)
Logical step number (vii) has now been explained.
2.11 Products of Concepts
The dimensions of products of concepts has been introduced in Section 1.3. From
the above example of Equation 2.1, those of area being H and of Length, L, then
those of the units-conversion factor A0 are HL2 . Also it was shown how dimensions
in products of concepts can be cancelled.
The ability both to cancel and to multiply dimensions is a requirement at a later
stage of dimensional analysis. It follows that concepts are required to be combined
only as products; no other form of combination allows dimensional cancellation.
With linear scales of measure, such products retain the idea of constancy of relative
magnitude. For example, with the units of pressure being FL2 , a change in the size
of the unit quantity for force means that the ratio of two pressures stays constant just
as they also would for a change in the unit quantity of length.
To show this, having the unit measure of force represented by x and that of length
by y, and using the scaling factors of kf and ks respectively, the two pressures can
be written as:
p1 D
kf1 x
2
.ks1 y/
I
p2 D
kf2 x
.ks2 y/2
:
Then the ratio of these two pressures is given by:
p1
kf1 ks2 2
D
:
p2
kf2 ks1
This ratio is independent of either of the unit measures of x or of y.
The equation for pressure that
p
F
`2
is thus unchanged in form by any change in the size of a unit of measure. It is
universally valid.
This part of the logic is the reverse of Bridgman’s when he specified retention
of absolute relative magnitude and then deduced the requirement of products of
concepts [1].
Logical steps numbers (viii), (ix) and (x) have now been amplified.
Administration when a space probe costing $125.106 flew 416.106 miles only to miss Mars because
one group measured force in Imperial pounds whilst another used Newtons [21]
36
2 Concepts, Dimensions and Units
2.12 Dimensional Equality in Functional Relations
To retain meaningful expressions, the additive terms in an equation containing products of concepts must all have both a dimensional and a units equality. For example,
consider the elementary relation for motion under constant acceleration that is:
s D ut C 12 f t 2 :
(2.2)
Then here each term has the following dimensions:
s LI ut LT1 T D LI
1
f t2
2
1 LT2 T 2 D L ;
where the factor of 12 has no dimensions and so is denoted by unity. Thus, with also
a uniformity of units, it remains meaningful to add these three terms of which two
are products.
Logical step number (xi) has now been explained.
2.13 Limitation to Functional Operations
Through the requirement to retain dimensional equality, it follows that the manipulation of complete equations during the generation of solutions is limited to certain operational rules. This has been discussed by both Bridgman [22] and by Taylor [11].
Addition and multiplication are permissible but other functional operations, such as
taking logarithms and forming binomial expansions, whilst generally acceptable numerically, dimensionally are only acceptable if all arguments are non-dimensional.
The latter restriction is seen from comparison of the dimensions of the terms of
a series. For example:
sin x D x x3
x5 x7
C
C :::
3Š
5Š
7Š
From the above discussion, and in particular from the right hand side of this equation, it is only dimensionally acceptable when x is dimensionless.
The measures of some concepts are not in the form of products of quantities. One
such is that of a regular scale of optical luminance which is related to surface area
by [23]:
ln L D Ka C constant :
(2.3)
A logarithm of a dimensional quantity is readily seen not to be dimensionally acceptable from the relation of:
ln s D .s 1/ .s 1/ 2
C
2
This shows that this equation is only dimensionally acceptable when s 1.
(2.4)
2.14 The Complete Equation
37
Equation 2.3 can be made dimensionally acceptable by rearranging it as:
log2 .L=L0 / D K.a a0 / ;
(2.5)
where the suffices indicate reference values. It follows that K 1=a.
A similar example arises in the expression for acoustic power which also is measured in a logarithmic scale. This obeys the relation for sound pressure intensity, p,
of:
n
log10 p D
C constant ;
(2.6)
10
where n is a sound level in decibels.
Both of these quantities have been expressed in these logarithmic scales because
they better represent the responses at regular intervals respectively of the human
sight and hearing [23, 24].
Another case where care has to be taken is in the use of functions of complex
variables. From the complex number of:
z D x C iy
we have
x2 C y2 D r 2
so that x and y each have the same dimensions as r and then i is non-dimensional.
With w D C i then the electrical field is given by [25]
dw
D Ex C iEy
dz
(2.7)
Denoting the dimensions of field by e, then dw=dz e so that w eL and then
eL which is consistent with the definition of the electrical potential, .
But also we have that:
ln
dw
D ln E C i. / :
dz
(2.8)
This, though it is a useful relation, does not satisfy an equality of dimensions.
Equations 2.3 and 2.8 are perfectly valid as representations of physical events and
as bases of calculation. They are only of unacceptable form for use in dimensional
analysis.
Logical point number (xii) has now been amplified.
2.14 The Complete Equation
It was pointed out that in dimensional analysis numerical addition of quantities has
to be of those quantities that are measured in both the same dimension and referred
38
2 Concepts, Dimensions and Units
to a common unit measure. Esnault-Pelterie required them to be of the ‘same physical nature’ which seems much too restrictive [18]. Where this type of addition is as
a multiplication then the multiplier must be a dimensionless number.
It has now been specified, for the purposes of dimensional analysis, that dimensional equality is required for the addition both of quantities and of products of
quantities. Such equations containing these were called ‘complete equations’ by
Bridgman [1].
The size of the unit measure of any dimension appearing in a complete equation
can be changed without either introducing or removing a units-conversion factor.
From Sections 2.8 and 2.11, if the unit measure is changed by a factor then, for
a linear scale, the equation is unchanged in algebraic form. In semantic terms, if an
equation is to provide a universal statement of a real event then this feature of the
complete equation is required so justifying the present derivation of it.
Both Buckingham and later Bridgman used the expression ‘complete equation’.
The former used it to imply that no variables are omitted, presumably within the limits of precision with which the equation models a phenomenon [26]. He introduced
the equality of dimensions as a separate idea. Bridgman, in contrast, clearly linked
the notation of a complete equation with that of equality of dimensions and gives
it the meaning, used here, that its algebraic form remains unchanged by changes in
the size of the unit measure. But Bridgman and other authors do not go as far as
specifying a complete equation as a requirement. For example, Bridgman and Sedov [27] said that physical regularities (laws) are ‘– generally independent of the
particular system of units of measurement selected from among a set of such systems – That this should be so is plausible — it is so exceedingly improbable as to
be practically impossible – if it did depend on one particular system of measurement.’
Bridgman’s order of logic is different from that presented here in that he started
with the acceptance of the complete equation, then derived the pi-theorem – which
is to follow here – and finally deduced equality of dimensions [1].
The present logic starts from the basic premiss of the requirement of meaningful
addition and leads on to the result of the complete equation. This procedure automatically arrives at Bridgman’s assumed result with no exceptions.
Bridgman’s example in support of his discussion [1] has been repeated by others
but seems not to be a powerful one. He added the linear-motion relations, v D f t to
s D 12 f t 2 to get v C s D f t C 12 f t 2 , stating correctly that it is numerically correct.
There is however no equality of either dimensions or units. Also, the latter equation
could be written as v D v.s; f; t/ which is analytically wrong there being a surplus
of independent variables. Also the two equations are uncoupled in that with f and t
specified the first equation can be solved for v and then the second independently for
s. This latter is an important point first dealt with elsewhere [13] and to be returned
to later.
It will be noticed that the present discussion does not specify the idea that it is
essential that an equation must be a complete one for it to represent validly a physical event; numerically this is not so. That idea was put forward by Buckingham.
Here, the existence of a complete equation is only derived as a consequence of the
2.15 Derived Concepts and Their Measure
39
dimensional equality that is specified for the requirements of dimensional analysis
and to retain a universal validity.
These matters amplify the logical point (xiii).
2.15 Derived Concepts and Their Measure
Table 2.1 lists a set of defined concepts. Each is defined in terms of the previously
defined primary concepts and each is in the form of products of quantities. Some
definitions introduce the appropriate units-conversion factor. In this table the defining relations are built up in a logically progressive order.
The first defining relation, number one in Table 2.1, is for the concept of velocity.
Column two gives the definition; column three indicates that this is derived from
two primary concepts which are those of length and of time; column six gives the
symbols to be used for the dimensions as being respectively L and T; column seven
gives the unit measures in the Système International d’Unités (SI units). Velocity
can require a units-conversion factor when, for example, velocity is measured in
knots, that is nautical miles per hour, and length and time are measured respectively
in metres and seconds.
The definition of angle, as the difference in orientation between two straight lines
observed in the plane containing both, has been left as part of Exercise 2.2. When
one line is rotated from a position of coincidence with the other until coincidence
is again achieved then that standard angle can be called any one of 360ı, 400 grad
or 2 radians; it is a matter of units. To extend the concept of length from that of
Table 2.1 Number in defining relation
No. Defining relation(s)
Measures
Primary
Derived
1
2
3
4
5
6
7
8
9
10
V s=t
! .1=ˇ0 /.s=r/
Er D hp f
Q D .1=J /F s
F D G0 m2 =r 2
q D r.F "0 /1=2
a V =t
I q=t
m D .F g0 / =a
Ek D 12 mV 2 W Ek Er
11
T D 12 mc 2 =kB
2
12
13
14
F D .1=P0 / .dE1 =dt / W E1 Er
H D F =0 sI
M0 D RT =p W Q m=M0
2
1
2
2
1
1
2
2
2
1
1
1
2
2
Units
Dimen- SI
Conversion sion
unit
Factors
Symbols
L, T
˛
m, s
rad
A
M
A
kg
1
K
1
1
1
C
cd
n
mol
1
1
1
1
1
1
1
1
2
1
40
2 Concepts, Dimensions and Units
a straight line to that of a curve we have to invoke the idea of the sum of an infinite
number of infinitesimal straight lines. Then alternatively, the measure of angle, ˛,
can be defined as proportional to the ratio of the length of the circular arc, s, to that
of the corresponding radius, r. To allow any of the above three acceptable units we
have to write this definition as:
!
1 s
;
ˇ0 r
(2.9)
where ˇ0 is the units-conversion factor for angle. This is shown as the second definition in Table 2.1. An example given later further illustrates the necessity for this
units-conversion factor.
Similarly the definition of solid angle is derived from those of length and area as
a=.0 r 2 / with 0 as the units-conversion factor: conventionally this latter is taken as
having a value of unity as it is the practice to take a single unit of length to measure
both a and r.
The third derived concept in Table 2.1 is defined by a physical law. It is Planck’s
law which relates energy of radiation to the frequency and so involves the primary
concept of time. The constant, hp , can be regarded as a units-conversion factor between those of time and those of energy.
The fourth concept in Table 2.1 relates heat to work and is again an expression
of the physical First Law of Thermodynamics together with mechanics [12]. Again
a units-conversion factor is introduced which is J and this is founded upon two
primary concepts. These are the primary definitions of length and force. Again it is
a statement of a physical law.
The fifth definition is the statement of Newton’s law of gravitational attraction. It
is an experimental law which gives a definition of mass. Now the universal gravitational constant, G0 , is the units-conversion factor.
The sixth definition is of the electrical charge in terms of the mutual force exerted.
This definition shows that electrical charge is a derived concept. This definition is
again based upon a physical law and founded upon the primary concepts of force
and length. Now the dielectric coefficient, "0 , is the units-conversion factor.
The seventh definition is of acceleration and so is an arbitrary defining relation.
Sometimes acceleration is quoted as a multiple of a standard gravitational acceleration. Otherwise, no units-conversion factor is involved.
The eighth definition, like the seventh, is an arbitrary definition of electrical current not involving a units-conversion factor. It is based upon one primary and one
derived concept.
The definition number nine shows that mass, m, is a derived concept. Accepting
Newton’s law, that acceleration, a, of a certain system is proportional to the causing
force, F , the constant of proportionality is a universal constant for that system. That
constant is called the mass of that system. To exclude relativity effects the constant
is specified as corresponding to the limit of the relative velocity of the observer
2.15 Derived Concepts and Their Measure
41
tending to zero. Thus it is written as:
m
g0
F;
a
(2.10)
where g0 is the units-conversion factor.
Thus there are two rigorous definitions of mass. To be consistent, there must then
be a relation between their units-conversion factors of G0 and g0 . It is usual to put
g0 D 1 thus fixing the experimental value of G0 . The so-called ‘english/imperial’
system of weights and measures did not do this and so it was an illogical one in this
respect. It is still used.
The tenth definition is of kinetic energy with the supplementary equating of this
energy with the Planck energy of definition number three. The first part is an arbitrary definition, the second comes from a physical law.
Definition number eleven gives the relation from the kinetic theory of gases that:
T D
1 mc 2
3 kB
(2.11)
and so is a physical law defining temperature. It contains the units-conversion factor
of kB , the Boltzmann constant.
The twelfth definition is of luminous flux in terms of luminous energy where
now the mechanical equivalent of light forms the units-conversion factor. It is supplemented by again identifying the energy with the Planck energy, which latter is
a physical law.
Definition number thirteen is of magnetic field strength in terms of electrical
current. The units conversion factor is now the magnetic permeability.
The final definition, number fourteen, is of quantity. The units-conversion factor
is the universal gas constant, R.
The molar quantity is introduced from the relations from gases. Those to be accounted for in dimensional analysis are:
p
R
D
T;
(2.12)
M0
m D M0 Am ;
(2.13)
Am D 1=Na ;
(2.14)
kB D R=Na ;
.1=2/mc
2
D .3=2/kB T :
(2.15)
(2.16)
Using again the symbol to mean ‘is dimensionally equal to’ and by assigning
m M then the dimensions of kB are, from Equation 2.16:
kB ML2 T2 1
(2.17)
42
2 Concepts, Dimensions and Units
Also from Equation 2.12
R=M0 L2 T2 1 :
(2.18)
From Equations 2.13, 2.14 and 2.15:
kB =m D R=M0
(2.19)
making these equations dimensionally consistent with Equations 2.12 and 2.13.
If the amount of substance is introduced as a fundamental dimension, and denoted by ‘n’, then assigning
Na n1
(2.20)
as required by the physical concept of Avogadro’s number and by the definition of
amount of substance. It follows from Equation 2.14 that
Am n
(2.21)
so that the so-called unit atomic mass does not have the dimensions of mass per
unit reference particle. Further, the molecular mass has, from Equation 2.13, the
dimensions of
M0 Mn1
(2.22)
so that it is not purely a numerical factor; it has dimensions. Then from Equation 2.15
R ML2 T2 1 n1 :
(2.23)
An illustration of the need for the unit of amount of substance to be included in the
SI set of dimensions is given later.
In thermodynamics, temperature is defined in terms of the properties of a gas by:
T p M0
;
R
(2.24)
which is entry number fourteen in Table 2.1. Similarly as with the definition of mass,
this relation is a limiting one taken as p ) 0. Equation 2.12 contains the units
conversion factor, R; We have to ask which of the two units-conversion factors, kB
and R applies to temperature and which to particle-quantity. A later example shows
that R is the one for particle-quantity and kB is that for temperature. This might
be expected from inspection of Equation 2.17 and 2.23; only the latter contains the
dimension of n.
The defining relations, listed in this table, are seen to be of two kinds. One, such
as the first one listed, is a definition. The other, such as the third one listed, is the
expression of a physical law.
2.15 Derived Concepts and Their Measure
43
Of the fourteen definitions given in Table 2.1, seven are listed in the SI system covering eight dimensions. In that system angle is classed as a supplementary
concept as once was quantity. The present discussion makes clear that both these
concepts are indistinguishable in status from the other five.
In summary the types of concepts are now listed:
a)
b)
c)
A primary concept where the definition of the concept is followed by the definition of its measure. These are those of extension, time and force.
A definition of a primary concept relying on some physical knowledge for the
specification of the particle, for examples, of atom, electron and molecule. This
is a definition of the measure and the example is that of molar quantity.
A definition of a derived concept of which there are two categories, viz.:
(i)
(ii)
A definition based upon primary concepts; examples are, velocity, acceleration and angle.
A definition based upon a physical law. Examples are mass, temperature,
electrical current and luminance.
The present approach differs from that used by others. Bridgman, for example, virtually proposed the definitions of all concepts to be inseparably linked with the
definition of the measure of them as is done here only for the derived concept [28].
It was because this approach raised difficulties for philosophers when considering
the meanings of time and of extension that the present line of argument for primary
quantities was advanced [13].
A primary concept is thus advanced as one that is independent of all others;
a derived one is either a definition of measurement in science or is an expression
of a physical law. This approach would appear to be consistent with Bridgman’s
view that “– operations which give meaning to our physical concepts should properly be physical operations, actually carried out –” [29] as long as the operations
leading to the present definitions of primary concepts are recognised as not, in
the first instance, requiring numerical measure. It is relevant that Bridgman goes
on to say that “It must not be understood that we are maintaining that as a necessity of thought we must always demand that physical concepts be defined in terms
of physical operations –” [29]. This is consistent with the present distinction just
mentioned, between defining relations formed from definitions and from physical
laws.
Whether a definition or a physical law, all these relations are single proportionalities with the constants of proportionality being simply units-conversion factors.
It is to be noted that all the equations in Table 2.1 are governing equations. That
is, they can be used to obtain solutions of processes that govern systems. That is
why the list excludes the Second Law of Thermodynamics. If the use of governing
equations leads to more than one solution, as can happen for example in the compressible flow of a gas [12], then the Second Law of Thermodynamics can be used
to determine which of the multiple solutions could occur in reality.
The logical steps numbers (xiv) and (xv) have now been explained.
44
2 Concepts, Dimensions and Units
2.16 Dimensions of Units-conversion Factors
From definition number seven in Table 2.1 we can write the dimensions of acceleration as a LT2 . Then from definition number nine we get g0 MLT2 F1 .
If we assign zero dimensions to g0 , that is g0 1, then one of two consequences
follow. Either we can put F MLT2 or m FT2 L1 . By making the unitsconversion factor non dimensional we remove one dimension from those needed to
describe the dimensions of the concept. Logical step number (xvi) has now been
covered.
This can be carried further. Several writers have in effect suggested that the number of primary dimensions is one more than the number of primary units-conversion
factors, so that the number of dimensions can be reduced to one. Buckingham suggested this in a little-known response to the Rayleigh–Riabouchinsky discussion to
be described later [30]. Later writers such as Wilson [31, 32] have repeated the idea
though without reference to Buckingham.
The total number of dimensions, and hence by the above demonstration, the total
number of units-conversion factors is, to some extent, at choice. For example, it has
long been common practice to assign zero dimensions to g0 and J . The commonest
form of current practice is to use the primary dimensions of:
M; L; T; ; A; C; n; ˛ ;
(2.25)
as listed in Table 2.1. A set, also listed in that table, totalling one less, of corresponding units-conversion factors can be, respectively:
G0 ; hp ; kB ; "0 ; p0 ; R; ˇ0 :
(2.26)
Inspection indicates that all other quantities that could also be regarded as unitsconversion factors can be expressed in terms of members of this set. For example,
there is the physical relation that,
0 "0 D 1=c 2 :
(2.27)
Now 0 , which is a units-conversion factor listed at item thirteen in Table 2.1, has
been assigned the value of 4107 H m1 . The entry number six in that Table
shows "0 as being the units-conversion factor for charge. Thus c becomes an alternative to "0 ; it is a units-conversion factor where physically c is a limiting value
for a perfect vacuum. In this sense it is analogous to R. It is related to Maxwell’s
derivation in 1860, from his studies of electro-magnetism, that the value of c is independent of the velocity of the observer. Einstein then made the general assumption
that c is a universal constant, independent of the choice of axes [33, 34]. This will
show later the comforting result that all units-conversion factors are of universal
value.
2.16 Dimensions of Units-conversion Factors
45
It follows that one of the entries in Table 2.1 could have been replaced by the
Einstein one of,
E D mc 2 :
(2.28)
Suppose that a problem in dynamics involved force, mass, length and time. The
units would be those respectively of F , M , L and T . By putting g0 1 then from
item number nine in Table 2.1, the dimensions of force now become,
F MLT2 :
(2.29)
By making the units-conversion factor, g0 , dimensionless the dimension F is not
required. This idea has been extended by the use of ‘atomic units’ as being convenient for the analysis of atomic structure. This adopts the following; e D 1, m D 1,
hp D 2. Another system, helpful in the analysis of electrons and positrons, is obtained by putting; m D 1, hp D 2 and c D 1. From this practice springs the
practice in astronomy of measuring extension in units of light-years simply to avoid
the continual inconvenience of recording large numerical values.
This process can be continued. If in item number five in Table 2.1, we put G0 1
then from that relation there is:
F M2 L2
Comparing this with Equation 2.29 gives:
M D L3 T2
(i)
Putting hp 1 in item number three gives:
(ii)
E T1
Also:
(iii)
E ML2 T2
Using item (i) gives that:
(iv)
E L5 T4
Comparing items (ii) and (iv) gives:
L T 3=5
Then from (i):
(vi)
M T1=5
46
2 Concepts, Dimensions and Units
From item number eleven in Table 2.1 and putting kB 1 gives:
ML2 T2
And using (v) and (vi) gives:
T1
(vii)
Again in item number 14 0f Table 2.1, by putting:
R1
Then:
M0 L2 T2
So that using items (v) and (vii) gives:
M0 T1=5
From Equation 2.22,
M n1 T1=5
So that:
(viii)
n T1=5 T1=5 D 1 :
From item number 12 in Table 2.1, there is:
P0 E 1
t C
so that putting P0 1;
(ix)
C ML2
D T16=5
T3
Finally, putting ˇ0 D 1;
(x)
!1
Thus all seven dimensions as listed in Equation 2.25 can, in principle be replaced
by just the single dimension of time. Equally, from this use of linear algebra, seven
could be replaced by any one of the eight dimensions.
However, carrying the reduction of dimensions as far as this is invariably not
acceptable for real analytical modelling of real events and processes; this will now
be explained.
2.17 The Inclusion of Units-conversion Factors
47
If the previous evidence, just mentioned, for the constancy of the measure of the
time and of the velocity of light, c is accepted, then it follows that there is a constancy of the measures of all concepts as they can, from the above demonstration,
be related to this single one of time or of c. All seems to rest on that one piece of
evidence.
The idea that the so-called universal constants of science are really just unitsconversion factors can be traced back to a paper by Kroon [35]. He wrote that
“physical constants are merely conversion factors”. He also included c as one of
them. Further, he makes the philosophical point that “... we can only work with
instantaneous length and time scales.”
Logical step number (xvi) has now been discussed.
2.17 The Inclusion of Units-conversion Factors
The discussion shows that the number of dimensions needed to describe concepts
is dependent upon the number of units-conversion factors retained in an analysis.
The problem that has exercised writers, such as Taylor [11] and Pankhurst [36],
is the determination of the number of units-conversion factors to retain in a general
functional statement describing a phenomenon; if indeed any. This has been stressed
in the comment by Prandtl that forms the heading to Chapter 6. Sedov also put the
matter of units-conversion factors as a prime problem [37]. He said that these factors
must be included whenever they are ‘essential’, but, as Kline pointed out [38], “–
the question of when the constants are essential is not simple; –”.
But there is a logical procedure. The following proposal is now made that in
part has been illustrated by the above examples. Philosophically, in any real phenomenon, to have absolute precision of understanding, all possible physical phenomena must be accounted for. This would require the inclusion in an analysis of all
those relations listed in Table 2.1 which contain the units-conversion factors listed
in Equation 2.26 and so involving the dimensions listed in Equation 2.25.
This can be amplified as follows. Forgetting for the moment the practical considerations which require an approximate modelling of a real event in order to make it
tractable, but rather consider the most general position as follows:
a)
b)
c)
d)
Real events on a continuum scale must be influenced to some degree by all
physical phenomena.
Therefore all concepts arising from the analytical expression of all these phenomena will appear in the list of variables in a general functional statement
governing the complete event.
Therefore all the units-conversion factors associated with all the concepts will
have to be included along with the physical variables.
The number of dimensions is of arbitrary choice; the number of units-conversion factors is one less.
48
2 Concepts, Dimensions and Units
e)
The minimum number of dimensions that can be used is one; the one to be used
is of arbitrary choice.
Before going further, there is an intriguing philosophical corollary to the above. Suppose that another completely independent
‘fundamental’ relation – like for example
Newton’s law of gravitation, F D G0 m2 =r 2 – is discovered, with its corresponding primary units-conversion factor – in this example G0 . Then either:
(i)
(ii)
we would have no dimension left for measurement; all that would be left to
science would be counting;
or
a completely new concept would have been discovered with its own unit of
measurement dimension and not related to any existing physical concepts except through its units-conversion factor.
If item (i) was true we would not be able to measure and so could not observe
anything; nothing would exist. Have we then reached the position where all ‘fundamental relations’ are now known?
The practical study of a phenomenon requires, as already stated, approximate
modelling and so not all physical phenomena are to be accounted for. Then only the
physical relationships chosen to give an acceptable solution of adequate precision,
with their physical concepts, such as are listed in column two of Table 2.1, and with
their corresponding units-conversion factors and dimensions, are to be included in
a list of the variables and universal constants – that is units-conversion factors –
required to describe the phenomenon. This gives a quite rigorous means of assessing which units-conversion factors are to be included in the list of variables: unlike
previous claims, the matter is neither one that is arbitrary nor a pragmatic choice
nor is the choice an art rather than a science. It is strictly controlled by the approximations introduced in the modelling of the true physical event. However care has to
be exercised in the procedure for listing the variables controlling a physical event.
Examples are given later.
Logical step number (xvii) has now been amplified.
2.18 Formation of Dimensionless Groups from Units-conversion
Factors
In the literature, and over a long period, a surprise has been hinted that a nondimensional group can be formed from hp , e, "0 and c. This group is e 2 ="0 hp c
and is called the fine-structure constant. This has even attracted philosophical attention. The fact that this non-dimensional group can be formed from units- conversion
factors is not a unique case as will now be shown.
If discussion is limited to phenomena requiring the seven dimensions of:
M; L; T; F; A; ; and n ;
2.19 Summarising Comments
49
then there are correspondingly six units-conversion factors. The choice of these factors is, to a degree, arbitrary. In this discussion they are now chosen to be:
g0 ; hp ; G0 ; "0 ; kB ; R
and are referred to as the ‘basic’ ones.
Other factors can be obtained from these. For example, these might be:
0 ; c; e; Na ; Am ; Fa ; me ; Ry ;
and these are referred to as ‘derived’ ones.
From this total of fourteen it is possible to derive seven non-dimensional groupings. This derivation is conveniently obtained by the cancellation technique used
in the present operation of the pi-theorem though it is not to be confused with the
application of that theorem because no functional relation is set out. The tabulation
is as follows in Table 2.2.
Thus the groups are as follows:
R
Fa c
; Na Am ;
;
Na kB
Na e
g02 "0 c 4
hp "0 c
m2e G"0
;
;
with
:
Ge 2 Ry2
e2
e2
"0 0 c 2 ;
Taking the square root of the reciprocal of the product of the fifth and seventh groups
and multiplying by the cube of the sixth group gives the group of:
Ry "20 h3p c
g0 me e 4
:
This with the next four of these non-dimensional groups are known relations from
analysis. Of numbers six and seven groups, the former is the reciprocal of the ‘fine
structure constant’ and the latter is analogous as another non-dimensional group.
Each of the last two groups can be given a physical interpretation. The former is
a measure of the strength of the electromagnetic force and the latter is a ratio of the
gravitational and Coulomb forces between electrons.
Thus, as Bridgman would have said, we see nothing esoteric about the finestructure constant being non-dimensional.
2.19 Summarising Comments
The discussion in this chapter has set out the foundation principles of dimensional
analysis. The whole logic starts from the self-evident premiss of meaningful addition. The basic requirements for further development in the coming chapters are
hp "0 c
e2
1
g02 "0 c 4
G0 e 2
1
L2
g02 "0 c 4
G0 e 2 Ry2
1
T
L
T4
L6
hp "0
e2
1
FL2
g02 "0
G0 e 2
"0
e2
1
R
Na kB
FL
FL
n
FL
R
FT4
L4
A2 T2
FL2
kB
R
Na
FL
M2
"0
FLT
FT2
ML
G0
g02
G02
hp
g0
Table 2.2
1
" 0 0 c 2
T2
L2
" 0 0
FT2
0 e 2
F
A2
0
L
T
c
AT
e
1
n
Na
1
Na Am
n
Am
1
Fa c
Na e
T
L
Fa
Na e
AT2
L
Fa
Na
AT2
nL
Fa
1
m2e G0 "0
e2
FL2
m2e G0
M
me
1
L
Ry
50
2 Concepts, Dimensions and Units
2.19 Summarising Comments
51
now available. In particular, the place of units-conversion factors has been rigorously laid down so that there is a clear method for deciding on their inclusion in
a general functional statement that governs a phenomenon.
Exercises
2.1
2.2
2.3
Define a straight line. As a hint, find out how Whitworth first made a flat surface [39] and so define a plane surface. Then consider the intersection of two
such planes [40]. Then define a circular arc. Consider how these definitions
are needed for a definition of angle. Then go on to demonstrate that a straight
line forms the shortest distance between two points. Satisfy yourself on the
rigorous logic of your argument. Out of interest, compare your definition of
a plane with the logic used in forming the Zeroth Law of Thermodynamics [12].
Define the concept of angle; use similar logical steps of Exercise 2.1. Bronowski rather suggested [41] that the ancients made use of the Pythagoras theorem
for a right-angle triangle, in constructing a right angle for their building construction. But such is the precision of the ancient temples and the medieval
cathedrals that the then scales of length might possibly be not sufficiently precise. Suggest means of constructing, in those times, a very large builders set
square to measure a right angle. Make use of some of the logic in Exercise 2.1.
If such a measuring device was made from a single metal, would a change in
uniform temperature change its precision?
As sketched in Figure 2.3, a pump of size, `, is pumping fluid of density ,
of viscosity , at a volume rate, Q, with an outlet velocity V , over a pressure
difference of p, against the gravitational constant g. Which of the following
Figure 2.3 The sketch of
a pump system
52
2 Concepts, Dimensions and Units
groups are non-dimensional?
QP
QP
QP
;
;
g`2
.g`/1=2
g1=2 `2:5
pV
p
p`6
;
;
`
V 2
V2
5
`
;
V `
V
2.4
2.5
2.6
2.7
Using the dimensions listed in column six of Table 2.1, determine the dimensions of the units-conversion factors in that Table.
Repeat the analysis of Exercise 2.4 but replacing the dimension of M with that
of F.
Repeat the analysis of Section 2.16. to express the dimensions of s, V , a, m,
and F in terms of L instead of those of T.
The induced electro-motive force (in volts) applied by a solenoid, ", is given
by:
"D
0 An2 di
;
` dt
where:
0
A
n
`
i
t
2.8
is the units-conversion factor (Table 2.1);
is the cross-sectional area of the solenoid;
is the number of coil turns;
is the solenoid length;
is the current;
is the time.
show that this is a complete equation.
The photo-electric equation is:
hp f W D
1
2
2
mvm
;
where:
hp
f
W
m
vm
2.9
is the Planck constant;
is the frequency;
is the work-function energy;
is the electron mass;
is the emitting velocity.
Show that this is a complete equation.
“Fan her head!” the Red Queen anxiously interrupted. “She’ll be feverish after
so much thinking.” Discuss!
References
53
References
1. P.W. Bridgman. Dimensional Analysis. Rev. Ed., Yale, New Haven, 1943.
2. H. Jeffreys. Units and dimensions, Phil Mag., Vol. 34, 7th Ser., No. 239, pp. 837–842, December 1943: (also, p. 839, Ll. 22–25).
3. J.C. Gibbings. The mole and extended dimensions, Int. Jour. Mech. Eng. Education, Vol. 10,
No. 2, p. 143, April 1982.
4. J.C. Gibbings. A logic of dimensional analysis, J. Phys. Author: Math. Gen., Vol. 15,
pp. 1991–2002, 1982.
5. Plotinus. Time and eternity, The philosophy of time, (Ed. R M Gale), pp. 24–37, Macmillan,
London, 1968.
6. F. Waismann. Analytic-Synthetic, The philosophy of time, (Ed. R M Gale), pp. 55–63,
Macmillan, London, 1968.
7. J.B.J. Fourier. Theorie Analytique de la Chaleur, Vol. 2, Ser.7, Ch. 2, Sec.9, pp. 135–140,
Firmin Didot, Paris, 1822.
8. M.W. Zemansky. Heat and Thermodynamics, 4th Ed., McGraw-Hill, New York, 1957.
9. J.H. Keenan, Thermodynamics, Wiley, New York, 1957.
10. J.C. Gibbings. Defining moments, Professional Engineering, I. Mech. E., Vol. 11, No. 14,
pp. 24–25, Wednesday 22 July 1998.
11. E.S. Taylor, Dimensional Analysis for Engineers, Clarendon Press, Oxford, 1974.
12. J.C. Gibbings. Thermomechanics: The Governing Equations, Pergamon, Oxford, 1970.
13. J.C. Gibbings. On dimensional analysis, J. Phys. A: Math. Gen., Vol. 13, pp. 75–89, 1980.
14. Augustine (Saint). Some questions about time, The philosophy of time, (Ed. R M Gale),
pp. 38–54, Macmillan, London, 1968.
15. B. Russell. History of western philosophy, George Allen & Unwin, London, 1961.
16. L.V. Morrison. We are just slowing down, The Times, No. 68587, p. 16, London, Tuesday
January 3 2006.
17. F. Bacon (Lord Verulam). The physical and metaphysical works of Lord Bacon, Trans J Devey,
Bell and Daldy, London, 1868.
18. R. Esnault-Pelterie. Dimensional Analysis (English edn), Lausanne: Rouge, 1950.
19. F. Bacon (Lord Verulam). The Novum Organon, Trans G W Kitchin, Oxford Univ. Press,
Oxford, 1840.
20. J.C. Maxwell. On the mathematical classification of physical quantities, Proc. Lond. Math.
Soc., Vol. 3, Pt. 34, pp. 224, March 1871.
21. N. Hawkes. How NASA put its foot in Mars blunder, The Times, No. 66635, p. 15, London,
2nd October 1999.
22. P.W. Bridgman. Dimensional Analysis, Encyclopaedia Britannica, 14th Ed. revised, 1959.
23. J.C. Gibbings. The systematic experiment, (Ed. J C Gibbings), Cambridge Univ. Press, 1986.
24. J.F. Dunn, G L Wakefield. Exposure Manual, 4th Ed., Fountain Press, England 1981.
25. J.C. Gibbings. The field-plane method for the design of two-dimensional electrostatic fields,
Jour. Electrostatics, Vol. 6, pp. 121–138, 1979.
26. E. Buckingham. On physically similar systems; illustrations of the use of dimensional equations, Phys. Rev., Vol. 4, Pt. 4, pp. 345–378, 1914.
27. P.W. Bridgman, L I Sedov. Dimensional Analysis, Encyclopaedia Britannica (Macropaedia)
15th Ed., Vol. 14, p. 422, 1974.
28. P.W. Bridgman. The Logic of Modern Physics, Macmillan, New York, 1927.
29. P.W. Bridgman. The nature of physical theory, Princeton Univ. Press, Princeton, 1937.
30. E. Buckingham. The principle of similitude, Nature, Vol. 96, p. 396, London, 9th December
1915.
31. W. Wilson. Dimensions of physical quantities, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (Philos. Mag.), Vol. 33, 7th Ser., No. 216, pp. 26–33,
January 1942.
54
2 Concepts, Dimensions and Units
32. W. Wilson. Note on dimensions, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (Philos. Mag.), Vol. 33, 7th Ser., No. 226, pp. 842–844, November 1942.
33. J C Maxwell. A treatise on electricity and magnetism, 3rd Ed., (1891), Dover, New York,
1954.
34. S. Hawking. (Ed). A stubbornly persistent illusion: the essential scientific writings of Albert
Einstein, Running Press, Philadelphia, 2007.
35. R.P. Kroon. Dimensions, J. Franklin Inst., Vol. 292, pp. 45–55, 1971.
36. R.C. Pankhurst. Dimensional analysis and scale factors, Chapman Hall, London, 1964.
37. L.I. Sedov. Dimensional and similarity methods in mechanics, Academic Press, New York,
1960.
38. S.J. Kline. Similitude and approximation theory, McGraw-Hill, New York, 1965.
39. L. Fox, R.V. Southwell. On the stresses in hooks and their determination by relaxation methods, Proc. I. Mech. E. (App. Mechs.), Vol. 155, pp. 1–19, 1946.
40. S.L. Green. Algebraic solid geometry, p. 17, Cambridge Univ. Press, 1941.
41. J. Bronowski. The ascent of man, British Broadcasting Corporation, London, pp. 157–162,
1976.
Chapter 3
The Pi-Theorem
In relation to engineering, the qualities required in
a mathematical process are utility, generality, and simplicity
and of these the greatest is simplicity.
Sir Charles Inglis
Notation
a
A
ai r
C
Cp
d
D
D1 . . . Di
e0 , e1
E
f
F
g
h
H
j
K1 , K2
k
L
`
m
nP n
n
N
p
QP
Q1 . . . Qi
R
s
Amplitude
Surface area
Powers of dimensions
Concentration
Coefficient of specific heat
Pipe diameter
Drag force
Dimensions
Elasticity coefficients
Electric field strength, elastic modulus
Acceleration
Force
Gravitational acceleration
Heat transfer coefficient
Magnetic field strength
Current density
Coefficients
Thermal conductivity
Pipe length
Scale size
Mass
Diffusion rate
Number of dimensions
Number of variables
Pressure, number of groups
Quantity flow rate, heat rate
Variables
Resistance
Surface area element; distance
J.C. Gibbings, Dimensional Analysis. © Springer 2011
55
56
3 The Pi-Theorem
S
t
T
u, v, w
U
v
W
x; y
X
Shape variable
Time
Temperature
Velocities
Energy per unit volume
Local velocity
Load
Variables
Body force
Fa
g0
hP
kB
me
Na
R
Ry
G0
Faraday constant
Inertia constant
Planck constant
Boltzmann constant
Electron mass
Avogadro number
Gas constant
Rydberg number
Gravitational constant
ˇ
ˇ0
ı
"0
P
, 0
˘
!
Angle
Units-conversion factor
Angle
Deflexion
Vacuum permittivity
Heat rate intensity
Conductivity
Viscosity, magnetic permeability
Non-dimensional product
Density
Surface tension
Shear stress
Electrical potential
Frequency
G
k
K
m
n
N
Number of non-dimensional groups
Number of cancellations for a non-dimensional group
Number of cancellation levels
Number of variables in a non-dimensional group
Number of dimensions
Number of variables
A
L
M
Current dimension
Length dimension
Mass dimension
3.2 The Basic Outcome
T
˛
57
Time dimension
Angle dimension
Temperature dimension
3.1 The Outline Form of the Theorem
The one basic theorem in dimensional analysis is known generally as the pi-theorem,
so called because it involves groups of products of quantities. It was introduced
in outline in Chapter 1 through a solution to two simple problems. It generates
the ability to derive, using a functional transformation, an equation containing only
products of variables and sometimes, a single non-dimensional product. The full
proof of this theorem, in a general and rigorous form [1], provides the final logical
step number (xviii) of the listing of the full logic of the subject given in Chapter 2.
This general proof is now given. Various examples illustrating procedures for its
correct use are then described here and in succeeding chapters. Some of these have,
in the past, either raised difficulties or even defied solution.
3.2 The Basic Outcome
The basis of dimensional analysis is that a natural phenomenon is described by the
very general equation:
f .Q1 ; Q2 ; : : : ; Qi ; : : : ; QN / D 0 ;
(3.1)
where each of the variables, Qj , j D 1; 2; j; ; N , is either a concept, a property in the generalised thermodynamic sense [2, 3] or a units-conversion factor.1
Equation 3.1 must exist as a functional relation and is required to meet the principle
of dimensional and units equality laid down in Chapter 2; it has to be a ‘complete’
equation.
It follows that the governing equations must be built up from the list in Table 2.1
and, or, formed from complete equations of properties such as [2]:
a)
The Newtonian equation for viscous shear in a fluid, that is:
D
1
@u
I
@y
Strictly speaking, a units-conversion factor is a universal constant and so should not properly be
called a variable; but to do so here is to follow a general practice by many writers: still it goes
against the grain not to accord with The Concise Oxford Dictionary.
58
b)
3 The Pi-Theorem
The relation for molecular diffusion, that is:
nP n D c D
c)
@ cn I
@x c
The relation for thermal conductivity, that is:
D k
d)
@T
I
@x
The relation for electrical current, that is:
j D @
@x
with other similar relations. It is noted that these physical relations are each
a complete equation and this is a requirement.
The formation of Equation 3.1 is the first step; upon this foundation all else rests.
It is essential, before the transformation is applied, to ensure that Equation 3.1 adequately describes the phenomenon to the desired precision in the sense of the discussion of Chapter 2. The physics must be fully understood so that the physical
variables together with the units-conversion factors are listed precisely and completely.2
But otherwise the form of the function in Equation 3.1 is not limited: it could
have a singularity; be a multi-valued one; even have a discontinuity. This generality
is important to the wide application of the pi-theorem.
The pi-theorem of dimensional analysis provides a transformation of Equation 3.1 into:
f ˘1 ; ˘2 ; : : : ; ˘i ; : : : ; ˘p D 0 :
(3.2)
In the latter equation each of the ˘i variables is in the form of a non-dimensional
product of some or all of the variables in Equation 3.1. A prime result is that p < N .
This transformation theorem is known as the pi-theorem.
There is a quite vital relationship between Equations 3.1 and 3.2. This is that
the pi-theorem transforms the former into the latter whilst still retaining a functional relation between the variables in the latter. Also Equation 3.2 retains the
format of being a ‘complete’ equation because all the variables are formed into
non-dimensional products. So important are these transformation features, yet being occasionally omitted in the literature, that this will be returned to again.
2
We recall again the prior discussion, that all practical analysis is an acceptable approximation.
This then is the criterion for an adequate selection of variables.
3.3 The Generalised Pi-theorem
59
3.3 The Generalised Pi-theorem
The preceding demonstration in Chapter 1 outlining the principles of the pi-theorem
will now be given as a generalised proof.
For N variables, there is Equation 3.1 which is to be a ‘complete’ equation. The
dimensions of each variable, Qj , can be expressed in terms of the dimensions, D1 ,
D2 , . . . , Dr , . . . , Dn , so that combining dimensions in products, as required from the
discussion of Chapter 2, tabulation is set out as Table 3.1.
Table 3.1
Variable
Dimensions
Q1
Da1 11 Da2 12 : : : Dar 1r : : : Dan1n
Q2
Da1 21 Da2 22 : : : Dar 2r : : : Dan2n
a
a
a
a
D1 i 1 D2 i 2 : : : Dr i r : : : Dni n
a
a
a
a
D1 N 1 D2 N 2 : : : Dr N r : : : DnN n
Qi
QN
Here the ai r are non-dimensional numbers. Some of the variables, Qi , may not
require all of Dn to describe their dimensions. That is, for one or more i , some or
all of ai r may be zero.
The variable Qi is now used to cancel the dimension in D1 . For the variable Q1
and considering just the dimension D1 ,
a
a ai 1
=D1 i 1
a
=D1 i1
Q1 i 1 =Qia11 D1 11
a a11
D 1;
(3.3)
or generally,
a
a
Qj i 1 =Qi j 1 D1 j 1
ai 1
a aj 1
D 1:
(3.4)
Equation 3.4 is effectively a transformation relation. The result of this transformation is tabulated in Table 3.2.
By writing Equation 3.1 as:
ai 1 a11 a11
˚ a
a
a a Q1 =Qi Qi D f Q2 i 1 =Qia21 Qia21 ; : : : ; Qi ; : : : ; QNi1 =Qi N 1 Qi N 1
(3.5)
or
ai 1 a11 a11
˚ a
a
Q1 =Qi Qi D f Q2 i 1 =Qia21 ; : : : ; Qi ; : : : ; QNi 1 =QiaN 1 :
(3.6)
60
3 The Pi-Theorem
Table 3.2
Variable
Dimensions
a
Q1 i 1 =Qia11
Qi
a
a
QNi 1 =Qi N 1
Da2 12 : : : Dan1n
ai 1 . a
a
D2 i 2 : : : Dni n
a11
a
a
a
D1 i 1 D2 i 2 : : : Dni n
aN 2
a . ai 2
a aN 1
a
D2 : : : DnN n i 1
D2 : : : Dni n
This retains the previously prescribed operational limits so that a ‘complete’ equation is retained.
Inspection of Table 3.2 shows that Qi is the only variable in Equation 3.6 that
contains a dimension in D1 . Following the argument previously set out in Chapter 1,
then for the dimension in D1 to balance, Equation 3.6 must take the form of:
ai 1
˚ a
a
Q1 =Qia11 Qia11 D Qia11 f Q2 i 1 =Qia21 ; : : : ; QNi 1 =QiaN 1 :
(3.7)
Thus the term in Qi cancels out as does D1 leaving only:
˚ a
a
f Q1 i 1 =Qia11 ; : : : ; QNi 1 =QiaN 1 D 0 :
(3.8)
This transformation process may be continued for successive cancellations of
D2 , D3 , . . . , Dn until there are no more dimensions left for cancellation and all the
groups of variables are in the form of products and are non-dimensional. There is
nothing mandatory about the completion of this transformation process; prior stages
give equally valid transformed equations. Also the order of the cancellation of the
dimensions, Di , is of arbitrary choice. This is because Di can represent any one of
the physical variables.
If each of the ai r are not zero then, with exceptions to be described, each time
a cancellation of a variable is made then:
a)
b)
one variable is added by multiplication to each group of variables;
one dimension is removed.
From item (b) there will be a total of n cancellations and after all possible cancellations each group will contain (1 C n) variables and there will be (N n) groups.
If one of ai r is zero, say ajr , then the dimensions of the group with Qj will not
contain Dr . A cancellation will not be needed so that this group will contain only
.1 C n/ 1 D n variables.
If at a set of cancellations, say the first one, there is the condition that:
ai 1 =ai r D aj 1 =ajr
3.4 Illustrative Examples
61
for all j D 1, 2, . . . , N , j D i being a trivial case, then this first cancellation will
cancel also the dimension Dr . The number of groups then becomes .N n/ 1 D
N n 1 because there will be one fewer set of cancellations. These results will
now be illustrated by examples.
This is the generalised proof of the pi-theorem and covers the final logical step
number (xviii) as listed in Chapter 2.
It is important to emphasise again that the pi-theorem is a mathematical transformation process. This ensures that, in the transformation from Equation 3.1 to
Equation 3.2, a functional relation is retained. Also, because only products are derived, then from the discussion of Chapter 2, a complete equation is retained. As is
to be described in Section 3.5, procedures by Rayleigh, Bridgman and Birkoff were
all clear about this. These matters are emphasised because there are many procedures that have been published in which a set of variables is merely reformed into
a set of non-dimensional groups with no reference to the outcome being a functional
relation between those groups nor that a complete equation is necessarily derived.
3.4 Illustrative Examples
Some general results can now be illustrated by some particular examples.
3.4.1 Linear Mass Oscillation
The phenomenon of a mass oscillating on a linear spring has been discussed in
Chapter 1 and is illustrated again in Figure 3.1. Returning to this example, that
analysis led to the result of:
!2m 1
D f .m; a/ :
e m
To balance the mass dimension in this equation it must then take the form of:
! 2m 1
1
D f .a/
e m
m
Figure 3.1 The oscillation of
a mass on a spring
62
3 The Pi-Theorem
so that:
!2m
D f .a/ :
e
(3.9)
It is not then possible to balance the dimensions. There are two possible reasons.
First, one or more independent variables might have been overlooked. But none
can be observed from the physics of this phenomenon. Secondly, the quantity a is
a false independent variable. Thus to obtain a balance of dimensions in Equation 3.9
the quantity a has to be removed. In any example where this sort of thing happens
it is always important to investigate the first possibility. However, inspection of the
listing of the dimensions of the four variables in Equation 1.6 shows immediately in
this case that the variable a is the only one with a length dimension so that it might
have been judged that it should be excluded at that stage.
3.4.2 Non-linear Mass Oscillation
We now consider the extension to the case of vibration against a non-linear spring.
This non-linearity can be described by the following relation between load and extension:
F D e0 ı C e1 ı 2 :
The transformation of this equation to a function of non-dimensional groups follows the procedure of the preceding proof. It can be tabulated as before as described
in Chapter 1.
Then the pi-theorem solution is of Compact Solution 3.1.
Thus the pi-theorem gives the result that:
!2m
e1 a
Df
:
e0
e0
Now the amplitude of the oscillation is correctly present in the functional relation.
The following notation is now used
G
k
K
m
n
N
number of non-dimensional groups;
number of cancellations in each group;
number of levels of cancellation;
number of variables in each group;
number of dimensions;
number of variables.
In the above Compact Solution 3.1 are included the numbers of levels of cancellation as denoted by K, and the number of cancellations for each group as denoted
by k.
3.4 Illustrative Examples
63
Compact Solution 3.1
K
!
m
a
e0
e1
1
T
M
L
M
T2
M
LT2
m
e0
e1
e0
T2
1
L
!2 m
e0
1
1
2
3
e1 a
e0
1
k
2
2
Conclusions from this example are as follows:
(a)
(b)
(c)
(d)
The number of variables, N , less the number of levels of cancellation equals
the number of non-dimensional groups, G; or, G D N K.
The number of cancellations for each group is less than the number of dimensions, n; or, k < n.
The number of variables in each group, m, is given by m D k C 1.
The value of K and hence of G is independent of the order of cancellation of
the n dimensions because each dimension requires one cancellation in turn.
3.4.3 Impact of a Jet
The next example is of the impact force, F , of a vertical jet of liquid from a nozzle, into the atmosphere and at the position where the jet is breaking up. This is
illustrated in Figure 3.2. The governing equations will be the following:
a)
b)
c)
the relation for the viscous force;
the relation for the surface tension force;
the Newton law.
The variables will be the surface tension, , the viscosity, , the density, , the jet
velocity at the point of break-up, V , and a size, `. The pressure does not appear in
the equations because the flow is in an atmosphere of uniform pressure and so there
are no pressure differences to give forces.
Thus we have that:
f .F; ; V; `; ; / D 0 :
(3.10)
64
3 The Pi-Theorem
Figure 3.2 The break-up of
a jet flow
Compact Solution 3.2
K
1
2
3
k
F
V
`
ML
T2
M
L3
L
T
L
M
LT
M
T2
F
¢
L4
T2
L2
T
L3
T2
F
v 2
V
V 2
L2
L
L
V 2 `
F
V 2 `2
V `
1
1
1
3
3
3
The transformation of this equation to a function of non-dimensional groups follows the procedure of Compact Solution 3.2.
From this solution Equation 3.10 is transformed to:
F
f
;
;
D 0:
(3.11)
V 2 `2 V ` V 2 `
For this example:
(i)
(ii)
(iii)
(iv)
G D N K;
m D k C 1;
K D n;
k D K.
3.4.4 Electromagnetic Field Energy
The next example is that given by Buckingham [4]. It has presented special difficulties. It is of the energy per unit volume, U , of an electromagnetic field. Buckingham
3.4 Illustrative Examples
65
Compact Solution 3.3
K
U
E
H
"
M
LT2
ML
AT3
A
L
A2 T4
ML3
ML
A2 T2
U
E
"
A
T
T2
L2
E" 2
A
L
1
A2
L2
1
2
1
2
U
H 2
3
k
1
E" 2
1
H 2
1
1
2
3
proposed that:
f .U; E; H; "; / D 0 :
The solution procedure is shown in Compact Solution 3.3.
The transformed equation becomes:
"
1=2 #
U
E
"
Df
2
H
H
(3.12)
(3.13)
Though the dimensions used to described the variables of Equation 3.12 are four
in number, that is, M, L, T, and A, in this case the dimensions can be taken as
effectively being those of the three cancelling quantities, that is:
ML=A2 T2 ; T2 =L2 ; A=L :
These can be simplified to:
M=L3 ; T=L; A=L :
(3.14)
For this example:
(i)
(ii)
(iii)
(iv)
G D N K;
m D k C 1;
K < n;
k K.
Now the condition number (iii) indicates the result that the number of effective
dimensions of Equation 3.14 is less than the number of those originally listed which
were M, L, T and A.
66
3 The Pi-Theorem
Again, the number of non-dimensional groups obtained is independent of the
order of cancellation of the three dimensions.
The discussion of this example is extended as the Exercise 3.1.
3.4.5 Heat Exchanger
Another example that has given trouble in the literature is that of the parallel-flow
heat exchanger. This is illustrated in Figure 3.3, showing the cold fluid in the central
tube receiving heat from the hot fluid in the annular passage over the area A.
The energy equation for the hot flow [2, p. 270], using mean values across each
stream is:
QP D m
P h Cph .Thi Th0 / m
P h Cph Th :
Similarly, for the cold stream and for the same heat rate to the cool flow, the
energy equation is:
QP D m
P c Cpc .Tc0 Tci / m
P c Cpc Tc :
The heat rate can be expressed by:
QP D hAT ;
where T is the temperature difference across the area A.
These equations show that the mass flow rate, m,
P and the coefficient of specific
heat, Cp are combined as mC
P p . Also the variables A and h are combined as the
product hA.
The Compact Solution 3.4 is set out as follows.
This shows that:
Tc
m
P h Cph hA
T
Df
;
;
(3.15)
Th
m
P c Cpc m
P c Cpc Th
Inspection of Compact Solution 3.4 shows that the effective dimensions are two
in number and not the original four. These two are seen to be:
ML2
;
;
T3 Figure 3.3 The contra-flow
heat exchanger
3.5 Prior Proofs of the Pi-theorem
67
Compact Solution 3.4
K
m
P h Cph
m
P c Cpc
Th
Tc
hA
T
ML2
T3 ML2
T3 ML2
T3 m
P h Cph
m
P c Cpc
1
1
2
k
hA
m
P c Cpc
1
Th
Tc
T
Tc
1
1
1
1
1
1
which reduce to:
ML2
;
;
T3
Now for this example:
(i)
(ii)
(iii)
(iv)
G D N K;
m D k C 1;
K < n;
k < K.
This example is returned to in greater detail in Chapter 6.
3.5 Prior Proofs of the Pi-theorem
A comparison is now made of the present derivation of the pi-theorem with existing
ones of which four basic variants are described.
The earliest version relies on the initial functional relation, such as that of Equation 3.1, being in the form of a single power product of the variables. This provision
was used by Rayleigh [5] in 1885 and still appears in several texts. Experiments
on all sorts of phenomena show that this simple form of the function does not at
all represent the experimentally determined one. The flow examples illustrated in
Figure 1.5. show where this assumption is clearly invalid.
The weakness of the Rayleigh assumption is here illustrated by a simple example.
For linearly accelerated motion there is the relation of:
s D ut C .1=2/at 2 :
68
3 The Pi-Theorem
This is normalised to the form:
s
1 at
D1C
:
ut
2 u
(3.16)
The dimensions of these variables are:
s
L
Variables
Dimensions
u
L=T
t
T
a
L=T2
Then Rayleigh’s assumption would be to propose that
sua t b ac D K ;
(3.17)
where K is a non-dimensional constant. Requiring the left hand side of Equation 3.17 to be also non-dimensional leads to:
for the dimension in L:
1CaCc D0
and for that in T:
a C b 2c D 0 :
These are two equations for three unknowns. So these equations are formed as:
a D 1 c
and
a C b D 2c ;
and then following the common practice in writings on dimensional analysis of
using determinants to obtain a solution gives the denominator as:
ˇ
ˇ
ˇ 1 0ˇ
ˇ
ˇ
ˇ 1 1 ˇ D 1 :
The first numerator is then:
ˇ
ˇ .1 c/
ˇ
ˇ 2c
and the second is:
ˇ
ˇ 1
ˇ
ˇ 1
ˇ
0 ˇˇ
D .1 c/
1ˇ
ˇ
.1 c/ ˇˇ
ˇ D c 1:
2c
3.5 Prior Proofs of the Pi-theorem
Substitution into Equation 3.17 then gives:
c
s
at
DK
:
ut
u
69
(3.18)
This, in comparison with Equation 3.16, is the wrong answer. Without even the
accompanying clarification that both s=ut and at=u are each dimensionless, commonly it has been assumed that all equations derived by this method of the form of
Equation 3.18 lead to:
s
at
Df
ut
u
with no justification expressed for this change.
It follows that the only case where Rayleigh’s assumption is valid is when only
a single non-dimensional group is the result.
This serious limitation was relaxed to a degree by Buckingham who in 1914
specified that the function be represented by a finite series of power products [4].
Both Focken [6] and Massey [7, p. 55] justify this approach by reference to the
Weierstrass approximation theorem, and so again this approach is approximate and
requires the existence of a function that is continuous and so is not general. In
presenting his version, Buckingham additionally separated all those variables having a common dimension and, dividing by one of them, expressed them as nondimensional ratios before applying the pi-theorem to the remaining variables. Neither the advantage nor the need for this procedure is clarified by such writers and is,
indeed, not required by the present proof and method. However, it should be noted
that Rayleigh was careful to retain a necessary functional relationship. It is not clear
how Buckingham’s mixed procedure retains this requirement.
A second approach is that discussed, for example, by Birkhoff [8, § 64] which is
to express the original function as an infinite Maclaurin series. But not all examples
are expressible in series form because of singularities, as indeed Birkhoff points out
[8, p. 94]. Further, convergence of a series does not necessarily exist nor can be
generally demonstrated and without convergence a series is meaningless.
Some phenomena are known analytically not to be represented by a Maclaurin
series. One example is the simple one of a vehicle coasting to rest. By writing the
resistance as being represented by:
R D K 1 V C K2 V 2 C K3 V 3 C and then putting:
m
dV
D R D K1 V K2 V 2 K3 V 3 C dt
70
3 The Pi-Theorem
gives on integration that:
"
2 #
t
1
K2
1
K3
K2
D
ln V C 2 V C
C
V 2 C C constant :
m
K1
2K1
K1
K1
K1
Thus the leading term is a logarithmic one and so cannot be represented by
a Maclaurin series in V . Equally, the use of the Weierstrass approximation theorem is precluded.
Another example of this is the flow at low Reynolds number past a sphere.
Brooke Benjamin discussed this and drew attention to the fact that analytical representation of the drag force involved a series containing logarithmic terms in the
Reynolds number [9]. So again, there is a limitation of application.
A third approach makes use of the concept of the invariance of a ‘complete’
equation. Birkhoff illustrates this by the example of the resistance on a closed body
in a flow [8]. Putting:
D D f .; V; `; / :
Following the example in Chapter 1, this can then be rewritten as:
D
f
;
;
;
V;
`
D 0:
V 2 `2 V `
(3.19)
A change in the size of the unit quantity can then be made in turn so as to give
a numerical value of unity to each of , V and `. Taylor [10] describes this as a use
of units that are intrinsic to the problem rather than a use of extraneous units. It is
then stated that Equation 3.19 can be replaced by:
D
f
;
; 1; 1; 1 D 0 ;
(3.20)
V 2 `2 V `
so that:
f
D
;
D 0:
V 2 `2 V `
(3.21)
This demonstration is basically that presented by Langhaar, who points out [11,
p. 55] that his proof both limits the independent variables to having positive values,
and [11, p. 58] requires that in the final formulation a non-dimensional group is to be
a single-valued function of the other ones. Neither of these very severe limitations
to applications appear in the present generalised proof.
A further difficulty with this third approach arises from the transformation between Equations 3.19 and 3.20. An equation that represents a real physical event is
understood to be a description of the relation between variables that retains truth as
these variables change in numerical value, these values being related to fixed values
of each and every unit quantity: it is a matter of semantics. It is also the requirement
3.5 Prior Proofs of the Pi-theorem
71
for a ‘complete’ equation as set out in Chapter 2. A fixed linear transformation applied to such an equation retains this meaningful representation by the transformed
equation. But the transformation between Equations 3.19 and 3.20 is one that must
change continuously as the last three variables of Equation 3.19 are each continuously changed: this has the effect of continuously changing the unit quantity of
these variables. Then the representation is no longer meaningful as just described
and also a constancy of relative magnitude is not retained.
Finally, a fourth approach, such as by Bridgman [12], relies on the function of
Equation 3.1 being differentiated in turn with respect to each and every one of the
variables. But the differentiability of a function is not generally demonstrable. Further, some of the ‘variables’ of Equation 3.1 can be units-conversion factors having
fixed values and so are not differentiable. Esnault-Pelterie [13] advanced further
detailed adverse criticisms of Bridgman’s proof by use of an example ascribed to
Villat. What is quite surprising is that Bridgman commenced his discussion [12] by
two simple examples, each of which resulted in a single non-dimensional group.
He basically introduced the present method of the removal of variables by applying
the condition of the balance of dimensions in his starting functional relation. And
yet, instead of following this through to produce a general proof, he developed the
limited one just described.
Staicu proposed a general dimensional analysis which claimed to derive more information than the standard application of the pi-theorem [14]. It has been adversely
criticised elsewhere [15] because of the severe limitations of the assumptions made.
For examples, only one pi group can be involved, the exponents of the variables are
“in some sense minimal” and the sign of all the exponents are all known a priori.
The present generalised and rigorous proof of the pi-theorem was signposted by
Ipsen in 1960 [16]. He used the procedure of sequential cancellation of the dimensions and was clear on the retention of a functional relation but he did not make the
basic statement that he had a new proof of the pi-theorem just describing the Buckingham demonstration as something different. He neither generalised his proof nor
set out the logical steps required as is done here in Chapter 2. His statement that
a variable, being the sole one having one particular dimension, cannot appear in
the functional relation is not as clear as the present one that it must cancel out. All
this may explain why Ipsen’s early valuable work has generally been overlooked by
later writers. For example, both Roberson and Crowe [17] and Barr [18] also used
the cancellation of dimensions technique but seemingly only as a means of obtaining the non-dimensional groups directly from a list of the variables, for they say that
it is used without resort to the pi-theorem. Taylor similarly uses the method also
only as a means of obtaining non-dimensional groups noting that the technique always results in the correct number of groups of the correct composition [10] but not
observing that it does so because it is an intrinsic part of the present demonstration
of the pi-theorem.
So again, these writers were only assembling variables into non-dimensional
groups without reference to the retention of a functional relation.
Taylor noted that the current technique avoids the difficulty of some other methods in the prior determination of the number of independent dimensions [10, p. 29]
72
3 The Pi-Theorem
and of the permissible dimensions [10, p. 349, lines 1–4]. Existing methods have,
for some examples such as those just given, raised practical difficulties in arriving
at the correct number of non-dimensional groups, and so have led to means for determining these groups such as that described by Van Driest [19]. But, as Taylor has
pointed out [10, p. 29], such discussions do not always lead to a determination that
is straightforward. Even Buckingham, for his example of Section 3.4.4, in order to
achieve a solution chose unusual combinations of dimensions to form his effective
ones but without explanation. Because of the intrinsic link between the present proof
and the operation of the pi-theorem, the latter always gives the correct number of
non-dimensional groups each of a correct form meanwhile automatically accounting
also for a variation in the number of variables from group to group.
From the pioneering idea by Ipsen, the present generalised and rigorous proof
overcomes these outstanding problems in dimensional analysis.
3.6 The Careful Choice of Variables
A further application of the pi-theorem is illustrated by the case of the steady flow
of a liquid along a smooth circular pipe. This is illustrated in Figure 3.4.
The momentum equation that governs this steady flow is:
u
@u
@u
@u
1 @p
Cv
Cw
D
C X C r2u ;
@x
@y
@z
@x
where:
r2 Figure 3.4 The steady incompressible flow in a pipe
@2
@2
@2
C 2 C 2
2
@x
@y
@z
3.6 The Careful Choice of Variables
73
Compact Solution 3.5
K
1
2
3
k
p
P
Q
L
d
g
L3
L
L
L
T2
M
LT2
M
L3
M
LT
p
L2
T2
L2
T
p
P2
Q
1
L4
P
Q
1
L
pd 4
P2
Q
d
P
Q
L
d
gd 5
P2
Q
1
1
1
1
3
3
T
g
P2
Q
1
L5
1
2
with similar equations for the other two direction components [20]. These equations
show that the pressure appears only as a difference so that the relevant variable is
p.
The pressure force in upward motion is against the weight so that further variables are and g. Also the pressure force operates against the surface friction so
that a further variable is . The pressure and friction forces vary with the flow rate,
QP and with the sizes of d and L.
Then,
P L; d; g/ :
p D f .; ; Q;
The solution is formed in Compact Solution 3.5.
From this solution there is:
pd 4
d L gd 5
Df
; ;
:
QP 2
QP d QP 2
(3.22)
(3.23)
Special cases can now be introduced into the general result of Equation 3.23.
First, if the axis of the pipe is horizontal then the gravity force is not affecting the
horizontal flow though the pressure does vary vertically according to the hydrostatic
relation. Therefore the variable g can be excluded so that Equation 3.23 is reduced
to:3
pd 4
d L
Df
;
:
(3.24)
QP 2
QP d
The simple combination of p with g as is done in elementary hydraulics has to be used with
care in a turbulent flow [20, 21].
3
74
3 The Pi-Theorem
Secondly, at positions well downstream of the pipe entry the flow pattern of velocity is unchanging along the pipe so that the wall friction is constant along the
length. Then from momentum considerations [2], a pressure difference is proportional to the associated local length [20]. This variable can then be replaced by the
pressure gradient so that:
p
@p
D
L
@L
and so Equation 3.24 is replaced by:
@p d 5
Df
@L QP 2
d
QP
:
(3.25)
This relation is correct for both laminar flow and turbulent flow. Thirdly, when
the flow is purely laminar, then downstream there are no accelerations of the flow
element, the flow being controlled by just viscosity. This means that the density does
not enter into the governing equations and so Equation 3.25 reduces to:
@p d 5 QP
@p d 5
D
D constant :
@L QP 2 d
@L QP
(3.26)
This equation is seen to be valid only as a specially restricted case of Equation 3.23. It illustrates the care to be taken in a choice of variables. Also, the result
has been obtained for a specified circular cross section. For a different shape, a variable, S , could be added. This could be a non-dimensional shape parameter.4
3.7 The Necessity for a Units-conversion Factor for Angle
The need for the units-conversion factor for angle, that was introduced in Section 2.15, can now be demonstrated. It is shown in Taylor’s example of the loading
of a structural frame as illustrated in Figure 3.5 [10]. Taylor put the deflection, ı,
as a function of the load, W , upon a triangular frame whose shape is controlled by
two angles of the triangle, ˇ and , and whose size is measured by the length of
one side, `. The further variables are the cross-sectional area of the members, A,
and Young’s modulus, E. Then assigning a dimension to angle, tabulation for the
pi-theorem is as in Compact Solution 3.6.
This gives that:
ı
ˇ W A
Df
; 2 ; 2 :
(3.27)
`
` E `
4
There is a long-standing habit amongst scientific authors of using the word ‘geometry’ when
they mean ‘shape’; surely both Fowler and Gowers would not have approved as ‘geometry’ is the
science of ‘shape’ as is defined in the Concise Oxford Dictionary.
3.7 The Necessity for a Units-conversion Factor for Angle
75
Compact Solution 3.6
K
ı
W
ˇ
`
A
E
L
ML
T2
˛
˛
L
L2
M
LT2
W
E
1
L2
2
ı
`
W
Eı 2
1
1
3
k
1
2
–
A
`2
1
ˇ
1
1
1
This is the result that Taylor obtained from the pi-theorem and which, as he
pointed out, is incorrect. For if ˇ and are each changed by the same factor then
their ratio is unchanged; but clearly this change in the angles would change the shape
of the frame and hence the deflexion. By introducing the units-conversion factor, ˇ0 ,
the extra group ˇ0 ˇ necessary to resolve Taylor’s difficulty is obtained; for now, the
solution comes from Compact Solution 3.7.
This gives the result that:
ı
ˇ W A
D f ˇ0 ˇ; ; 2 ; 2
(3.28)
`
` E `
A change in the angles is now reflected by a change in ˇ0 ˇ and hence in the
deflexion.
It is now seen that the dimension and units of angle, with its units-conversion factor, is required in dimensional analysis so that it is not to be regarded as subservient
in the SI system of units.
In this example the effective dimensions are:
M=T2 ; ˛ and L ;
that is, three in number.
Figure 3.5 The deflexion of
a structure under load
76
3 The Pi-Theorem
3.8 General Results from the Pi-theorem
There are some general results given by the pi-theorem for the formation of nondimensional groups.
The results from inspection of the operation of the pi-theorem together with the
examples are now listed in Table 3.3.
From the foregoing general demonstration of the pi-theorem and the illustrative
examples just given, general conclusions are:
(i)
(ii)
(iii)
(iv)
G D N K;
m D k C 1;
K n;
k K.
In Table 3.3, the last three examples are cases where the effective number of
dimensions is less than the original tabulated number. This is indicated by K < n,
which is condition (iii) above, for each of these cases. More similar examples are
given as Exercises 3.4–3.7 and 3.12 and 3.17. The above four conditions form some
general rules for the formation of non-dimensional groups to satisfy the pi-theorem.
Compact Solution 3.7
K
ı
L
1
2
W
ˇ
ML
T2
˛
˛
A
E
ˇ0
L
L2
M
LT2
1
˛
W
E
L2
–
ı
`
W
Eı 2
A
`2
1
1
–
1
–
3
k
`
1
2
ˇ0 ˇ
1
ˇ0 1
1
1
–
–
–
–
1
Table 3.3
Example
G
k
K
m
n
N
Linear spring
Non-linear spring
Pipe flow
Vertical jet
Field energy
Structure
Heat exchanger
1
2
4
3
2
5
4
2
2
1, 2, 3
3
2, 3
1, 2
1
2
3
3
3
3
3
2
3
3
2, 3, 4
4
3, 4
2, 3
2
2
3
3
3
4
4
4
3
5
7
6
5
8
6
–
–
3.9 Summarising Comments
77
3.9 Summarising Comments
The final stage in the full logic of dimensional analysis has been considered in this
chapter. It is completed by a proof of the pi-theorem that is both general and rigorous. Operation of it thus gives non-dimensional groups that are automatically of the
correct number, each being of the correct size and correct composition.
The various examples given bring out the problems that have exercised writers
in the past particularly when the effective dimensions are different from those originally set out: this has been resolved here by straightforward procedures in operating
the present demonstration of the pi-theorem.
An illustration of the care to be taken in the choice of units-conversion factors
comes from the demonstration that has been given of the case for inclusion of the
dimension of angle to be placed alongside the other standard dimensions.
Thus one is with Bridgman who wrote; “– and we are for the present secure in our
point of view which sees nothing mystical or esoteric in dimensional analysis.” [12].
Exercises
3.1
Noting the three effective dimensions of Equation 3.14, write them as:
X D M=L3 I
3.2
3.3
Y D T=LI
Then re-do that problem of Equation 3.12 but now using these dimensions
of X , Y , and Z to show that the answer is still given by Equation 3.13.
An aerofoil; is in forced pitching oscillation within the flow of a low-speed
air-stream. The mean value of the oscillating force on the aerofoil, FN , is
a function of the oscillation frequency, !, the viscosity, , the density, , the
stream velocity, V , and the scale size, `. Show that:
FN
!` pV `
D
f
;
:
pV 2 `2
V
The emitted energy of a black body per unit area, per unit time, E, is a function of the temperature, T , the velocity of light, c, the Planck constant, hP ,
and the Boltzmann constant, kB . Show that:
Eh3p c 2
kB4 T 4
3.4
Z D A=L
D constant ;
which is Stefan’s law.
In a liquid dielectric the Zeta potential, , across the Helmholtz double-layer
of charges along an electrode, is a function of the charge per unit area, , the
thickness of the double-layer, d , and the dielectric constant of the liquid, ".
78
3 The Pi-Theorem
Figure 3.6 The end-heated
rod in the atmosphere
Show that:
& "
D constant :
d
3.5
3.6
What are the effective dimensions in this derivation?
Figure 3.6 shows a circular rod in an atmosphere at a temperature of Ta
that is attached to a wall which applies a temperature of T0 so that a heat
rate of QP is applied. The end temperature is Te . Inspection of both the governing equation for heat transfer along the rod and that to the atmosphere,
shows that only temperature differences occur. Thus .Te Ta / is a function
of .T0 Ta /, the thermal conductivity of the rod, k, the heat transfer coefficient to the air, h, the rod diameter, d and the rod length, L. Show that:
Te Ta
k L
Df
;
:
T0 Ta
hd d
What are the effective dimensions?
The chemical constant, k, has the dimensions of M= LT2 5=2 . It is a function of the Boltzmann constant, kB , the Planck constant, hP , and the molecular mass, m. Show that:
kh3P
m3=2 kB5=2
3.7
D constant :
What are the effective dimensions?
Figure 3.7 shows a parallel plate condenser which exerts a force fe between
the plates. This is a function of the charge per unit area on each plate, q, the
dielectric coefficient of the intervening space, ", the area of each plate, A,
and the spacing between the plates, z. Show that:
2
fe "
z
D
f
:
2
q A
A
What are the effective dimensions?
There are edge effects upon the plates so that q is not uniform there. If the
plate spacing is sufficiently small these edge effects become negligible. For
3.9 Summarising Comments
3.8
79
this case simplify the above result. Hence show that then the force is independent of the plate spacing.
The entropy per unit area, of the event horizon of an astronomical black hole,
s, is a function of the Planck constant, hP , the Boltzmann constant, kB , the
universal gravitational constant, G0 and the speed of light, c. Show that:
shP G0
D constant :
kB c 3
3.9
The radius of the event horizon of an astronomical black hole, r, is a function of the mass, m, the velocity of light, c, and the universal gravitational
constant, G0 . Show that:
rc 2
D constant :
G0 m
3.10
An object of mass, m, is suspended vertically on a spring and is oscillating
vertically at a frequency, !, with an amplitude of a. Show that the mean
kinetic energy of the object, E, is given by:
E
D constant :
m! 2 a2
3.11
3.12
The armature of a DC motor has a diameter, d , a length, `, and carries a current i . It rotates at an angular velocity of ! in a magnetic field of flux density,
B. Show that the output power, P , is given by:
P
`
D
f
:
2
iB!d
d
Ignoring end effects of the magnetic field at the ends of the armature, reduce
the above result to one non-dimensional group.
A fine tube is inserted vertically into a still liquid so that the meniscus in
the tube rises up by the height, h. With this height being a function of
the weight per unit volume of the liquid, w, the surface tension, , and the
Figure 3.7 The parallel plate
condenser
80
3 The Pi-Theorem
diameter of the tube, d , using the MLT system of units show that:
h
wd 2
Df
:
d
3.13
3.14
What are the effective dimensions? Repeat the analysis using the FLT dimensional system where F is the unit of force. Compare the two methods of
solution.
Fans and windmills produce high sound levels because the fundamental
operating mechanism is to produce a fluctuating pressure field. The sound
power, as energy per unit time, P , is a function of the rotational speed, !,
the diameter, d , the gas density, , and the speed of sound, a. Show that:
P
!d
D
f
:
! 3 d 5
a
The non-dimensional group on the right-hand side of this equation is a measure of the Mach number at the fan tips. If this is limited to a specified subsonic value then show that the acoustic power is proportional to the square
of the diameter.
An electron of mass, m, and charge, e, passes across a magnetic field of
magnetic flux density, B, at a velocity, v, so that its path has a radius of
curvature, r. Show that:
Ber
D constant :
mv
3.15
The effect of an atomic explosion initially is from the rapid expansion of an
extremely strong spherical shock wave. The radius of curvature of this shock
wave, r, is a function of the explosion energy, E, the initial air density, 0 ,
and the time, t. Show that:
Et 2
D constant :
0 r 5
3.16
The stability of a co-axial spray generated by an inner co-axial jet inside
a co-axial electrode, is controlled by the ratio of the electric relaxation time
to the hydrodynamic capillarity time. with the notation of:
r
tc
te
"
jet radius
capillarity time
electrical relaxation time
dielectric coefficient
electrical conductivity
liquid density
surface tension coefficient
References
81
and with:
tc D f Œ; ; r ;
te D f Œ"; show that:
te
" 1=2
/
:
tc
r 3
3.17
A liquid electrolyte is contained in a conductivity cell. This is rectangular in
shape with electrodes of area A on opposite facing sides spaced apart a distance `. An electric field is applied across the electrodes. Above a critical
value of this field, Ec , the liquid is set in motion under electro-hydrodynamic
forces. With the notation of:
A
Ec
kB T
`
"
0
Electrode area
Critical field
Thermal energy
Electrode spacing
Dielectric coefficient
Zero-charge conductivity
and then:
Ec D f .A; kB T; `; "; 0 /
3.18
show that Ec is not a function of 0 . In this application of the pi-theorem
what are the effective dimensions?
A pipette is used to issue the liquid slowly, a drop at a time, from the sharp
nozzle outlet of diameter, d . The drop size is then controlled by the surface
tension of the liquid, . Show that the weight of each drop, W is given by:
W
D constant :
d
References
1. J C Gibbings. A logic of dimensional analysis, J. Phys. A: Math. Gen., Vol. 15, pp. 1991–2002,
1982.
2. J C Gibbings. Thermomechanics, Pergamon, Oxford, 1970.
3. J H Keenan. Thermodynamics, Wiley, New York, 1957.
4. E Buckingham. On physically similar systems: illustrations of the use of dimensional equations, Phys. Rev., Vol. 4, Pt. 4, pp. 345–376, 1914.
5. J W S (Lord) Rayleigh. Review: Professor Tait’s ‘Properties of matter’, Nature, Vol. 32, p. 314,
6th August 1885.
6. G M Focken. Dimensional methods and their applications, Edward Arnold, London, 1953.
82
3 The Pi-Theorem
7. B S Massey. Units, dimensional analysis and physical similarity, Van Nostrand Reinhold,
London, 1971.
8. G Birkhoff. Hydrodynamics, Princeton University Press, Princeton, 1960.
9. T B Benjamin. Note on formulas for the drag of a sphere, Jour. Fluid Mech., Vol. 246, pp. 335–
342, January 1993.
10. E S Taylor. Dimensional analysis for engineers, Clarendon Press, Oxford, 1974.
11. H L Langhaar. Dimensional analysis and theory of models, Wiley, New York, 1951.
12. P W Bridgman. Dimensional analysis, (Rev. Ed.), Yale, New Haven, 1943.
13. R Esnault-Pelterie. Dimensional analysis, (English edn.), Rouge, Lausanne, 1950.
14. C I Staicu. General dimensional analysis, J. Franklin Inst., Vol. 292, Pt. 6, pp. 433–439, 1971.
15. E de St Q Isaacson, M de St Q Isaacson. Dimensional methods in engineering and physics,
Edward Arnold, London, 1975.
16. D C Ipsen. Units, dimensions, and dimensionless numbers, McGraw-Hill, New York, 1960.
17. J A Roberson, CT Crowe. Engineering fluid mechanics, Houghton Mifflen, Boston, 1975.
18. D I H Barr. The proportionalities method of dimensional analysis, J. Franklin Inst., Vol. 292,
No. 6, pp. 441–449, December 1971.
19. E R Van Driest. On dimensional analysis and the presentation of data in fluid-flow problems,
ASME, J. Appl. Mech., Vol. 68, pp. 34–40, 1946.
20. S Goldstein (Ed.). Modern developments in fluid dynamics, Vol. 1, Chap. 3, § 35, Oxford,
1938; Dover, New York, 1965.
21. P H Sabersky, A J Acosta. Fluid flow, MacMillan, New York, 1964.
Chapter 4
The Development of Dimensional Analysis
There is a yearly battle going on for students’ minds; history
might help to convince them that the use of the Reynolds number
as an independent variable is an application of a basic truth,
and not just a useful convention for a handy diagram.
N. Rott
Notation
b
d
K
`
n
Q
Re
Rf
V
Plate breadth
Pipe diameter
Resistance coefficient
Plate length
Resistance exponent
Volume flow rate
Reynolds number
Resistance force
Velocity
Viscosity
Fluid density
Surface-friction shear
4.1 The Case for the History of Dimensional Analysis
The history of science and engineering often can reveal useful lessons for improving
current practice. For example, the study of the steady decline in British engineering
industry from the Great Exhibition of 1851 onwards [1] still has not always been
acted upon in many instances up to this day.
As Rott has written, “ – the peculiar history of the ideas on hydrodynamic similarity in the epoch around 1900. The ideas were initiated by the great men around
the turn of the century, but were originally met with indifference, for an unbelievable
long period.” [2]. This history, then, illustrates that remarkable tardiness in adopting
the use of dimensional analysis which continues in some branches of science and
engineering.
J.C. Gibbings, Dimensional Analysis. © Springer 2011
83
84
4 The Development of Dimensional Analysis
4.2 The Onset of Similitude
The idea of similar systems long preceded the development of dimensional analysis. Pomeranz goes so far as to propose relevant contributions by Ptolomy and
Galileo [3], whilst Szucs adds the name of Kepler [4]. However most of the concern
of these early workers was with the dimensions of measurement [5].
Long before this, Euclid in his studies of geometry had dealt in detail with the
idea of similarity in the shape of geometrical figures, variation being in size only so
that relative representation could be by the value of a single ratio of lengths. Yet, as
seen in Exercise 4.1, Pythagoras could have benefited from an elementary knowledge of dimensional analysis in deriving his famous theorem, which Bronowski has
described in the words of; “To this day, the theorem of Pythagoras remains the most
important single theorem in the whole of mathematics.” [6].
Newton took all these ideas much further in his famous Principia [7]. He considered similarities in the trajectories of bodies in motion under his laws of motion,
going on to derive the similarity between accelerations, momenta and forces. He
thus introduced the ideas of similarity between kinematic and force systems. This
part of Newton’s work was not acted upon and so rather lapsed from use until taken
up by Thomson in the late 19th century who then applied it to problems in hydrodynamics and structures [8]. In the meantime earlier in that century, the idea of
similarity of biological systems was put to extensive use by D’Arcy Thompson in
his studies of animal structure and locomotion [9, 10].
There was a long lapse of some fifty years after Newton’s Principia before Euler,
in 1736, made the next step. He effectively introduced a units-conversion factor into
Newton’s law of motion. But again, as Macagno says; “It is remarkable that Euler’s
ideas on dimensions and homogeneity did not have repercussions, – ” [5].
4.3 The Onset of Dimensional Analysis
Almost a whole further century passed before the first real stage in dimensional
analysis was created by Fourier [11]. He made the basic point that an equation that
represents a real physical event must have the same dimensions for all its additive
terms. Correspondingly, quantities on both sides of an equation had to have the
same dimensions. He then went on to point out that these forms of equation remain
unchanged when a change is made in the size of the unit measure. He used this
principle as a means of checking the validity of an equation. But he did not extend
this finding further to derive results.
An early signpost to dimensional analysis was set up by Stokes. In 1823 Navier
had set out the governing equations for a viscous flow and Stokes produced a further derivation in 1845 [12]. From this, in 1856 Stokes applied these equations to
the problem of viscous friction from the air upon the swing of a pendulum. He
showed that there was a single independent non-dimensional group controlling this
phenomenon; this group we now call the Reynolds number [13].
4.4 The Developing Use of the Pi-theorem
85
Then in 1871 Maxwell introduced the symbolism, that is used to this day, to
denote dimensions of a quantity. That is that of M for the dimension of mass, L for
length and T for time [14].
It was a further half a century after Fourier’s lead before the use of dimensional
analysis really got started. The great breakthrough was made by Lord Rayleigh who
in 1871 used the ideas of similitude in a study of the colour of the sky [15]. He later
advocated dimensional analysis in 1885 [16] and though he usually referred to it as
a method of similitude he also used the description of ’method of dimensions’ [17,
18]. Lord Rayleigh’s extensive early contributions are described in detail by Rott [2].
4.4 The Developing Use of the Pi-theorem
The first attempt to give a general statement of what is now called the pi-theorem
was made by Vaschy in 1892 [19]. He followed this by a more general statement
in 1896 [20]. Yet this work was seemingly ignored as described by Macagno [5]
who pointed out that as a consequence Riabouchinsky rediscovered this theorem
independently in 1911 [21]. His assumption of a series solution would seem to have
been influenced by Rayleigh’s work. Even then Buckingham, who produced his
proof of the pi-theorem in 1914 [22], did not acknowledge Riabouchinsky’s priority
until 1921 [23].
There still seemed to be some reluctance by others to follow Rayleigh’s lead.
A famous paper was written by Reynolds in 1883 [24]. He had performed experiments on the flow through pipes using a visualisation method to determine a criterion that distinguished whether the flow was a steady laminar one or an unsteady
turbulent one.1 This work was and remains of considerable practical importance because the friction loss in the flow greatly increases in going from a laminar flow to
a turbulent one. [25].
In either of these types of flow we have the relation of:
D f Œ; Q; ; d :
The pi-theorem solution is set out in Compact Solution 4.1.
This gives the result that
d 4
Q
D
f
D f .Re / :
Q 2
d
(4.1)
(4.2)
This shows that, when the friction jumps in value in going from laminar flow to
a turbulent one, then the criterion dividing these two regimes is given by the corresponding value of the Reynolds number. This was Reynolds most significant dis1
Because Reynolds’ pipe-flow experiment was such a seminal contribution to the early development of dimensional analysis, it is important that this flow is described correctly. This is done in
Appendix 4.1
86
4 The Development of Dimensional Analysis
Compact Solution 4.1
Q
d
M
LT2
M
L3
L3
T
M
LT
L
L2
T2
L2
T
Q2
Q
1
L4
1
L
d 4
Q2
d
Q
1
1
covery; it was a finding that could only be revealed by dimensional analysis both
then and to this day.
There is another important parameter which is the presence of disturbances in the
form of residual turbulence in the incoming flow entering a pipe. Reynolds recognised this as well but found that with a high degree of such disturbance the Reynolds
number fell to a minimum value of about 2000. This priority of dimensional analysis
was acknowledged by Foppl in 1910 [26].
Prior to this, Sommerfeld had, in 1908, given the name of Reynolds number to
this non-dimensional group not having taken account of Stokes original observation
of some three decades earlier as mentioned above [27].
Dimensional analysis was first used extensively in aerodynamics [28]. Following
Reynolds’ work, Rayleigh made a clear application of dimensional analysis to aerodynamics when he used it in 1904 to correlate values, measured by Zahm, of the
drag of a flat plate at zero incidence [29]. Before this the forces of both lift and drag
on aerofoils were given in the literature as forces per unit wing area. This practice
went back to at least Wenham who carried out the very first model tests in his wind
tunnel in 1871 [30].
The first attempt to evaluate a Reynolds number scale effect between model tests
and full-scale was made by Wilbur and Orville Wright who tethered one of their
full-scale man-carrying gliders in a steady wind and compared the measured forces
on it with those that they had obtained in their wind tunnel [31]. This provided them
with a form of correction factor which they applied in their calculations with their
considerable success [32].
Rayleigh brought his influence to bear in two ways. First, he was regarded in his
day as a pre-eminent scientist, certainly in Great Britain. Secondly, he was the first
chairman of the Advisory Committee for Aeronautics. This enabled him to bring
the application of dimensional analysis to aerodynamics to the attention of the staff
of the Aeronautical Department of the National Physical Laboratory at Teddington.
Lanchester was aware of the application of dimensional analysis at least by 1909 for
in the Committee’s Report and Memoranda No. 1 it is made clear that at 12th May
4.4 The Developing Use of the Pi-theorem
87
1909 Prandtl was not properly using this analysis for the correlation of the drag of
airships [33]. He was using Froude’s law and it was pointed out in a footnote by
Lanchester that this law was not applicable.
Yet in his lengthy paper of 1910 Prandtl gave an equation for the measured frictional resistance on a plane surface of [34]:
Rf D Kb`nV nC1 :
Figure 4.1 Reproduction of Fig. 1 of ACA R&M No. 15. (See [38])
(4.3)
88
4 The Development of Dimensional Analysis
Here Rf is the frictional resistance, K is a coefficient, b is the breadth and ` is the
length of the surface, V is the velocity and n is an exponent which for smooth surfaces was quoted as n D 0:80 0:85. What is now intriguing is that this expression
Figure 4.2 Reproduction of Fig. 2 of ACA R&M No. 38. (See [35])
4.4 The Developing Use of the Pi-theorem
89
can be readily rewritten as:
kf Rf
/ Ren1
b`V 2
which is the standard non-dimensional form.
Figure 4.3 Reproduction of Fig. 6 of ACA R&M No. 40. (See [37])
(4.4)
90
4 The Development of Dimensional Analysis
Doubts still existed about the validity of dimensional analysis to aerodynamics.
The evidence indicates that these were finally resolved in 1911 by Bairstow and
Booth in R&M No. 38 [35]; by Rayleigh in R&M No. 39 [36]; and in R&M No. 40
by Melville Jones [37].
The source of these doubts can be illustrated by the data for the drag of flat plates
normal to the flow which had been correlated on the basis of size alone. This result
is shown in Figure 4.1 by the original figure from R&M No.15 [38]. Then this was
compared with the less successful correlation in terms of the Reynolds number as
shown in Figure 4.2 [35]. Rayleigh obviously felt the need for an apologia which
was issued in R&M No. 39 [36]. This was followed by Melville-Jones’ successful
correlation for the drag of smooth wires given in R&M No. 40, Pt.1 [37]. His original figure is shown as Figure 4.3. It seems quite likely that this latter figure was
the means of finally convincing those working on aerodynamics of the validity of
dimensional analysis to their studies [28].
From then on dimensional analysis was gradually developed. The advances in
developing the final general proof of the pi-theorem are described in Chapter 3. The
first attempt to set out a logic of the full analysis was by Bridgman in his classic and
pioneering book of 1922 [39]. This was taken further to give a full logic sixty years
later [40].
4.5 The Place of Dimensional Analysis
The place of dimensional analysis in science has been put by Rott who described this
subject; “ – as a (or as ’the’) discipline of science.” [2]. Pomerantz, in his introduction to the Franklin conference, wrote of this subject as “ – an intellectual technique
– ranging from the simplest – on to the complex and esoteric methodology – ” [3].
But what a salutary lesson from this history. It took almost one hundred years
from Fourier for it to be widely adopted and then only in aerodynamics. Yet, as
explained in Chapter 3, further developments in the theory continued for over half
a century. Yet still and so often, the analysis is not used by the physicist, the physicalchemist and the electrical engineer in cases where it could be of considerable benefit.
Appendix 4.1 The Reynolds Pipe-Flow Experiment
Reynolds’ experiment was upon the flow of water through a glass pipe. This flow
was visualised by means of a filament of dyed water introduced into the pipe entry.
A diagram of the behaviour of this filament in many texts gives a misinterpretation
of the onset of flow turbulence.
The present writer repeated this experiment in his department. The flow was
from a tank into an analytically profiled entry cone set flush into a side wall of the
tank [41]. This avoided the sharp inlet edge to Reynolds’ nozzle with its possibility
Appendix 4.1 The Reynolds Pipe-Flow Experiment
91
Figure 4.4 (a) Sketch showing tank vorticity. (b) Sketch
showing laminar flow.
(c) Sketch showing fluctuating laminar flow. (d) Sketch
showing turbulent flow
of localised flow separation which could introduce upstream disturbances. The glass
tube was specially drawn to have a bore of precision diameter and straightness and
the metal inlet was machined to match this glass bore quite accurately. The outlet
from the nozzle for the stream of dye was located in the position of high acceleration
of the main flow at entry to the main-flow nozzle so that the dye flow was laminar
there.
As found by Reynolds, the water in the tank had to stand for many hours at
least to allow all the vorticity from the filling of the tank to die out. If this was not
done, a false flow pattern was obtained, the dye line having a waviness in it which
travelled with flow whilst retaining its distinct outline. Typically, the picture was as
sketched in Figure 4.4(a). Clearly the vorticity in the tank remained for this long
time. Additionally, the opening of the dye inlet tube had to be operated with care to
avoid a significant input disturbance to the flow.
Initially, as the flow was speeded up from a laminar one, the dye line sometimes
developed from a straight line form, as shown in Figure 4.4(b) to one with a regular
waviness, each wave travelling downstream with the flow; typically as sketched in
Figure 4.4(c). But the dye filament remained quite distinct. This was an indication of
a fluctuating laminar flow. This is the diagram that appears in many texts but it was
not the turbulent pattern. That it was not a rotational flow was further confirmed by
a bang on the side of the tank, when the filament was straight, resulting in a single
wave form of the dye travelling along the flow as in Figure 4.4(c). Such an impulse
would, by potential flow theory, have set up a potential flow disturbance and not any
vorticity ( [25], Section 119).
As the velocity was increased, eventually the distinct dye line was quite suddenly diffused into a cloud of dye across the pipe flow with no discernable and
distinct pattern as sketched in Figure 4.4(d). This was described by Piercy as “it
92
4 The Development of Dimensional Analysis
presented a frosted appearance of the outlet jet” and was the true picture of turbulence. It seemed to originate quite suddenly in the pipe inlet region. Further, when
the velocity was decreased, this dispersion of dye remained to be swept downstream
by the flow. The vorticity did not dissipate within the flow but by Kelvin’s theorem
was transported with it [25].
It was not found possible to move the transition front up and downstream by
variation of the flow velocity suggesting that the onset of turbulence was controlled
by the initial development of the pipe flow at entry coming from the wall boundary
layers. Prandtl described vortices entering at inlet which break up into separate ones,
saying that; “Apparently events of this kind are responsible in most cases for the
production of turbulence in straight pipes with well rounded mouths. The initial
vortices whose existence has been postulated owe their origin to disturbances which
occur before the fluid enters the pipe.” [42].
Exercises
4.1
4.2
Using dimensional analysis, prove the famous result of Pythagoras concerning
a right-angle triangle. (Hint; from the point of the right angle, drop a perpendicular on to the horizontal hypotenuse.)
If for a floating body, the volume displaced is V , the density of water is ,
and the acceleration due to gravity is g, show by dimensional analysis that
the buoyancy force is proportional to the weight of water displaced. Consider
then how a single experiment could derive Archimedes principle.
References
1. D S L Cardwell. The organisation of science in England, Heinemann, London, (Rev. Ed.),
1972.
2. N Rott. Lord Rayleigh and hydrodynamic similarity, Phys. Fluids A Vol. 4, Pt. 12, pp. 2595–
2600. December 1992.
3. M A Pomerantz. Forward, Jour. Franklin Inst., Vol. 292, No. 6, p.389, December 1971.
4. E Szucs. Similitude and modelling, pp. 32–33, Elsevier, Amsterdam, 1980.
5. E O Macagno. Historico-critical review of dimensional analysis. J. Franklin Inst., Modern
dimensional analysis, similitude and similarity, Vol. 292, No. 6, pp. 391–402, Pergamon Press,
1971.
6. J Bronowski. The Ascent of man, British Broadcasting Corporation, London, 1973.
7. I Newton Philosophiae naturalis principia mathematica, Lib. ii, Prop. 32, 1686. (Trans. U
Calif. Press, 1962).
8. J Thomson. Collected papers in physics and engineering, Cambridge Univ. Press, 1912.
9. D’Arcy W Thompson. On growth and form, Cambridge, 1948.
10. D’Arcy W Thompson. The principle of similitude, Nature, Vol. 92, p. 202, 22nd April 1915.
11. J B J Fourier. Theorie Analytique de la Chaleur, Vol. 2, Sec. 7, Ch. 2, Sec. 9, pp. 135–140,
Firman Didot, Paris 1822.
12. S Goldstein. Modern developments in fluid dynamics, Vol. 1, pp. 95–97, Dover, New York,
1965 (1938).
13. G G Stokes. On the effect of the internal friction of fluids on the motion of pendulums, Trans.
Camb. Philos. Soc., Vol. 9, Pt. 2, No. 10, pp. 8–106, 1856 (Read 9th December 1850).
References
93
14. J C Maxwell. On the mathematical classification of physical quantities, Proc. Lond. Math.
Soc., Vol. 3, No. 34, p. 224, March 1871.
15. Lord Rayleigh. On the light of the sky, its polarization and colour, Philos. Mag., Vol. 41 Pt.4,
pp. 107, 274. 1871.
16. Lord Rayleigh. Review: Professor Tait’s ’Properties of matter’, Nature, Vol. 32, p. 314, 6th
August 1885.
17. Lord Rayleigh. On the question of the stability of the flow of fluids, Philos. Mag., Vol. 34,
pp. 59–70, 1892.
18. Lord Rayleigh. On the viscosity of argon as affected by temperature, Proc. R. Soc., Vol. 66,
pp. 68–74, 1900.
19. A Vaschy, Sur les lois de similitude en physique, Annales Telegraphiques, Vol. 19, pp. 25–28,
1892.
20. A Vaschy. Theorie de electrite, Paris, 1896.
21. D Riabouchinsky. Methode des variables de dimensions zero et son application en aerodynamique, L’Aerophile, pp. 407–408, 1911.
22. E Buckingham. On physically similar systems; illustrations of the use of dimensional equations, Phys. Rev., Vol. 4, pp. 345–376, 1914.
23. E. Buckingham. Notes on the method of dimensions, Philos. Mag., Ser. 6, Vol. 42, pp. 696–
719, 1921.
24. O Reynolds. An experimental investigation of the circumstances which determine whether the
motion of water shall be direct or sinuous and the law of resistance in parallel channels, Philos.
Trans. R. Soc., Vol. 174, p. 935, 15th March 1883. (See also Br. Assoc. Rep. 1889).
25. N A V Piercy. Aerodynamics, Sect. 196, English Univ. Press, London, 1937.
26. Technical report, Advisory Committee for Aeronautics 1910–1911, p. 113.
27. N Rott. Note on the history of the Reynolds number, Ann. Rev. Fluid Mech., Vol. 22, pp. 1–11,
1990.
28. J C Gibbings. The use of dimensional analysis in aerodynamics: an historical note, Aer. J.,
Vol. 86, No. 855, pp. 176–178, May 1982.
29. Lord Rayleigh. Fluid friction on even surfaces, Philos. Mag., Vol 8 (6th Ser.), p. 66, July–
December 1904.
30. Annual Report of the Aeronautical Society of Great Britain, Vol. 1, pp. 75–76, London, 1871.
31. M W McFarland (Ed.), The papers of Wilbur and Orville Wright, McGraw-Hill, New York,
1953.
32. J C Gibbings. Achievement of aerial flight: an engineering assessment, Aer. J., Vol. 85,
No. 846, pp. 257–265, July/Aug., 1981.
33. R H Bacon. Report of the Advisory Committee for Aeronautics 1909–1910, R & M No. 1,
p. 132, 12th May 1909.
34. R Giacomelli, E Pistolesi. Historical Sketch, Aerodynamic theory, (Ed. W F Durand), Vol. 1,
Div. D, p. 363, Springer, 1934.
35. L Bairstow, H Booth. The principle of dynamical similarity in reference to the results of
experiments on the resistance of square plates normal to a current of air, ACA Report 1910–
1911, R & M No. 38, 21st March 1911.
36. Lord Rayleigh. The principle of dynamic similarity in reference to the results of experiments
on the resistance of square plates normal to a current of air, ACA Report 1910-1911, R & M
NO. 39, 26th March 1911.
37. B Melville-Jones. The resistance of wires and ropes in a current of air, ACA Report 19101911, R & M No. 40, Pt. 1, p. 40, March 1911.
38. Lord Rayleigh. Note as to the application of dynamical similarity, Report of the Advisory
Committee for Aeronautics, 1909–1910, R & M No. 15, Pt. 2, p. 38, 23rd June 1909.
39. P W Bridgman. Dimensional analysis, Yale Univ. Press, 1922 (2nd Ed. 1931).
40. J C Gibbings. A logic of dimensional analysis, J. Phys. A: Math. Gen. Vol. 15, pp. 1991–2002,
1982.
41. J C Gibbings. The throat profile for contracting ducts containing incompressible irrotational
flows, Int. J. Mech. Sci., Vol. 11, Pt. 3, pp. 293–301, March 1969.
42. L Prandtl. The mechanics of viscous fluids, (W F Durand, Ed.), Aerodynamic theory, Vol. 3,
Div. G, Sec. 2.6, p. 189, Springer, Berlin, 1935.
Chapter 5
The Choice of Dimensions
– the principle known as Occam’s Razor: essentia non sunt
multiplicanda praeter necessitatem (hypotheses are not to be
multiplied without necessity).
R.V. Jones
Notation
a
A
c
c
CV
CH
d
D
E
F
g
g0
G
H
i
I
kB
`
`m
L
m
m
P
M
M0
n
N
p
P
P0
Area, amplitude
Amplitude
Velocity of light
Mean molecular velocity
Coefficient of specific heat; mass based
Coefficient of specific heat; volume based
Tube diameter
Drag
Energy; electromotive force
Force
Gravitational acceleration
Gravitational constant
Number of non-dimensional groups; modulus of rigidity
Magnetic field strength
Electric current
Luminous flux intensity
Boltzmann constant
Length scale, tube length
Mean-free path length
Self inductance
Mass, electron mass; mass per unit length
Mass-flow rate
Dipole moment
Molecular ‘weight’
Number of dimensions
Number of variables; number density
Radiation pressure
Power output
Mechanical equivalent of light
J.C. Gibbings, Dimensional Analysis. © Springer 2011
95
96
5 The Choice of Dimensions
Q
QP
r
R
Re
t
v
z
Heat rate
Quantity flow rate
Distance
Gas constant
Reynolds number
Time
Velocity
Distance
p
@p=@x
ˇ0
!
0
L
S
'
!
Pressure drop
Pressure gradient
Units-conversion factor for angle
Angle
Thermal conductivity
Frequency
Frequency
Viscosity
Magnetic permeability
Liquid density
Density of solid
Temperature, torque
Potential difference
Frequency.
5.1 Care in Choosing Dimensions
It has already been indicated that the use of the dimensions such as M, L and T is by
no means mandatory. For example, in problems in statics the mass dimension might
more conveniently be replaced by the dimension of force. In gravitational problems
the replacement may well be by the weight force.
These changes sometimes require care because replacements such as those can
control the inclusion of units-conversion factors. Such changes are not always
straightforward so that examples are now given where care is required.
5.2 The Number of Non-dimensional Groups
It has been shown in Chapter 3 that the number of non-dimensional groups that are
derived from application of the pi-theorem is given by:
G N n:
(5.1)
5.3 Mass and Force Dimensions
97
This suggests that the more dimensions available for the parameters in a problem,
the fewer would be the non-dimensional groups. Such proposals are in danger of
being unduly influenced by a prior knowledge of the solution existing from formal
analysis of the governing equations. Arising from this observation, there have been
several proposals aimed at increasing the number of dimensions in any problem.
These proposals are four-fold in type.
5.3 Mass and Force Dimensions
The first proposal now considered was to use both the dimension of mass and separately that of force in one phenomenon. An example dealt with the fall under gravity
of a very small solid sphere through a liquid. The forces acting are the drag, D, the
weight, W , and a buoyancy force, FB . The event is illustrated in Figure 5.1. This
requires the variables of:
v; `; S ; L ; ; g :
A straight attack on this problem without any further reference to the governing
equations, and using the dimensions of M, L, and T results in the Compact Solution 5.1 of the pi-theorem.
This gives:
2
L `
S
Df
;
(5.2)
g` L
In this equation the first non-dimensional group is a Reynolds number and the second is a Froude number.
In an endeavour to reduce this result to the known analytical solution it has been
proposed that the dimensions should have that of force, F , added. Certainly this
reduces the non-dimensional groups to two in number but the falsity of this approach
is seen by the discussion of Chapter 3 where in Newton’s law of motion, when the
dimensions of force and of mass are separate then there is a need for the introduction
of a units-conversion factor, g0 . So this extra variable occurs and then the solution
Figure 5.1 Illustration of the
slow fall of a small sphere in
a fluid
98
5 The Choice of Dimensions
Compact Solution 5.1
v
`
S
L
g
L
T
L
M
L3
M
L3
M
LT
L
T2
S
L
L
1
L2
T
L g
2
L
1
L
L v`
g`
2
1
1
Compact Solution 5.2
`
S
L
g
g0
L
T
L
M
L3
M
L3
FT
L2
F
M
ML
FT2
g0
gg0
M
LT
L
T2
S
L
g0
L
1
L2
T
g0
L gg0
2
L
1
L
g0
L `
gg0 `
2
1
1
reverts again to one containing three non-dimensional groups. With the variables as:
v D f .`; S ; L ; ; g; g0 /
(5.3)
the result is in Compact Solution 5.2.
This shows that:
L `
Df
g0
2 S
;
g0 g` L
:
(5.4)
Putting g0 D 1 in this equation reduces it to Equation 5.2 as it should.
The problem with the proposal leading to Equation 5.4 arises as follows: for
the very small scale and slow flow, and following the discussion of Chapter 1, the
5.3 Mass and Force Dimensions
99
Compact Solution 5.3
`
gS
gL
L
T
L
F
L3
F
L3
FT
L2
S
L
L g
1
LT
L g
L2
L g`2
1
density, L , is excluded from the expression for the drag, so that both densities are
related to weight forces in the combination (g/. Then Equation 5.3 reduces to;
D f .`; S g; L g; / :
It is now proper to use the dimensions of F, L, T without the need of the unitsconversion factor, g0 , giving the Compact Solution 5.3.
This gives:
S
D
f
(5.5)
2
L g`
L
By recognising from the initial assessment of the physics that no momentum effects
are acting but only weight forces, the three groups of Equation 5.2. are now reduced
to two in number. Also the use of just the dimensions of F, L, and T gives a neater
solution over that of Equation 5.4.
A full inspection of the equations governing this phenomenon shows the way
forward. There are two sets of equations which are found to be uncoupled. One set
is the hydrodynamic one for incompressible flow given in Chapter 1 but now with
the density excluded for the reasons just given. In this set there is for the drag on the
sphere:
D D f .; ; `/ :
(5.6)
As has been seen in Chapter 1, this reduces to:
D
D constant :
`
(5.7)
Then there is the equation of a balance of forces acting on the sphere. There are
two cases. For a sphere of comparatively high density falling at a steady and low
100
5 The Choice of Dimensions
velocity through a gas the drag force is equal to the weight of the sphere, W , as the
buoyancy force from the air is negligible. Because W / gS `3 , now the densities
are required in the independent variables. Then substitution into Equation 5.7 gives:
gS `2
D constant :
(5.8)
When the sphere is falling through a liquid, then the weight is balanced by the
significant buoyancy force from the liquid given by gL `3 , plus the drag force. Substituting into Equation 5.7 gives:
g .S L / `2
D constant :
(5.9)
Equations 5.8 and 5.9 are the full solution.
This is another example where the physics and its governing equations have to
be fully assessed before application of the pi-theorem: but in this case there is the
further important feature of these equations that they are uncoupled, a most useful
character that will recur later.
If the sphere is accelerating after its initial release then a further term arises. This
comes from the inertia effect of the acceleration of the fluid around the sphere which
is called the virtual inertia term [1].
5.4 Mass and Volume Dimensions
In the phenomenon of heating in solids, so that there is no motion and so no application of momentum, then the physics of such events would suggest that the dimensions used should be those of L, T, ˛ and H where H is the dimension of heat.
Figure 5.2. illustrates a case where a semi-infinite solid bar is initially at a uniform
temperature. Then one end is raised by a fixed temperature difference of so that
the temperature along the bar rises with the time, t, resulting in the heat applied to
the end face being a function of time. Putting this input Q as the heat per unit area
Figure 5.2 Illustration of the
end heating of a bar
5.4 Mass and Volume Dimensions
101
Compact Solution 5.4
Q
t
k
CV
H
L2
T
H
LT
L2
T2
tk
CV
k2
H
L
L4
H2
Q
tk
t 2 CV
L
L2
Q
tk
t 2 CV 1
L
L2
Q2 CV
k2 1
then:
Q D f .; t; k; CV / :
(5.10)
The solution is then as in Compact Solution 5.4.
This gives the result that:
Q 2 CV
D constant :
k2
(5.11)
This result presents a serious omission. For it does not contain the time, t. It would
be most surprising if this were to be correct as it would be quite expected that the
total heat would steadily increase as time passes. So now Equation 5.10 has to be
inspected for any fault.
It is noted that, though it does not contain a dimension in M , the coefficient of
specific heat is defined through the First Law of Thermodynamics which is a massbased expression. Here the mass dimension has been excluded in favour of that of
heat. Thus there are physical grounds for defining a coefficient of specific heat based
upon volume. Thus writing this coefficient, CH , replaces Equation 5.10 by,
Q D f .; t; k; CH / :
(5.12)
The Compact Solution 5.5 gives the solution.
This gives the result that:
Q2
D constant :
tkCH 2
(5.13)
102
5 The Choice of Dimensions
Compact Solution 5.5
Q
t
k
CH
H
L2
T
H
LT
H
L3
tk
H
L
Q
tk
CH
tk
L
1
L2
Q2
tkCH
2
Q2
tkCH 2
1
Formal analysis shows that this is the correct result. But herein lies a difficulty.
First it is to be observed that previously existing texts mostly have solved problems
for which formal solution exists. If that had not been so here, resolution of the
discrepancy between Equation 5.11 and Equation 5.13 would have had to rely on an
adequate understanding of the subtle physics.
Expanding on this explanation, it is noted that dimensionally, in the M, L, T system mCV Q ML2 =T2 so that CV L2 =T2 . Alternatively, with Q H
then CV H=M . Thus if CV is chosen then both H and M dimensions are required. By a similar reasoning CH M=LT2 or CH H=L3 . Now if CH is
chosen then either M or H dimensions are required. The conclusion is that in using
the heat dimension in place of the mass one then the coefficient of specific heat must
be based upon a volume and not on a mass. For this example that necessary understanding has been described. Yet it is clear that this physical understanding can be
quite subtle and has relied here not only upon a suitable choice of variables but of
also the appropriate dimensions.
It is noted that this approach of using CH instead of CV was used by Rayleigh
which enabled him to obtain a single group for the problem described later in Chapter 6.
5.5 Temperature and Quantity Dimensions
A similar problem arises in the example of the thermal conductivity of a gas as
described again in the discussion of the kinetic theory of gases in Chapter 8. In that
5.5 Temperature and Quantity Dimensions
103
Compact Solution 5.6
N
`m
c
m
R
ML
T3
1
L3
L
L
T
M
L2
T2 m
L
T3 mR
1
LT
mRc
1
L2
2
`m
mRc
N `3m
1
1
discussion, the expression for the thermal conductivity will be derived as:
D constant :
N kB c`m
(5.14)
That derivation relies on the use of kB as the units-conversion factor because the
variables have the dimension of temperature. Also mass is involved to describe the
motion of the elementary particles. If instead from inspection of item 14 in Table 2.1
it is considered that the gas constant, R, is to be used as the units-conversion factor
for temperature and no account is taken of the mole as a unit, M0 being dimensionless by Equation 2.13, then the solution would go as in Compact Solution 5.6.
This gives that:
`2m
D f N `3m :
mRc
As in the discussion of Chapter 8, putting / N , then:
D constant :
mRcN `m
Now:
Rm D kB M0
104
5 The Choice of Dimensions
so that:
D constant :
M0 kB cN `m
Comparing this equation with Equation 5.14 shows that it is incorrect. This illustrates again a case where care has to be taken in the choice both of the variables and
hence of the dimensions and then of the appropriate units-conversion factors.
5.6 Mass and Quantity Dimensions
The flow through a pipe, as illustrated in Figure 5.3, was discussed in Section 3.6.
P , and p had a dimension in
The variables were chosen carefully and of them, Q,
M. Thus when as in an approximation for laminar flow the density, was excluded
there were still three variables left with a dimension in M so that there was no
problem in cancelling this dimension in application of the pi-theorem.
It is thus acceptable to use the mass flow rate, m
P as the dependent variable. Then:
m
P D f .d; `; p; ; / :
(5.15)
The pi-theorem goes as in Compact Solution 5.7.
This gives that:
m
P
` d 2 p
Df
;
d
d
2
or rewritten as:
m
P
Df
d 3 p
` d 2 p
;
d
2
(5.16)
As in Section 3.6 this solution is the correct one for the full pipe flow and applies to
both laminar and turbulent flow. Again, as in Section 3.6, far downstream from the
entry and for laminar flow in a horizontal pipe the density, , is excluded from the
Figure 5.3 Illustration of the
flow into a circular tube
5.6 Mass and Quantity Dimensions
105
Compact Solution 5.7
m
P
d
`
p
M
T
L
L
M
LT2
M
LT
M
L3
m
P
p
p
p
LT
T
T2
L2
m
P
p
2
L
1
L2
m
P
d
`
d
d 2 p
2
1
1
1
list of variables. It now becomes appropriate to consider a flow rate as a volume rate
and not a mass rate. Thus the variables are:
QP D f .; d; @p=@x/ :
Also it is now appropriate to use the dimension of force rather than to use the mass
dimension. The solution then goes as in Compact Solution 5.8.
This results in the single non-dimensional group of:
P
Q
d 4 .@p=@x/
as obtained in Section 3.6.
Compact Solution 5.8
P
Q
d
@p=@x
L3
T
FT
L2
L
F
L3
@p=@x
1
LT
L
–
–
P
Q
d 4 .@p=@x/
1
–
–
–
P
Q
@p=@x
4
D constant
(5.17)
106
5 The Choice of Dimensions
This is another example where care is required in choosing the variables to accord
with the physics of the phenomenon and hence a suitable choice of dimensions
follows. The association in the above tabulation of the force dimension with only
the terms respectively in the viscosity and the pressure is consistent with the fluid
mechanics of this flow where the pressure and viscous forces are in balance.
5.7 The Angle Dimension
The need to assign a dimension to angle was discussed in Chapter 2 where the
requirement of a corresponding units-conversion factor, ˇ0 was demonstrated. This
need is now repeated for the case of torsion of a cylinder of diameter, d and length,
` under a torque, T .
The modulus of rigidity, G is defined in terms of the angular twist of the cylinder,
, so that its dimensions are M=.LT2 ˛/. Then:
T D f .G; d; `; / :
(5.18)
This involves the four dimensions of M, L, T, ˛. It has been proposed that a desirable
result is obtained by using these four dimensions in Equation 5.18 thus resulting in
just two non-dimensional groups. The error here is that Equation 5.18 in effect puts
the units-conversion factor for angle at unity and so the angle is measured in radians
and then is of zero dimensions. This reduces the number of dimensions to three
and so results in three non-dimensional groups. The way forward is to add the unitsconversion factor to Equation 5.18 so that the solution is as in Compact Solution 5.9.
This gives:
Tˇ0
`
Df
; ˇ0 :
(5.19)
Gd 3
d
Compact Solution 5.9
T
G
d
`
ˇ0
ML2
M
LT2 ˛
L
L
˛
1
˛
G
ˇ0
ˇ0
M
LT2
1
T2
Tˇ0
G
L3
Tˇ0
Gd 3
`
d
1
1
5.8 Electrical Dimensions
107
The effective dimensions are just three in number: that is; M=T 2 , L and ˛ thus
resulting in the three non-dimensional groups. Three groups are still obtained with
ˇ0 D 1. Equation 5.19 is the correct solution and progress can only be made by
introducing two approximations. First, to be compatible with the definition of G the
deflexion is assumed to be purely elastic so that the angular deflexion is proportional
to the torque. Secondly, if there are no significant end effects, which is not always
the case, then there will be a distance uniformity so that the angular deflexion will
be proportional to the length. These two approximations then reduce Equation 5.19
to:
T`
D constant :
Gd 4
(5.20)
Thus Equation 5.20 becomes a special case of Equation 5.19 being then an approximate solution.
5.8 Electrical Dimensions
Taking as an example the force, P , acting between two parallel plates of a condenser
each of area, a, spaced apart a distance, z, and under a potential difference, '. This
is illustrated in Figure 5.4. Then:
P D f ."; '; a; z/ :
Using the M, L, T, A, system of units gives the following result in Compact Solution 5.10, which gives that:
hai
P
Df
2
"'
z2
(5.21)
In this phenomenon there is no mechanical motion and no current flow. Thus it
seems more appropriate to replace the dimensions of M and A by those respectively
of force, F and of charge, Q. Using the dimensions of F, Q and L, gives Compact
Solution 5.11, which again gives Equation 5.21.
Figure 5.4 Sketch of the field lines between the two plates of a condenser
108
5 The Choice of Dimensions
Compact Solution 5.10
P
"
'
a
z
ML
T2
A2 T4
ML3
ML2
AT3
L2
L
"' 2
ML
T2
"' 2
P
1
a
z2
1
Compact Solution 5.11
P
"
F
Q2
'
a
z
L
Fl2
FL
Q
L2
"' 2
F
P
"' 2
1
a
z2
1
This is the correct solution. There are edge effects because there the field lines
are not straight and perpendicular to the condenser plates. This is illustrated in Figure 5.4. When a z 2 such effects can be negligible so that from the uniformity
across the plates, then P / a. This reduces Equation 5.22 to:
P z2
D constant :
"' 2 a
(5.22)
This again is a relation that embodies an approximation. The latter use of the dimensions for the variables of this problem is again consistent with the physical
significance of the variables.
5.9 Use of Vectorial Dimensions
109
5.9 Use of Vectorial Dimensions
In 1892 Williams proposed an extension to dimensional analysis by adding vectorial
identities to the numerical ones though he appeared not to have made use of this
idea in deriving results [2]. Bridgman quoted Williams paper and his work has been
quoted in support of this proposal [3]. However, when Bridgman applied this idea he
showed that the desired result was obtained without its use but instead by a proper
assessment of the physics of the problem together with a proper use of a unitsconversion factor.1 A further example by Bridgman is unfortunately incorrect and
so cannot be adduced in support of this idea.2
Brooke Benjamin discussed the application of vectorial identities by describing
in detail the case of the drag resistance of a sphere in a flow at a very low Reynolds
number [4]. This was described in Chapter 1 where the result derived analytically
by Stokes was obtained as:
D / V `
writing this as:
D D kV ` :
(5.23)
Brooke Benjamin observed that this equation can be taken as a vectorial identity
between D and V , both being aligned in the direction of the free stream so that
Equation 5.23 was, as he said, ‘generalised to a vector formulation’.
However, he went on to point out that Equation 5.23 is a limiting case as the
Reynolds number tends to zero. A more detailed analytical solution he quoted as
being:3
2
3
9 2
D D kV ` 1 C Re C Re lnRe C O Re ;
(5.24)
8
40
where the Reynolds number is given by:
Re V `
(5.25)
so that the solution given by dimensional analysis would be:
D
D f .Re / :
V `
1
See pp. 59–65 of [3].
See pp. 65–67 of [3].
3
This is a good example where the assumption of a simple power series cannot be made in order
to develop the Rayleigh–Buckingham version of the pi-theorem.
2
110
5 The Choice of Dimensions
Retaining V as a vector in Equation 5.25, then Equation 5.24 cannot be a vectorial
identity.
It has to be concluded that the result of using dimensional analysis gives an answer that can refer only to numerical equality. This is consistent with the full logic
presented here in Chapter 3 which is developed from the starting point of just meaningful numerical addition.
There have been the strongest of objections raised against assigning vectorial
characteristics to length dimensions in dimensional analysis. An early advocate of
this idea referred to the extended lengths as vector lengths [5].
Gessler has strongly queried this as, for just one example of many, some authors
have allocated various directional length dimensions to the essentially scalar quantity of viscosity [6]. Further, in the literature there are examples of the assignment by
a single author of different directions to viscosity depending upon the phenomenon
being considered, this despite the accompanying statement that “. . . there should be
a one-to-one correspondence between physical quantities and dimensional formulae.” Further, even different assignments for viscosity have been given by different
authors for the same phenomenon. Some quite ignore the physics of viscosity in
that, for example, in a laminar flow the viscous stress acts in all three directions.
Other writers have more strongly opposed this extension of the length dimension.
For example, Massey is severe in his criticism though while he notes that he does
not “. . . refute the fallacy rigorously” [7]. Yet later he condemned the practice as
“unsound and delusive” [8].
In a later paper the proposal of extended lengths is refuted in detail [9]. That paper
recorded a critical review of twenty five examples of supposed use that had appeared
in three different texts. It was stated that every single example had been successfully
resolved without the use of extension of the length dimension. It is relevant to note
that for all these examples the solution was initially known from standard analysis.
Furthermore each example fell under one or more of the following headings:
a)
b)
c)
d)
e)
f)
g)
h)
The derivation given had either an incorrect or a poor formulation of the
physics in drawing up the initial functional statement.
The result was only valid for certain approximations which were not inherent
in the original functional statement.
The result was only valid for a special case which was not specified in the
functional statement
An inadequate expression of the physics had preceded the formulation of the
original functional statement.
There was an uncoupling of the equations governing the phenomenon which
was not reflected in the original functional statement.
Variables were missed from the original functional statement.
Results were obtained that incorporated expressions of either definitions or of
basic physical laws neither of which, by their natures, are provable by dimensional analysis.
The answer given was incorrect.
5.9 Use of Vectorial Dimensions
111
A discussion of one particular example was given by Focken [10].4 but the lesson
has not been quoted by later writers.
In using this idea of extended length dimensions, different authors adopt different
dimensional representations of the same variables. For example, in considering the
problem of the energy of vibration of a stretched wire for the approximation of
a small amplitude, it is noted that the physics of this phenomenon is of the total
mass, m, of the wire oscillating over an amplitude of a. The energy will be kinetic
and so of the form of a mass times the square of a velocity represented by the
frequency. Thus the length of the string will not be a relevant variable. Then the
following Compact Solution 5.12 gives the known answer:
Compact Solution 5.12
!
E
a
m
1
T
ML2
T2
L
M
E
m
L2
T2
E
m! 2
L2
E
m! 2 a2
1
Thus:
E
D constant :
m! 2 a2
For this problem, the following dimensional scheme introducing orthogonal length
dimensions, Lx and Lz , has been adopted in the literature:
E
Lax Lbz MT2 ;
!
T1 ;
m M;
a
4
Lcx Ldz :
See pp. 105–108, particularly p. 107, of [10].
112
5 The Choice of Dimensions
A single power product would take the form of:
h
iC
Lax Lbz M 1
B
c d
M
L
L
1:
x
z
T2
TA
(5.26)
Meeting the required dimensions of the individual variables as listed in Compact
Solution 5.12, gives that:
In E,
In a,
Lax Lbz D L2
Lcx Ldz
DL
so that
a C b D 2;
(5.27)
so that
c C d D 1:
(5.28)
B D 1 ;
A D 2 ;
(5.29)
(5.30)
Then for:
MI 1 C B D 0
TI 2 A D 0
Lx I
Lz I
so that
so that
a C cC D 0 ;
b C dC D 0 :
(5.31)
(5.32)
From Equations 5.31 and 5.32 there is:
a C b C C.c C d / D 0
and so from Equations 5.27 and 5.28 we find that:
C D 2 :
Inserting this with Equations 5.29 and 5.30 into Equation 5.26 gives that:
E! 2 m1 a2 D constant ;
which is the correct answer.
Thus of the four indices, from Equations 5.27 and 5.28, it is possible to arbitrarily
choose either a or b and either c or d and still get the correct answer. To illustrate
further, two solutions by two different authors to the problem of natural thermal
convection, which both come under the cases (b) and (d) above, used the following
scheme of dimensions:
(i)
(ii)
For the kinematic viscosity, ,
Dimensions
Lx
Ly
Author A
1
1
Author B
1
1
Lz
0
1
T
1
1
H
0
0
0
0
For the thermal conductivity, k,
Author A
k
0
0
Author B
k
1
1
1
1
1
1
1
1
1
1
5.10 Concluding Comments
113
It is seen that of the six sets of extended lengths three are different yet both sets
purported to give the ‘correct’ answer.
This reveals the misconception of the use of extended lengths.
Examples of specific problems are discussed in detail in the paper cited [9]. Many
are otherwise derived in the present book. In that paper it is concluded that “In the
absence of a general supporting argument, the case for the extension of the length
dimension has rested on the validity of examples. – there is no need for the extension
of the length dimension in dimensional analysis; indeed, it is positively harmful.”
To do so is to contravene the principle of Occam’s Razor. As Russell interpreted this
maxim; “ – if everything in some science can be interpreted without assuming this
or that hypothetical entity, there is no ground for assuming it” [11].
5.10 Concluding Comments
The examples described illustrate the care that can be needed in the initial determination of both the variables and their appropriate dimensions. The discussion shows
that when dimensional analysis is used to determine an unknown solution, considerable care and understanding must be exerted: the initial steps of dimensional
analysis are not always simple ones.
Exercises
5.1
5.2
A magnetic field of strength, H , in a medium of magnetic permeability, 0 ,
is imposed by a magnetic dipole of moment, M . Show that, at a fixed point in
the field, the field strength is proportional to the moment and in the field the
field strength varies inversely as the cube of the distance, r, from the dipole.
Noting that a field strength is related to a force, choose suitable dimensions.
Light applies a radiation pressure, p, that depends upon the reflectance, r, of
the receiving surface. With radiation pressure dependent upon the luminous
flux intensity of the source, I; the distance of the source, `, the velocity of
light, c, and the mechanical equivalent of light, P0 , show that:
pc`2
D f .r/ :
P0 I
5.3
Use the dimensions of force, luminous flux intensity, velocity and length.
The power output of a dynamo, P , is a function of the electromotive force,
E, the current, i , the self inductance, L, and the current frequency, !. Show
that:
PL!
Li !
Df
:
E2
E
Use dimensions of power, current and time.
114
5 The Choice of Dimensions
References
1. M M Munk. Fluid mechanics, Part 2, Aerodynamic theory, (Ed. W F Durand), Vol. 1, Div. C,
p. 257, Springer, Berlin, 1934.
2. W Williams. On the relation of the dimensions of physical quantities to directions in space,
Philos. Mag., 5th Ser., Vol. 34, p. 234–271, 1892.
3. P W Bridgman. Dimensional analysis, Yale University Press, 1931.
4. T Brooke Benjamin. Note on the formulas for the drag of a sphere, J. Fluid Mech., Vol. 246,
pp. 335–342, January 1993.
5. H E Huntley. Dimensional analysis, Macdonald, London, 1952.
6. J Gessler. Vectors in dimensional analysis, Proc. ASCE, J. Eng. Mech. Div., Vol. 99, (EM1),
pp. 121–129, February 1973.
7. B S Massey. Units, dimensional analysis and physical similarity, Van Nostrand Reinhold,
London, (pp. 71–72), 1971.
8. B S Massey. Directional analysis? Int. J. Mech. Eng. Educ., Vol. 6, Pt. 1, p. 33–36, 1978.
9. J C Gibbings. Directional attributes of length in dimensional analysis, Int. J. Mech. Eng. Education, Vol. 8, No. 3, pp. 263–272, 1981.
10. C M Focken. Dimensional methods and their applications, Arnold, London, 1953.
11. B Russell. History of Western philosophy, pp. 462–463, George Allen & Unwin, London,
1969.
Chapter 6
Supplementation of Derivations
The task of discovering the physical magnitudes connected with
the phenomena which are decisive and of eliminating those
magnitudes which are of subordinate importance, is the very
core of the problem.
L. Prandtl
Notation
a0
c
Cp
CV
e
E
f
F
g
Gr
h
k
kB
`
`m
m
ms
Ma
M0
N
NU
Speed of sound
Mean molecular speed
Specific heat, constant pressure
Specific heat, constant volume
Specific internal thermal energy
Young’s elastic modulus
‘functional’
Force
Gravity acceleration
Grashof number
Height
Thermal conductivity
Boltzmann constant
Length scale
Mean free molecular path length
Mass of molecule, of planet, of projectile
Mass of the sun
Mach number
Molecular mass
Number of molecules/unit volume
Nusselt number
J.C. Gibbings, Dimensional Analysis. © Springer 2011
115
116
6 Supplementation of Derivations
p
Pr
q
Q
QP
R
Re
s
t
T
Tw
T0
u, v, w
U
v
V
w
x, y, z
Pressure
Prandtl number
Resultant velocity
Heat
Heat rate
Gas constant, range
Reynolds number
Distance
Time
Temperature
Wall temperature
Reference temperature
Velocity components
Reference velocity
Specific volume
Resultant velocity
Width
Coordinates
ˇ
ı
˚
Bulk modulus
Cp =CV
Deflexion
Second coefficient of viscosity
Coefficient of viscosity
Density
Dissipation function
6.1 Information from the Physics
Often a solution using the pi-theorem can be supplemented by further knowledge
of the physics of the phenomenon being studied. Also inspection of the governing
equations can sometimes enable a simplification in setting out the variables or in
combining variables into independent sets. Some examples of this are now given;
others appear in later chapters.
6.2 The Bending of a Beam
Figure 6.1 illustrates the bending of a beam under a transverse end load when it is
fixed at the other end. The following notation is used:
6.2 The Bending of a Beam
E
I
`
w
W
ı
117
Modulus of elasticity;
Second moment of cross-sectional area;
Length of beam;
Cross-sectional dimension of beam;
Concentrated end load;
End deflexion.
Then, for deflexion within the elastic limit, there is the relation of:
ı D f .W; `; E; w/ :
The pi-theorem solution is set out in Compact Solution 6.1.
This gives the relation that:
ı
W w
Df
;
:
`
E`2 `
(6.1)
(6.2)
This is the general solution. But consideration of the physics enables some approximations to be introduced. They are:
a)
When the thickness of the cross-section in the plane of bending is not very
small compared with the width then there is negligible transverse bending
of the cross-section. If this transverse bending is significant then a further
variable, the Poisson ratio, is added to Equation 6.2 forming its own nondimensional variable.
Figure 6.1 Sketch illustration
of the bending of a cantilever
beam
Compact Solution 6.1
ı
W
`
E
W
L
ML
T2
L
M
LT2
L
W
E
L2
ı
`
W
E`2
W
`
1
1
1
118
b)
c)
6 Supplementation of Derivations
When in addition to approximation (a), the deflexion is small then plane crosssections remain plane under load.
When the deflexion is small, and within the elastic limit for the beam, then
ı /W.
With these approximations, item (c) above requires that Equation 6.2 must take the
form of:
w ı
W
D
f
`
E`2
`
or,
hwi
ıE`
Df
:
W
`
(6.3)
The second moment of area, I / w 4 . Thus Equation 6.3 becomes:
4
ıE`
w
I
Df
Df
:
4
W
`
`4
Subject to the above approximations (a) and (b), E and I appear in the governing
equations in the combination EI . This last equation must then take the form of:
ıE`
`4
/
W
I
so that:
ıEI
D constant :
W `3
(6.4)
This is a well recognised result but it is now made clear, and will be returned to
later, that it is a special case of Equation 6.2, being seen here to be subject to several
limiting approximations.
6.3 Planetary Motion
Applying dimensional analysis to the motion of the planets the following notation
is used.
F
`
m
ms
t
Gravitational force of the sun;
Size of the orbit;
Mass of the planet;
Mass of the sun;
Time of planetary orbit.
6.3 Planetary Motion
119
Compact Solution 6.2
t
F
m
`
T
ML
T2
M
L
F
m
L
T2
F t2
m
L
F t2
m`
1
Then there is:
t D f .F; m; `/ :
The solution from the pi-theorem is given in Compact Solution 6.2.
This gives that:
F t2
D constant :
m`
(6.5)
There is extra information available in the form of Newton’s law of gravitation
which says that:
F /
mms
:
`2
The mass of the sun being a constant then insertion of this law into Equation 6.5.
gives the result that:
t2
D constant :
`3
This is Kepler’s famous law which he derived from extensive observation. The above
derivation again makes use of extra information though not in this case requiring any
approximations.
It is fascinating that had Kepler known of dimensional analysis when he derived the above result in 1619 and after lengthy calculation, he could have deduced
Newton’s law of gravitation before the latter finally confirmed the accuracy of it in
1685 [1].
120
6 Supplementation of Derivations
6.4 Extrapolated Solutions
Examples have been given where a non-dimensional group has been excluded because a variable in that group has been extrapolated to a value of either zero or
infinity so that the group then has a fixed value. This asymptotic approach has to be
done with care as is now illustrated.
The example of natural convection, to be discussed later, involves finally an
asymptotic solution as Re tends to zero as Churchill has also discussed [2]. The
idea of an asymptotic solution was used by Prandtl in 1904 for his concept of the
thin boundary layer [3].
Care has to be taken because misunderstanding of the physics can arise. The
asymptotic approach can also fail. For example, if one tries to obtain an incompressible flow from a compressible flow by letting the Mach number tend to zero
one simply loses the solution because, for example p=p0 ! 1. A further example is of the laminar flow through a pipe as considered in Sections 3.6, 5.6, and by
Pankhurst [4].
With the notation of:
L
p
QP
d
pipe length
pressure drop
flow rate
pipe radius
viscosity
then:
P ; L :
p D f L; Q;
Compact Solution 6.3 gives the solution as:
Compact Solution 6.3
p
L
P
Q
d
M
LT2
L
L3
T
L
M
LT
p
1
T
p
P
Q
1
3
L
d 3 p
P
Q
L
d
1
1
6.5 Uncoupled Equations
121
Thus,
d 3 p
Df
QP
L
d
(6.6)
which is correct because it takes account of the entry region. It would now be incorrect to make the entry effect negligible by letting L ) 1 because in Equation 6.6
this gives the incorrect asymptotic result that:
pd 3
D constant :
QP
(6.7)
This suggests that the pressure drop is independent of the pipe length which clearly
contravenes the physics. As explained previously, that physics shows that away from
the entrance region the balance between the pressure and viscous forces requires
that, p / L. Then Equation 6.6 gives the previous correct result of:
pd 4
D constant :
LQP
(6.8)
Some writers have advocated a criterion for an asymptotic solution as being ˘ 1.
This is clearly a invalid assumption as is seen in Figures 1.3 and 1.5 for the four
flows described there. For these flows, the asymptotic solution is not obtained in
Equation 1.18 by putting Re D 0. Another writer adopted the assumption that ˘ D
ŒO1. This assumption can be seriously misleading as some magnitudes of the pigroups can be as high as fifteen orders of greatness or twelve orders of smallness.
This also illustrates why dimensional analysis cannot be used to give numerical
values of the non-dimensional groups.
6.5 Uncoupled Equations
Another example where supplementary information comes from the governing
equations is of the motion of a projectile under gravity in a vacuum as illustrated
in Figure 6.2.
Figure 6.2 Sketch illustration
of the trajectory of a projectile
122
6 Supplementation of Derivations
Compact Solution 6.4
R
g
h
m
u0
v0
t
L
L
T2
L
M
L
T
L
T
T
u20
g
v02
g
gt 2
L
L
L
R
h
u20
gh
v02
gh
gt 2
h
1
1
1
1
With the notation of:
g
h
m
R
t
u0
v0
acceleration under gravity
height of launch point
mass of projectile
range
time of flight
horizontal component; launch velocity
vertical component; launch velocity
it is incorrect to put:
R D f Œg; h; m; u0 ; v0 ; t :
(6.9)
Doing this then solution of the pi-theorem would lead to Compact Solution 6.4.
This results in:
2 2
u0 v0 gt 2
R
Df
;
;
h
gh gh h
and so gives four non-dimensional groups. It is noted that dimensional analysis has
shown that the mass, m, does not appear in this solution.
The governing equations are, for the horizontal component of the motion,
R D u0 t
and for the vertical component,
h D v0 t .1=2/ gt 2 :
Inspection of these governing equations shows that the horizontal and vertical motions are uncoupled so that the latter can first be solved alone for t and then the
former gives R. Thus, from inspection of the equation for the vertical motion,
t D f .g; h; v0 /. Then the pi-theorem solution gives Compact Solution 6.5:
6.5 Uncoupled Equations
123
Compact Solution 6.5
t
g
h
v0
T
L
T2
L
L
T
t2g
L
L
t2g
h
v02
gh
1
1
v02
g
which gives:
t 2g
Df
h
v02
gh
then:
t 2 g gh
Df
h v02
or:
tg
Df
v0
gh
v02
gh
v02
:
(6.10)
Then for the horizontal component of the trajectory, R D f .u0 ; t/ leading to the
Compact Solution 6.6.
Compact Solution 6.6
R
u0
t
L
L
T
T
R
u0
T
R
u0 t
1
This gives:
R=u0 t D constant
(6.11)
124
6 Supplementation of Derivations
Eliminating t between Equations 6.10 and 6.11 gives the two groups as:
R tg
gh
Df
u0 t v0
v02
or:
Rg
Df
u0 v0
gh
:
v02
For h D 0, this finally reduces to the single non-dimensional group of:
Rg
D constant :
u0 v0
A similar case of motion occurs for the stability of an aeroplane. By limiting motion
to small perturbations of disturbance, then the solution for the longitudinal motion
can be separated from that for the lateral one. In this case progress has again been
made by noting a particularity of the governing equations in that the two motions
are uncoupled. Other examples follow.
6.6 Forced Convection of Thermal Energy
An illustration of the need for care in introducing units-conversion factors will now
be described in the considerable detail which is found necessary to resolve a long
standing problem. It will also show another case of the uncoupling of equations
which again influences the subsequent use of dimensional analysis.
6.6.1 Compressible-flow Energy Transfer
We now consider the case of the transfer of thermal energy between a uniform compressible stream of an ideal and Newtonian gas, and a solid body. This is illustrated
in Figure 6.3. It is a matter of concern, for example, in the design of gas and steam
turbines and of aeroplane cooling radiators.
With u, v, and w, the x, y, and z velocity components and q the resultant velocity
then for a continuum and compressible flow and with:
div q D
@u
@v
@w
C
C
@x
@y
@z
(6.12)
with:
q 2 D u2 C v 2 C w 2
(6.13)
6.6 Forced Convection of Thermal Energy
125
Figure 6.3 Illustration of an
example of forced thermal
convection
the continuity equation is [5]:
div q D @
:
@t
(6.14)
Here is the density and t the time.
There are three Navier–Stokes momentum equations without body forces and for
compressible flow. The one for the u component of velocity is [5]:
du
@p
@
@q
D
C
. div q/ C div C div . grad u/ :
(6.15)
dt
@x
@x
@x
The one for the v component is:
dv
@p
@
D
C
. div q/ C div
dt
@y
@y
@q
C div . grad v/
@y
(6.16)
@q
C div . grad w/ ;
@z
(6.17)
and for the w component is:
dw
@p
@
D
C . div q/ C div
dt
@z
@z
where p is the pressure, is the viscosity and is the second coefficient of viscosity.
The energy equation for this compressible flow is:
de
D div kT p div q C ˚ ;
dt
(6.18)
126
6 Supplementation of Derivations
where e is the specific internal energy, k is the thermal conductivity and ˚ is the
dissipation function. This latter function contains terms like:
@u
@x
2
(6.19)
and so on [6].
A further equation required for a solution is the equation of state which, for an
ideal gas, is written as [7]:
p
RT
D
;
M0
(6.20)
where R is the universal gas constant and M0 is the molecular molar mass. The
particular gas is specified by the value of M0 .
Also the specific internal energy, e, is given by:
e D CV T ;
(6.21)
where CV is the coefficient of specific heat at constant volume.
There are further relations like [7]:
D f .T /I
0
k
D f .T /I
k0
CV D f .M0 ; T / ;
(6.22)
where the suffices of ‘0’ on 0 and k0 indicate reference values usually taken as
boundary values. For extremes of high pressure these functions can include the pressure as an independent variable. Finally, a commonly made assumption, not always
valid, is that,
/ :
(6.23)
Also, limiting study to steady flows excludes the time t.
There are then twelve equations which are Equations 6.12–6.18, then Equations 6.20 and 6.21, then the three of Equation 6.22 and finally Equation 6.23. Correspondingly there are twelve possible unknown dependent variables which are:
u; v; w; q; T; p; ; ; k; cv ; e; :
(6.24)
Thus with the dependent variable of q, there are eleven independent variables giving:
q D f .U; p0 ; 0 ; 0 ; k0 ; `; Cv0 ; R; M0 ; T0 ; Tw / :
(6.25)
Here, the suffices of 0 indicate reference values and Tw is the body wall temperature.
These variables require five dimensions which are M, L, T, and n. The pitheorem solution then is of Compact Solution 6.7.
6.6 Forced Convection of Thermal Energy
127
Compact Solution 6.7
q
L
T
U
L
T
0
M
L3
0
k0
M
LT
ML
T3 `
CV
R
M0
T0
Tw
p0
L
L2
T2 ML2
T2 n
M
n
M
LT2
R
M0
L2
T2 k0 T0
cv T0
RT0
M0
Tw
T0
ML
T3
L2
T2
L2
T2
1
0
0
k0 T0
0
p0
0
T
L2
L2
T2
1
T
g
U
U0
0
k0 T0
0 U 2
CV T0
U2
RT0
M0 U 2
p0
0 U
1
1
L
1
1
1
1
L
U0 `
0
p0 `
0 U
1
1
this shows that:
q
Df
U
0 U ` k0 T0 CV T0 RT0 Tw p0 `
;
;
;
;
;
0 0 U 2 U 2 M0 U 2 T0 0 U
:
(6.26)
The last group in Equation 6.26 comes from the independent variable of p0 . But
inspection of Equation 6.20 shows that it is a superfluous variable because the other
variables in that equation have been included in the listing of Equation 6.25. thus
the corresponding last group in Equation 6.26 can be excluded.
Inverting the second independent group of Equation 6.26, dividing the fourth by
the third and also dividing the third by the second produces the following:
q
0 U ` 0 U 2
R
CV 0 Tw
Df
;
;
;
;
:
(6.27)
U
0
k0 T0 M0 CV
k0
T0
Of these independent groups the first is the Reynolds number, Re , and the fourth is
the Prandtl number, Pr . Using the ideal gas relation of:
R
D Cp CV
M0
(6.28)
and indicating the ratio of the specific heats by:
Cp =CV :
(6.29)
128
6 Supplementation of Derivations
Then the third group is:
R
D 1
M0 C V
(6.30)
so that the third group can be replaced by .
Noting Equations 6.18 and 6.19, the dissipation function, ˚, can be represented
by:
0 U 2
`2
and with the other terms in this equation, all have the dimensions of ML1 T3 . Then
from that equation, in non-dimensional form it can be expressed by:
˘˚ 0 U 2 `2
0 U 2
D
`2
k0 T0
k0 T0
which is the second independent group in Equation 6.27.
In summary, Equation 6.27 is now written as:
q
Tw
D f Re ; ˘˚ ; ; Pr ;
:
U
T0
(6.31)
(6.32)
An alternative formulation is obtained by requiring that the compressibility of the
gas, @=@p is related to a process that is independent of the viscosity which latter is
represented by the variable, . Such a process is a sound wave as the strength tends
to zero for then the process tends to an isentropic one [8]. In this limit,
1=a02 D
@
;
@p
where a0 is the velocity of sound. This is given by [9]:
a02 D R
T0 :
M0
(6.33)
Then by combining the second, third and fourth independent groups in Equation 6.32 there is:
˘˚
1 0 U 2 M0 CV k0
D
. 1/Pr
k0 T0 R CV 0
U2
D
.R=M0 /T0
U2
D 2
a0
D Ma2 ;
where Ma is the Mach number.
(6.34)
6.6 Forced Convection of Thermal Energy
129
Replacing then the variable ˘˚ by Ma in Equation 6.32 reforms this equation
into:
q
Tw
D f Re ; Ma ; ; Pr ;
(6.35)
U
T0
which is the commonly quoted form. Yet it is now seen that the representation of
the dissipation function is subsumed in the variables of Equation 6.35.
This is an example in which it is important to consider the full set of governing
equations so that the full set of independent variables of Equation 6.25 can be deduced. It is also important to understand the physics represented by each individual
non-dimensional group.
Equation 6.26 has another point of interest in that it contains non-dimensional
groups of three different sizes; that is, groups containing two, three or four variables.
Again the present demonstration of and operation of the pi-theorem produces these
groups in a quite straightforward manner.
There is a thermodynamic requirement that an ideal gas that obeys Equation 6.20
needs three thermodynamic properties to fully describe it; this is satisfied here. The
independent variables in Equation 6.25 are classified as:
Properties;
Units-conversion factor
Boundary conditions
Derived variables
T0 , 0 , M0
R
U , Tw , `
, k, CV .
In forming this table it has been recognised that 0 , k0 , and Cv0 are known from the
gas properties as listed.
For the case of determination of heat transfer, an alternative dependent group can
be the Nusselt number, Nu , which is:
P Œk0 ` .Tw T0 / :
Nu Q=
(6.36)
It is noted that kB does not appear among the variables of Equation 6.25 despite it
being the units-conversion factor for T . It is seen that it does not appear amongst
the above governing equations. These contain R and M0 in the combination R=M0 .
However,
R
kB
D
M0
m
(6.37)
so that R could be replaced by kB whilst m would then replace M0 . This would
give an unsatisfactory formulation because, there being one less dimension required,
that of quantity, there would be one extra non-dimensional group. This would be
superfluous as will now be shown. Tabulating with these revised variables gives the
Compact Solution 6.8.
130
6 Supplementation of Derivations
Compact Solution 6.8
q
L
T
U
L
T
0
M
L3
0
M
LT
k0
`
CV
kB
m
T0
Tw
p0
L
L2
T2 ML2
T2 n
M
M
LT2
k0 T0
CV T0
kB T0
Tw
T0
ML
T3
L2
T2
ML2
T2
1
ML
T3 0
0
k0 T0
0
kB T0
0
m
0
p0
0
T
L2
L2
T
L3
T
LT
1
T
q
U
U0
0
k0 T0
0 U 2
cv T0
U2
k0 T0
M0 U 2
mU
0
p0
0 U
1
1
L
1
1
L2
L2
1
L
U0 `
0
kB T0
0 U `2
mU
0 `2
p0 `
0 U
1
1
1
1
This gives the result that:
q
U0 ` k0 T0 CV T0 kB T0 mU p0 `
Df
;
;
;
;
;
:
U
0 0 U 2 U 2 0 U `2 0 `2 0 U
(6.38)
Dividing the third non-dimensional group of this equation by the second gives:
CV T0 0 U 2
CV 0
D
2
U
k0 T0
k0
which is the Prandtl number.
Combining the first, the second and the fourth independent groups of Equation 6.38 gives:
0 0 U 2 kB T0
kB 0
D
:
U0 ` k0 T0 0 U `2
k0 0 `3
(6.39)
This group can be assessed by substituting, from the kinetic theory of gases (Section 8.28), the relation:
0 / 0 c`m
and the one for the conductivity (Section 8.29) which is:
k0 / N k B c`m
6.6 Forced Convection of Thermal Energy
131
then:
kB 0
1
D
:
k0 0 `3
N `3
This is a measure of the reciprocal of the quantity of gas.
Usually this is modelled as being infinite in value and so this group would be
omitted as being zero in value. Should it be small in an enclosed space then heating would give a time dependent phenomenon so that the time, t , becomes another
independent variable. It would bring in the non-dimensional group of .U t/=`.
The final independent group containing p0 can be excluded for the reason given
in the case of Equation 6.26.
It is now seen that this different formulation leads to the same result of Equation 6.26.
6.6.2 Incompressible-flow Energy Transfer
The energy transfer in an incompressible flow is relevant, for examples, to the performance of heat exchangers and road vehicle radiators.
For incompressible flow the continuity equation becomes:
div q D 0 :
(6.40)
The momentum equations for incompressible flow reduce to:
du
@p
D
C 2 u ;
dt
@x
dv
@p
D
C 2 v ;
dt
@y
dw
@p
D
C 2 w :
dt
@z
(6.41)
A further assumption made is that temperature changes are small so that Equations 6.22 are not required. Then, inspection of these four equations for the four
unknown independent variables, that is; p, u, v and w, shows that now they can
be solved on their own. This is a further example of the importance of equations
becoming uncoupled. In this case the momentum and continuity equations are uncoupled from the energy equation which now is:
cv
dT
D k2 T C ˚ :
dt
(6.42)
Thus in principle the continuity equation and the momentum equations can be
solved for the velocity field and then the energy equation can successively be solved
for the temperature field.
132
6 Supplementation of Derivations
Compact Solution 6.9
q
U
0
`
L
T
L
T
M
L3
M
LT
L
0
T
L2
q
U
U
0
1
1
L
U `
0
1
This incompressible assumption means also that a solution does not require the
equation of state, Equation 6.20, and so the units-conversion factor of R does not
appear amongst the variables. Inspection of the dimensions of the variables in the
pi-theorem tabulation following Equation 6.25 shows that only the two variables of
R and M0 contain the dimension of quantity and so excluding R excludes also M0 .
The flow solution then gives:
q D f .U; ; `; 0 / :
The pi-theorem solution is in Compact Solution 6.9.
This leads to:
q=U D f .U `=0 / :
Inspection of the momentum equations, Equations 6.41, shows that the pressure appears only as a difference. Thus this dependent variable occurs only as the combination .p p0 / so that p0 separately is not present amongst the independent variables.
Again, as for the compressible-flow case, p0 is excluded as a separate variable but
now the reason is a completely different one. So an alternative solution to the continuity and momentum equations is:
.p p0 /=U 2 D f .U `=0 / :
(6.43)
The dissipation function now being known, the energy equation can be solved for
the temperature. So the independent variables for this are:
T D f .U; ; `; 0 ; CV ; k0 ; T0 ; Tw / :
(6.44)
The first four independent variables come from the momentum equations, the fifth
and sixth from the energy equation and the last two are boundary conditions to the
energy equation.
6.6 Forced Convection of Thermal Energy
133
Compact Solution 6.10
T T0
CV
0
U
Tw T 0
k0
`
L2
T2 M
LT
M
L3
L
T
ML
T3 L
T T0
Tw T0
CV .Tw T0 /
k0 .Tw T0 /
1
L2
T2
ML
T3
0
k0 .Tw T0 /
0
T
L2
L2
T2
CV .Tw T0 / U2
1
U
0
k0 .Tw T0 /
0 U 2
1
L
1
U `
0
1
However, in Equation 6.42 the temperature appears as only a difference. Thus
the variables T0 and Tw are combined into the independent variable of .Tw T0 / so
that Equation 6.44 reduces to:
.T T0 / D f .CV ; 0 ; ; U; .Tw T0 / ; k0 ; `/ :
(6.45)
Proceeding with the solution of the pi-theorem gives Compact Solution 6.10.
The pi-theorem thus gives:
T T0
CV .Tw T0 / U ` k0 .Tw T0 /
Df
;
;
:
(6.46)
Tw T0
U2
0
0 U 2
Dividing the first independent group by the third and also taking the reciprocal of
the third gives:
T T0
CV 0 U `
0 U 2
Df
;
;
:
(6.47)
Tw T0
k0
0 k0 .Tw T0 /
In this equation the first independent group is the Prandtl number, the second is the
Reynolds number and the third represents the dissipation function.
Now for the incompressible flow, the dissipation function in the energy equation
appears as a separate non-dimensional group. As before, commonly in formal analysis it is taken as being negligible. This is equivalent to neglecting this last group in
Equation 6.47 so that finally:
T T0
CV 0 U `
Df
;
:
(6.48)
Tw T0
k0
0
134
6 Supplementation of Derivations
Compact Solution 6.11
P
Q
k0
`
Tw T 0
CV
0
U
ML2
T3
ML
T3 L
L2
T2 M
L3
M
LT
L
T
k0 .Tw T0 /
CV .Tw T0 /
ML
T3
L2
T2
P
Q
k0 .Tw T0 /
k0 .Tw T0 /
0
k0 .Tw T0 /
L
T3
L4
T2
L2
CV .Tw T0 /
U2
U 3
k0 .Tw T0 /
0 U 2
k0 .Tw T0 /
1
1
L
1
P
Q
k0 `.Tw T0 /
`U 3
k0 .Tw T0 /
1
1
P The soAlternatively, the dependent variable can be chosen as the rate of heat, Q.
lution is in Compact Solution 6.11.
This gives:
QP
CV .Tw T0 /
`U 3
0 U 2
Df
;
;
:
k0 ` .Tw T0 /
U2
k0 .Tw T0 / k0 .Tw T0 /
Multiplying the first independent group by the third and dividing the second by the
third gives:
QP
CV 0 U `
0 U 2
Df
;
;
:
k0 ` .Tw T0 /
k0
0 k0 .Tw T0 /
(6.49)
This again has the same independent groups as in Equation 6.47.
It is seen that the results given in these equations have only been obtained as
a result of an extended investigation. The discussion has particularly covered the
approximations involved, the care needed over the determination of the use of the
appropriate units-conversion factors and the use of the uncoupling of the governing
equations all consequent upon the assumption of incompressible flow.
6.7 The Rayleigh–Riabouchinsky Problem
In the early development of dimensional analysis a problem arose in the application
to the problem of forced convection heat transfer. In the original study [9], Rayleigh
P He had omitted 0 from the list
had considered the relation for the heat rate, Q.
6.7 The Rayleigh–Riabouchinsky Problem
135
Compact Solution 6.12
P
Q
k0
`
Tw T0
CV U
ML2
T3
ML
T3 L
M
LT2 L
T
k0 .Tw T0 /
CV .Tw T0 /
ML
T3
M
LT2
P
Q
k0 .Tw T0 /
CV .Tw T0 /
k0 .Tw T0 /
L
T
L2
CV U
k0
1
L
P
Q
k0 `.Tw T0 /
CV U `
k0
1
1
of independent variables. He also combined CV and into the single variable of
.CV /. His retention of U amongst the variables can be justified because in the
energy equation, Equation 6.42, it is noted that [7],
D
@
@
D
Cq
Dt
@t
@s
so that the velocity appears in the last term.
Setting out the corresponding pi-theorem solution gives Compact Solution 6.12.
Rayleigh thus obtained only two non-dimensional groups so getting [9],
QP
CV U `
Df
k0 ` .Tw T0 /
k0
(6.50)
a result repeated by Bridgman [10].
By excluding the variable 0 Rayleigh excluded the dissipation function from
his result and also did not take account of the momentum equations. Inspection of
Equation 6.42 shows a justification for his combination of and CV as a single variable. But as described in Section 5.4, this example comes under the same heading
of being able to use a dimension of heat, H, rather than that of M. Then the combination of CV is the equivalent as before of basing CV upon a volume rather than
on a mass. It might be thought that it is equally acceptable to also divide through
this reduced energy equation by k0 to form a single variable from these three. But
this would require the approximation at this stage of putting the value of the dissipation function at zero. Also, interestingly, doing this would prohibit a solution
because in the above tabulation of dimensions an attempted solution would show
the temperature difference as the only variable with a dimension in . This becomes
a case where combination of variables obtained from inspection of the governing
136
6 Supplementation of Derivations
equations cannot be taken as always acceptable for developing a solution obtained
by dimensional analysis.
Riabouchinsky then proposed using the result of the kinetic theory of gases to
measure the temperature in terms of the kinetic energy of the molecules [11]. This,
he claimed, would remove the dimension of temperature so resulting in a further
non-dimensional group. That is one more non-dimensional group than was obtained
by Rayleigh. It might be expected that this removal of one dimension would correspond with removal of the corresponding units-conversion factor, which in this case
is the Boltzmann constant. But the equation:
T / mcN2 =kB
(6.51)
is uncoupled from the others so that the energy equation can, in principle, be solved
in isolation from the Boltzmann one. Riabouchinsky’s proposal was a subtle one.
The correct solution to this controversy was outlined in 1980 [12] and is now
enlarged to a full explanation.
Riabouchinsky did not give his full analysis but it appears that he used Equation 6.51 in which he set kB 1. He also seems to have used the energy equation 6.42.
With QP ML2 T3 , then from Equation 6.51,
k0 ML2 T3 L2 L 1 D MLT3 1
From Equation 6.51,
ML2 T2
and then,
k0 L1 T1
From Equation 6.42
CV L1 T1 L2 T 1 D L3
So Riabouchinsky’s solution is in Compact Solution 6.13.
This gives:
P
Q`
k0 `2
3
D
f
;
C
`
:
V
mcN2 U
U
Dividing the first group by the second and then inverting the second one gives:
QP
U
3
D
f
;
C
`
V
k0 `mcN2
k0 `2
(6.52)
6.7 The Rayleigh–Riabouchinsky Problem
137
Compact Solution 6.13
P
Q
k0
`
mcN 2
U
CV
ML2
T3
1
LT
L
ML2
T2
L
T
1
L3
P
Q
mcN 2
1
T
P
Q
mcN 2 U
k0
U
1
L
1
L2
P
Q`
mcN 2 U
k0 `2
U
CV `3
1
1
1
Compact Solution 6.14
P
Q
ML2
T3
CV
k0
T
L2
`
mcN 2
U
L
ML2
T2
L
T
P
Q
mcN 2
1
T
P
QC
V
mcN 2 k0
UCV
k0
1
L2
1
L
2
P
QC
V`
mcN 2 k0
UCV `
k0
1
1
which is the result put forward by Riabouchinsky, differing from Rayleigh’s solution of Equation 6.50. But Rayleigh was solving just the energy equation whilst
Riabouchinsky was dealing with both the energy and the temperature equations.
As with Rayleigh’s solution, the exclusion of the variable 0 has the same consequences for both.
Riabouchinsky’s form of the energy equation, Equation 6.42, would be:
d mcN2
CV
D kr 2 mcN2 :
(6.53)
dt
From this equation the grouping of CV =k can be taken as a single variable. Then
the solution is given in Compact Solution 6.14.
138
6 Supplementation of Derivations
This gives:
P V `2
QC
UCV `
Df
:
mk0 cN2
k0
(6.54)
This has just the two groups as were obtained by Rayleigh in Equation 6.50. In
Riabouchinsky’s solution there is a difference from the procedure in Rayleigh’s
because now combining .CV /=k0 to form a single variable does not change the
number of dimensions. It also now results in the two groups that Rayleigh obtained
and not the three that Riabouchinsky found. Their two approaches are now seen to
give a common result. However, whilst this corrected result of Rayleigh’s applies to
Newtonian fluids in general, Riabouchinsky’s is limited, through Equation 6.51, to
perfect gases. The full correct solution is that of Equation 6.49.
To complete the discussion of this famous controversy the full solution is now
set out but using Riabouchinsky’s approach. As before, the temperature terms in
the energy equation are as differences so the variable is written as .cNw2 cN02 /. The
pi-theorem solution now becomes Compact Solution 6.15:
Compact Solution 6.15
m cNw2 cN02
CV
0
U
ML2
T2
1
M
M
L3
M
LT
L
T
P
Q
m.cNw2 cN02 /
CV 0
L5
T3
L5
T2
1
L3
L2
T
0
U
L
P
Q
k0
`
ML2
T3
1
LT
L
m.cNw2 cN02 /
U 2
P
Q
U 3
k0
U
L2
1
L2
P
Q
U 3 `2
k0 `2
U
m.cNw2 cN02 /
U 2 `3
CV `3
0
U `
1
1
1
1
1
L3
This gives the result that:
QP
Df
U 3 `2
"
#
k0 `2 m cNw2 cN02
3 0
;
; CV ` ;
:
U
U 2 `3
U `
By dividing the first group by both the second and the third ones, then inverting the
fifth one, then multiplying the fourth and the fifth and dividing by the second, then
6.8 Natural Thermal Convection
139
dividing the fifth by both the second and the third all results in:
"
#
QP
U ` CV 0
0 U 2
3
; CV ` :
Df
;
;
0
k0 k0 m cNw2 cN02
m cNw2 cN02 k0 `
(6.55)
The dependent group is the one obtained by Rayleigh: the first independent one
is the Reynolds number; the second is the Prandtl number, the third represents the
dissipation function and the fourth is the extra one found by Riabouchinsky.
Dividing this extra one by the Prandtl number gives the group
0
:
k0 `3
As Riabouchinsky apparently put kB 1 then the above group is the one previously
discussed in Sect 6.6.1 as representing the reciprocal of the total quantity of gas and
so can be excluded as being zero in value.
Thus adopting Riabouchinsky’s approach in the full solution gives the identical
result to that of Equation 6.49.
This famous example, which dates back to 1915, is here fully resolved. It is
of considerable value in that it brings forward several important matters. First is
the care needed in the initial choice of variables so that all the relevant physics
is accounted for; secondly is the care to be taken in combining variables through
reference to the various governing equations; thirdly is the care to be taken in an
initial inspection of the dimensions of the variables and in particular the care in
checking the effect on the presence of the needed dimensions as resulting from an
initial combination of variables; fourthly is the value in a careful inspection of the
form and the meaning of the non-dimensional groups obtained and fifthly is the need
of inspection for equations being uncoupled.
In discussing this controversy, Bridgman made the perceptive comment, over the
choice of variables, that [10]:
“We will probably find ourselves able to justify the neglect of all these quantities, but the
justification will involve real argument and a considerable physical experience with the
physical systems of the kind which we have been considering.”
6.8 Natural Thermal Convection
The heat transfer by natural thermal convection is illustrated in Figure 6.4. This
phenomenon is associated with the application to buildings, to heating radiators and
to the thermal convection interchange between the earth surface and the atmosphere.
It is again governed by the equations for forced convection. These are Equations 6.12–6.18, and 6.20. The standard formal analysis of these flows introduces
several approximations to these equations in order to make reasonable progress:
these have to be addressed.
140
6 Supplementation of Derivations
The first approximation is to assume a one-dimensional flow so that only the
equation of momentum, Equation 6.15 is retained. However it now has a gravityforce term and so becomes:
du
@p
@
@q
D
g C
.div q/ C div C div .grad u/ :
(6.56)
dt
@x
@x
@x
Secondly, the pressure gradient is approximated to by the hydrostatic value so that:
@p
D g0
@x
Then Equation 6.56 reduces to:
du
0
@
@q
D g
1 C
.div q/ C div C div .grad u/ :
dt
@x
@x
(6.57)
The coefficient of bulk thermal expansion is defined in terms of the specific volume
by:
ˇD
1 @v
:
v0 @T
Then as v D 1=:
@.1=/
@T
0 @
D 2
:
@T
ˇ D 0
Figure 6.4 Illustration of an
example of natural thermal
convection under buoyancy of
a gas
6.8 Natural Thermal Convection
141
This is approximated by:
1 0
ˇD
0 T T0
1
D 1
:
0 T T0
(6.58)
With this and the assumption that / , then the momentum equation, Equation 6.57 becomes:
Du
D gˇ .T T0 / C f .u; `/ :
Dt
(6.59)
Making the assumption of incompressible flow in only the continuity and energy
equations, and again with those of negligible temperature effects in Equations 6.22,
reduces the energy equation to:
0 Cv0
dT
D k0 2 T C 0 f .u; `/ :
dt
(6.60)
It further follows from these approximations that Equation 6.20 is not required for
a solution so that the variable R=M0 is excluded. Thus a solution comes from the
two simultaneous Equations. 6.59 and 6.60. Choosing U as the dependent variable,
inspection of these two governing equations gives:
U D f Œ0 ; gˇ; Tw T0 ; 0 ; `; k0 ; CV :
(6.61)
The pi-theorem solution is in Compact Solution 6.16.
Compact Solution 6.16
U
L
T
0
M
L3
gˇ
Tw T0 0
`
k0
CV
L
T2 L
ML
T3 L2
T2 gˇ .Tw T0 /
L
T2
k0 .Tw T0 /
ML
T3
CV .Tw T0 /
L2
T2
M
LT
0
0
k0 .Tw T0 /
0
L2
T
L2
T2
gˇ.Tw T0 /
U2
0
0 U
k0 .Tw T0 /
0 U 2
CV .Tw T0 /
U2
1
L
L
1
1
gˇ.Tw T0 /`
U2
0
0 U `
1
1
142
6 Supplementation of Derivations
Thus the relation is:
0
gˇ .Tw T0 / ` k0 .Tw T0 / CV .Tw T0 /
Df
;
;
:
0 U `
U2
0 U 2
U2
By inverting the first group, multiplying the square of this times the second group,
and inverting the third group and also multiplying this by the fourth group, all leads
to:
gˇ02 `3 .Tw T0 / CV 0
0 U `
0 U 2
Df
;
;
:
(6.62)
0
k0 k0 .Tw T0 /
20
In this equation the dependent variable is the Reynolds number; the first independent group is called the Grashof number and the second one is again the Prandtl
number. The third independent group is the previously described representation of
the dissipation function.
Again in this case, it is usual practice to regard the dissipation function as a negligible term in the energy equation. Making this assumption then reduces Equation 6.62 to:
gˇ02 `3 .Tw T0 / CV 0
0 U `
Df
;
:
(6.63)
0
k0
20
Alternatively, the heat rate can be taken as the dependent variable so that Equation 6.63 becomes:
QP
D f ŒGr ; Pr :
k0 ` .Tw T0 /
(6.64)
There is now an interesting comparison between the three thermal convection cases
considered here. For the compressible flow case the group representing the dissipation function has been shown to be contained within the other independent groups
that are considered significant. For the other two cases where it is not so contained,
the dimensional analysis reproduces the necessary separate group which is usually
insignificant.
At very low speeds the acceleration term on the left-hand side of Equation 6.59
is negligible. Inspection of that equation with Equation 6.60 gives the variables as:
U; gˇ0 ; 0 ; k0 ; 0 CV ; Tw T0 ; ` :
The pi-theorem solution is in Compact Solution 6.17.
This gives that:
gˇ0 `2 .Tw T0 /
k0 .Tw T0 / 0 CV ` .Tw T0 /
Df
;
:
0 U
0 U 2
0 U
6.8 Natural Thermal Convection
143
Compact Solution 6.17
U
gˇ0
0
k0
0 CV
Tw T0 `
L
T
M
L2 T2 M
LT
ML
T3 M
LT2 gˇ0 .Tw T0 /
k0 .Tw T0 /
0 CV .Tw T0 /
M
L2 T2
ML
T3
M
LT2
L
gˇ0 .Tw T0 /
0
k0 .Tw T0 /
0
0 CV .Tw T0 /
0
1
LT
L2
T2
1
T
gˇ0 .Tw T0 /
0 U
k0 .Tw T0 /
0 U 2
0 CV .Tw T0 /
0 U
1
L2
1
1
L
gˇ0 `2 .Tw T0 /
0 U
0 CV `.Tw T0 /
0 U
1
1
Of these three non-dimensional groups, dividing the third by the second, taking the
reciprocal of the second and then multiplying the first and the third and dividing this
product by the second finally gives:
gˇ02 `3 .Tw T0 / CV 0
0 U ` CV 0
0 U 2
Df
;
:
(6.65)
0
k0
k0 k0 .Tw T0 /
20
The dependent group is the product of the Reynolds number and the Prandtl number;
the first independent group is the product of the Grashof number and the Prandtl
number whilst the second independent group is representative of the dissipation
function.
Or this is written as:
0 u2
Pr Re D f Pr Gr ;
:
(6.66)
k0 .Tw T0 /
If the Dissipation Function is again neglected and putting QP as the dependent variable, then:
QP
D f ŒGr Pr :
k0 ` .Tw T0 /
(6.67)
This last equation together with that of Equation 6.64 are commonly given in thermodynamic texts. It is now clear that their derivation depends upon the many approximations to the full governing equations. These approximations have to be introduced to get these two valid results.
144
6 Supplementation of Derivations
Certainly the exact solution without these assumptions is the much more complex
one of Equation 6.26 which, together with also the Grashof number, contains six
non-dimensional groups.
6.9 Summarising Comments
The discussion of this chapter emphasises the care needed in supplementing the
original functional statement by further knowledge of the physics or by the introduction of valid approximations or both. As shown here, the original functional
statement can enable dimensional analysis to derive a general solution. Then supplementation leads to special cases. This latter result is relevant to the previous
discussion in Section 5.9.
Exercises
6.1
A bi-metallic strip deflects under a change in its temperature. With the notation of:
B
E1 , E 2
H
`
R
˛1 , ˛2
T
Breadth of the strip;
Young’s moduli of the two components;
Height of the strip;
Length of the strip;
Radius of curvature of the deflected strip;
coefficients of thermal expansion of the
two components;
Temperature change.
show that:
r
Df
`
h b
E1
; ; ˛1 T; ˛2 T;
` `
E2
:
Set down physical conditions so that this result reduces to:
r
E1
D f ˛1 T; ˛2 T;
:
`
E2
6.2
A liquid meniscus can rise or be depressed under the action of surface tension
in a vertical tube. The direction depends upon the contact angle. With the
following notation:
6.9 Summarising Comments
g
h
R
145
Acceleration of gravity;
Displacement of meniscus;
Tube radius;
Contact angle;
Liquid density;
Surface tension coefficient.
show that:
h
Df
R
; :
gR2
Consider the physical condition that enables this equation to be reduced to the
approximate relation of:
hgR
D f ./ :
6.3
The current density from the cathode of a valve oscillator, j , is a function of
the electron mass, m, the electron charge, e, the cathode-anode distance, `,
the applied potential, , and the permittivity, ". Show that:
j 2 m`6
"`
D
f
:
e3
e
Noting that the equation governing the path of an electron is:
m
6.4
d2 x
d
D e
:
dt 2
dx
can you simplify the above result?
The frequency of a vibrating string, !, is a function of the elasticity, E, the
mass per unit length, m, the amplitude of oscillation, a, the length, `, and the
tension in the string, . Show that:
hai
D
f
:
m! 2 `
`
For small deflexions of the string the restoring force is proportional to the
deflection. Then, bearing in mind the previous discussion of linear vibration,
simplify this relation for the frequency. [Note that this case is unlike that described in Section 9.5. Whilst the added strain in the string due to the deflexion
also gives a cube relation between the corresponding restoring force and the
amplitude, this strain is very small compared with the pre-tension in the string
of musical instruments so that a linear relation is a good approximation: but it
is an approximation.]
146
6 Supplementation of Derivations
Then go on to derive an expression for the energy of vibration and show that
this is not a function of the tension. Finally show that:
E`
D constant :
a2
6.5
Waves on the surface of a liquid can be due to gravity forces and to surface
tension effects. With the notation of:
c
g
h
Wave speed
Gravitational acceleration
Liquid depth
Wave length
Liquid density
Surface tension coefficient
show that:
c2
Df
g
6.6
g2 h
;
:
Deduce expressions for very deep liquid when gravity effects dominate and
again for when surface tension effects dominate.
The temperature, T , of an astronomical black hole is a function of the following:
c
G
hp
kB
M
Speed of light
Gravitational constant
Planck constant
Boltzman constant
Mass of black hole
Show that:
T 2 GkB2
Df
hp c 5
M c2
kB T
:
In the equation for the hydrostatic equilibrium in stellar structures the variables M and G appear in the combination of the single variable of (M G).
Use this to simplify the above equation.
References
1. M White. Isaac Newton, Fourth Estate, London, 1997.
2. S W Churchill. Similitude: Dimensional analysis and data correlation. C R C Handbook of
Mechanical Engineering, Sec. 3.3, pp. 3-28–3-43, 1998.
3. L Prandtl. Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg, 1904, Leipzig, 1905.
References
147
4. R C Pankhurst. Alternative formulation of the Pi-theorem, J. Franklin Inst., Vol. 292, No. 6,
pp. 451–462, 1971.
5. A M Kuethe, J D Schetzer. Foundations of aerodynamics, John Wiley, New York, 1950
6. S Goldstein (Ed.). Modern developments in fluid dynamics, Vol. 2, §. 263, p. 613, Dover
Publications, New York, 1965, (Oxford 1938).
7. J C Gibbings. Thermomechanics, Pergamon, Oxford 1970.
8. H W Liepmann, A E Puckett. Aerodynamics of a compressible fluid, John Wiley, New York,
1947.
9. Rayleigh (Lord). Nature, Vol. 95, p. 66, 1915.
10. P W Bridgman. Dimensional Analysis, Yale Univ. Press, 1931.
11. D Riabouchinsky. Nature, Vol. 95, p. 591, 1915.
12. J C Gibbings. On dimensional analysis, J. Phys. A: Math. Gen., Vol. 13, pp. 75–89, 1980.
Chapter 7
Systematic Experiment
– the partial information given by dimensional analysis may be
combined with measurements on only a part of the totality of
physical systems covered by the analysis, so that together all the
information needed is obtained with much less trouble and
expense than would otherwise be possible.
P.W. Bridgman
Notation
CH
CV
d
DC , D
F
g
ic
k
k0
`
n
P
Q
QP
s
t
T
v
V
w
x
xk
xT
z
Filter coefficient
Coefficient of specific heat
Fan diameter: pore size
Diffusion coefficients
Faraday constant
Gravitational acceleration
Electrical convection current
Trip rod diameter
Thermal conductivity
Scale length
Rotational speed
Fan power
Torque
Heat rate
Electrode spacing
Time
Thrust
Flow velocity
Flow velocity
Filter thickness
Distance
Trip position
Position, start of transition
Valency
˛, ˇ
ˇ
ˇ0
ık
Coefficients, dissociation, recombination
Gas expansion coefficient
Units-conversion factor for angle
Boundary-layer displacement thickness
J.C. Gibbings, Dimensional Analysis. © Springer 2011
149
150
7 Systematic Experiment
"
'
Dielectric coefficient
Efficiency
Blade angle
Electrical conductivity
Viscosity
Density
Electrical potential
H
p
T
Hydraulic head difference
Pressure difference
Temperature difference.
7.1 The Benefits of Dimensional Analysis
The elementary introduction to dimensional analysis given in Chapter 1 gave but
a glimpse of its power when in partnership with experiment. In this chapter this
great benefit will be illustrated in detail as it is a prime case for the usefulness of
this analysis.
Dimensional analysis can markedly organise well the design and the economy
of an experiment; it can greatly clarify and order the output of the data; and it can
considerably extend the applicable range of investigation. These features, which can
have many valuable consequences for experiment, are now listed to be followed in
turn by detailed description.
a)
b)
c)
d)
e)
f)
g)
h)
There can be a great reduction in the amount of experimental investigation
through a reduction in the number of independent variables that are required to
be adjusted.
The effect of a variable can most conveniently be determined by the experimental variation of another one.
An experiment can provide results that apply to the range of a variable that is
much greater than the range set in the experiment. In the extreme case, this
applicable range can be achieved by the experimental measurement of only
a single value of one independent variable.
The dimensional analysis can show in some cases that a variable has no effect
upon the phenomenon and so can be excluded as an experimental variable.
The oversight of an independent variable in the planning of an experiment can
be revealed.
The experiment can be organised so that data acquisition is set in an ordered
and convenient manner.
A standard validity test can be applied so that the relationship for the number
of non-dimensional groups is properly satisfied.
The presentation of the data is greatly compacted and clarified.
7.2 Reduction of Variables
i)
151
The cost of an experiment can be reduced, or even sometimes experimentation
can be made feasible, by enabling tests to be made on reduced-scale models
of the full-size system. Or, it can ease experimental difficulties by enabling
experiments to be performed on larger-scale models of very small systems.
These admirable features will now be described in detail.
7.2 Reduction of Variables
The great reduction of experiment whilst achieving desired results is now illustrated
by the example of a test of a fan in a duct. Suppose interest is in the power to drive
such a fan which is pumping air at a fairly low speed along a duct. This is illustrated
in Figure 7.1.
It could be reasonable to specify initially that the power, P , is a function of the
fan diameter, d , the air density, , the air velocity along the duct, V , and the fan
rotational speed, n. This is then written as:
P D f .d; ; V; n/ :
(7.1)
If an experiment is designed on the basis of Equation 7.1, then a variation of only
d would give a single curve on a graph of P plotted against the values of d such as
is sketched in Figure 7.2(a). Repeating this experiment with a range of values of would lead to the graph with a family of curves as sketched in Figure 7.2(b). Further
variation of the duct velocity, V would result in the family of graphs of Figure 7.2(c)
and finally variation of the fan velocity, n would result in a family of sets of graphs.
With a minimum of five points to determine each curve the full experiment would
require 54 D 625 experimental points or 1250 readings. This has been described as
going from a page, to a book, to a shelf of books and on to a library.
However, from Equation 7.1 solution of the pi-theorem is as in Compact Solution 7.1.
Figure 7.1 Diagram of an axial-flow fan in a duct
152
7 Systematic Experiment
Figure 7.2 Illustration of the complexity of the basic data
Compact Solution 7.1
P
d
V
n
ML2
L
M
L3
L
T
1
T
T3
P
L5
T3
P
n3
V
n
L5
L
P
n3 d 5
V
nd
1
1
This leads to:
h v i
P
D
f
:
n3 d 5
nd
(7.2)
7.3 Further Reduction of Non-dimensional Groups
153
Figure 7.3 Axial-fan power
coefficient
Now the whole experiment gives a single line on a single graph as sketched in
Figure 7.3. This results in a reduction of the 1250 readings in the experiment to
just ten which is an enormous reduction in experimental effort.
7.3 Further Reduction of Non-dimensional Groups
Continuing with the example of the flow through a fan, suppose interest is in the
fan efficiency, , as the dependent variable. Aeroplane airscrews and many ship propellers have the facility of varying the blade angle, , so that this becomes a further
independent variable. Thus, with the units-conversion factor of ˇ0 we write:
D f .d; ; V; n; ; ˇ0 / :
(7.3)
Reduction is shown in Compact Solution 7.2.
This gives:
V
Df
; ˇ0 ;
nd
(7.4)
Compact Solution 7.2
d
V
n
ˇ0
1
L
M
L3
L
T
1
T
˛
1
˛
V
n
L
V
nd
1
ˇ0 1
154
7 Systematic Experiment
Figure 7.4 Axial-fan efficiency plots
and, as described in Chapter 1 one variable is shown by this application of dimensional analysis to be excluded so that experiment would not require its measurement.
In this case this exclusion is of the air density, . A typical plot of experimental values is shown in Figure 7.4.
If there is interest in only the peak values of the efficiency, p , then these are
marked in Figure 7.4. This implies a relation at this criterion between and V =nd
as illustrated by the chain line in this diagram. Thus Equation 7.4 reduces to either
V
p D f
(7.5)
nd
or, as an alternative, to:
p D f .ˇ0 / :
(7.6)
The chain-line curve in Figure 7.4 now represents Equation 7.5.
If interest is in only the absolute maximum value of the efficiency, max , then this
is represented by a single point in Figure 7.4 so that Equation 7.4 finally is reduced
to
max D constant :
(7.7)
Accompanying this relation are constant values of V =nd and of ˇ0 .
This search for certain values such as maxima, minima and zeros is a common
experimental requirement which, as shown, reduces the number of non-dimensional
groups.
7.4 Alternate Dependent Variables
155
Compact Solution 7.3
p
d
V
n
M
LT2
L
M
L3
L
T
1
T
p
L2
T2
p
n2
V
n
L2
L
p
n2 d 2
V
nd
1
1
7.4 Alternate Dependent Variables
In the design of an experiment, care has to be taken in carefully distinguishing between dependent and independent variables. It might be assumed that in this example of the axial-flow fan the power depends also upon the pressure rise across the
fan, p, so that Equation 7.1 should be:
P D f .d; ; V; n; p/ :
(7.8)
This is incorrect. Having set up the four independent variables of Equation 7.1 in
an experiment, then the value of p is fixed. Thus it is not an independent variable
that can be varied in complete independence of all the others but is an alternative
dependent one so that this is correctly expressed by:
p D f .d; ; V; n/ :
The solution takes the form of Compact Solution 7.3.
This alternative solution is thus:
p
V
Df
:
n2 d 2
nd
(7.9)
(7.10)
In the initial assessment of the physics of a phenomenon it can be difficult to determine which of the variables are truly independent and which are possible dependent
ones. It can be helpful then to visualise what happens in the process. In this case of
the flow through a fan, for example, once the size of the system as measured by the
duct diameter, d , and then the air density, , and the duct velocity, V , are set up,
then running up the fan to a speed of n necessarily not only fixes the power, P , but
also the pressure rise, p. This also fixes the Fan thrust, T , and the torque, Q. The
156
7 Systematic Experiment
latter is obvious from the relation between power and torque given by:
P D nQ :
This shows that there are four possible choices of dependent variable in the design
of this experiment; that is; P , p, T , and Q.
If, however, the test is on a complete fan and duct system pumping to and from
the atmospheric pressure, then the duct velocity is no longer an independent variable
but becomes yet another possible dependent one. This arises because running the fan
at a speed, n, now fixes the value of the duct velocity, V , the value of p across the
whole duct now being set at a value of zero.
7.5 Parameter Variation
In the example of Equation 7.2 there is a choice of which variables to vary in the experiment. First, there is a need to change only one of the four independent variables
listed in Equation 7.1 and to read values of the dependent variable, P . Secondly,
this independent variable can be chosen as that which is most convenient to vary
in the experiment. For this experiment this could be the velocity, V , as it could be
convenient to have a simple throttling device at the end of the duct. This could be
easier experimentally than varying the fan speed and certainly more convenient than
altering the size of the system as measured by values of the diameter, d . So varying V and reading values of P gives a range of values of both the non-dimensional
groups of Equation 7.2 and hence provides the graph of Figure 7.3.
Thus the effects of all the four independent variables of Equation 7.1 are determined by varying only one of them.
7.6 Range of Application
The range of application of an experiment for each variable is limited only by the
corresponding experimental ranges of the non-dimensional groups containing the
variable under consideration. For the example just given, when a fan diameter and
the rate of rotation is specified then the range of application of the duct velocity,
V; is given by the corresponding range of experimental values of V =nd . This gives
a considerable flexibility of application of the results.
On this last point care has to be taken over the physics of the flow. Experimentally, when V =nd D 0 then the power is found to be finite in value. If this equation
is interpreted mathematically as n D 1 then so also, from the dependent group,
would be the power. For Equation 7.1 to be valid, this condition must be interpreted
as being for V D 0. For n very large the flow becomes a compressible one so that
Equation 7.1 is an inadequate statement. The same form of argument applies to the
other end of the range shown in Figure 7.3.
7.8 Missing Variables
157
Compact Solution 7.4
d
V
n
1
L
M
L3
L
T
1
T
M
LT
T
L2
1
V
n
V
1
L
1
L
Vd
nd
V
1
1
7.7 Superfluous Variables
Two examples have been given for which dimensional analysis shows that a variable
does not enter into the problem thus avoiding wasted experimentation.
The first example was given in Chapter 1 where in the case of linear elastic
vibration the amplitude of the vibration is not relevant. The second example, that
has just been shown, is that for the expression for the efficiency of a fan where the
fluid density is excluded.
7.8 Missing Variables
Dimensional analysis can reveal the oversight of an independent variable.
The flow through an airscrew satisfies the statement of Equation 7.1. For the efficiency Equation 7.3 then applies. Results of a test on a small airscrew of 12 in
(30.5 cm) diameter in a wind tunnel are shown in Figure 7.5. The curves are for two
values of the wind-tunnel speed and indicate a failure of Equation 7.3. The conclusion is that in this case of small scale flow the variable that has been excluded is the
viscosity, . This brings the density into the list of variables so that the dimension
of mass, introduced by the variable, , can be dealt with by the pi-theorem as is now
shown in the tabulation below.
Thus the variables now show that:
D f .d; ; V; n; ; / :
Solution of the pi-theorem comes from Compact Solution 7.4.
(7.11)
158
7 Systematic Experiment
So that:
Df
Vd nd
;
V
(7.12)
or:
Df
V nd 2
;
nd
(7.13)
In Equation 7.12 the first independent group represents the Reynolds number as
does the second non-dimensional group in Equation 7.13. Comparing these equations with Equation 7.4. shows that now the Reynolds number appears as the extra non-dimensional group. Also, because the extra variable, , is introduced then
also the density, is added as an independent variable thus enabling the mass dimension to be cancelled. The two curves of Figure 7.5 are now seen to represent
two values of the Reynolds number because Equation 7.12 represents a family of
curves.
Figure 7.5 Efficiency plot of a small-scale airscrew: for Re D .Vd /=; lower curve, Re D
1:3 105 , upper curve, Re D 2:5 105
7.9 Influence of Variables
159
Figure 7.6 Illustration of the
electrical boundary layer
7.9 Influence of Variables
Another example comes from experiments to measure the electrical boundary layer
or diffuse double layer in an electrolyte liquid. This is illustrated in Figure 7.6. As
sketched, the positive potential on the electrode tends to attract the negative ions in
the electrolyte and repel the positive ones. There is then a conductivity effect tending
to move the ions which is counteracted by a diffusion effect in the opposite direction.
As a result, under a steady state continuum condition,1 there is a distribution of the
ion concentration and hence of the potential away from the electrode.
The potential, ', at a position, x, from one electrode, was taken as a function of
the applied potential between the electrodes, '0 , the electrical conductivity at zero
charge density, 0 , the dielectric coefficient of the electrolyte, ", the time of application of the potential, t , the distance between the two electrodes, `, the coefficients
of diffusion of the positive and negative ions, DC and D , the coefficient of dissociation of the ions, ˛, the coefficient of recombination, ˇ, and the product of the
valency with the Faraday number, zF , which latter is an alternative units-conversion
factor.
Inspection of the governing equations [1] shows that:
' D f Œ'0 ; 0 ; "; t; `; DC ; D ; x; ˛; ˇ; zF :
(7.14)
The solution is then Compact Solution 7.5.
1
In physics it is usual to distinguish two regimes, the microscopic and the macroscopic: in fluid
mechanics of gases three regimes are set as the free-molecule flow, the slip regime and the continuum regime.
160
7 Systematic Experiment
Compact Solution 7.5
'
'0
ML2
AT3
ML2
AT3
zF
0
"
AT
n
A2 T3
ML3
A2 T4
ML3
t
`
DC
D
x
˛
ˇ
L
L2
L2
L
T
T
A
n
AL3
n2
˛
zF
ˇ
z2 F 2
1
T
L3
AT2
T
0 '02
"'02
ˇ
z 2 F 2 '0
ML
T3
ML
T2
TL
M
"
0
ˇ0 '0
z2 F 2
L2
T2
'
'0
1
T
"
0 t
DC t
D t
˛t
zF
ˇ0 '0 t 2
z2 F 2
L2
L2
1
L2
DC t
`2
D t
`2
x
`
ˇ0 '0 t 2
z 2 F 2 `2
1
1
1
1
1
This gives the result that:
'
" DC t D t ˛t x ˇ0 '0 t 2
Df
;
; 2 ;
; ;
:
'0
0 t `2
`
zF ` z 2 F 2 `2
This can be rearranged as:
'
0 t DC " DC x ˛" " 2 ˇ'0
Df
;
;
;
;
;
:
'0
" 0 `2 D ` zF 0 zF ` 0
(7.15)
(7.16)
Figure 7.7 shows results of measurements of the distribution of potential in a liquid
electrolyte under an applied potential difference between two plane electrodes. The
results show the distribution through the electrical boundary-layers adjoining each
electrode surface [2, 3]. These layers are sometimes called diffuse double layers
after the Debye discrete double layers of charges, one in the electrode and one in
the electrolyte: this terminology seems a contradiction in terms. In a gas it is called
a plasma sheath. They are shown for two values of the electrode spacing, `, using
staggered origins.
7.9 Influence of Variables
161
Figure 7.7 Potential distribution through electrical boundary layers for values of '0 . Codes:
upper curves; ı, 48.5 kV, , 87.5 kV, , 173 kV; lower curves; , 55 kV, , 110 kV, ı,
215 kV (see [2, 3])
Table 7.1
'0 V
` 102 m
'0 =`2 104 V m2
48.5
87.5
173
55
110
215
2.95
3.0
3.05
10.8
10.8
10.8
5.57
9.72
18.6
0.47
0.94
1.84
For a test on a single liquid, the value of DC =D is constant and the readings at
infinite time reduce Equation 7.16. to,
x DC " ˛" " 2 ˇ'0
'
Df
;
;
;
:
(7.17)
'0
` 0 `2 zF 0 zF ` 0
The experiments covered the ranges of the parameters as now given in Table 7.1.
162
7 Systematic Experiment
The upper set of results in Figure 7.7 shows those obtained in one liquid at one
electrode spacing and three values of '0 . Close agreement between these three sets
in this non-dimensional plot is seen so that '='0 D f .x=`/. As Equation 7.17
is applicable two conclusions can be drawn. First, ` being held closely fixed, the
second independent group is held constant and so is excluded from this correlation.
Secondly, the third independent group is also a constant. Thirdly, as there is variation
of '0 in the fourth group, then the results show that this group, and hence ˇ, is
numerically insignificant in the correlation of this set of data.
The lower set of results in Figure 7.7 shows a similar set for a larger value of the
electrode spacing. Now there is agreement between only two sets and so for these
two, as for the upper set at ` D 3 cm, there is no numerical effect of ˛ and of ˇ.
The mean curve for the upper set of results is reproduced as a dotted curve for
the origin of the lower set. There is now seen to be a slight difference due to the
difference in the value of `. Thus having excluded an influence of ˛ and ˇ, inspection of Equation 7.17 indicates this to be an effect of the diffusion group. Thus with
a ratio in `2 of 13 between the upper and lower sets this ratio of the group containing
DC was found to change the sum of the non-dimensional potential drops across the
electrical boundary layers by a factor of 1 3.
The set of results at '0 D 55 V is divergent from the other five sets. This suggests
an effect of ˇ at this lowest value in Table 7.1 of '0 =`2 .
This study shows that when there is careful design of an experiment to accord
with a prior dimensional analysis, then useful deductions can be made when experimental results are studied in the form of non-dimensional groups in conjunction
with the physics of the phenomenon. This advantage is particularly of help when in
this example the phenomenon is a complex one not being amenable to a full classical
analytical solution [2].
A similar example is described in Chapter 10.
7.10 Measurement Limitation
In experiment, when determining the values of the variables that form a nondimensional group, it can happen that it is not possible to obtain a meaningful measure of one of them. An example is in filtration through a finely porous medium.
A simple arrangement of a porous filter is sketched in Figure 7.8. When a fluid
passes through such a porous medium having very small pores, then the viscous
stresses are large so that the flow is a very slow one. This means that the Reynolds
numbers of these flows are small so that, following the discussion of Chapter 1, the
density can be excluded from the list of independent variables.
With the following notation of:
7.10 Measurement Limitation
163
Compact Solution 7.6
p
d
w
v
M
LT2
L
M
LT
L
L
T
p
1
T
p
v
1
L
dp
v
w
d
1
1
CH
d
H
p
v
w
Filter coefficient
Pore cross-section dimension
Hydraulic head across filter
Pressure drop across filter
Mean inlet velocity
Filter thickness
Viscosity
then with:
p D f .d; ; w; v; / :
The pi-theorem solution is as Compact Solution 7.6.
Figure 7.8 Sketch of the flow
through a porous filter
164
7 Systematic Experiment
Thus:
hw i
dp
Df
:
v
d
(7.18)
For the simple arrangement of a filter as shown in Figure 7.8, and when d w then
over most of the length, w, the average flow conditions will be uniform through the
filter. Then any distortions in the flow pattern from a uniform one at both inlet and
outlet will be of negligible influence. Thus the pressure through the filter will drop
uniformly so that p / w. Then Equation 7.18 reduces to:
d 2 p
D constant :
vw
(7.19)
When the flow direction has a vertical component then there is an added gravitational force component along the flow so that p is replaced by the change in
hydraulic head given by:
gH D p C gz ;
where z is the vertical distance between inlet and outlet to the filter and with
z > 0 for a downward flow.
Then Equation 7.19 becomes:
gd 2 H
CH ;
vw
(7.20)
where the constant, CH , is the non-dimensional filter coefficient.
In many cases d is not measurable in a meaningful way. For examples, there
are the cases when there is seepage of water through a bed of gravel which contains a distribution of sizes of the gravel stones together with a distribution of their
arrangement or similarly when an air filter is formed from a pack of felt material.
This problem of an application of dimensional analysis is readily overcome as
follows. By performing a test on the chosen filter material in which H , w and v
are measured gives, from Equation 7.20, a measure of .CH =d 2 /. This enables the use
of that equation for evaluating any filter of that chosen material and inlet velocity
and for which values of and of the fluid are specified.
Historically, Darcy derived the empirical formula of:
v D kJ ;
where
J D
H
w
7.11 Effectiveness of Experimental Variables
165
so that
J 1 :
From Equation 7.20:
k
gd 2 L
:
T
It is unfortunate that the Darcy form, which makes no use of dimensional analysis,
is still used because extensive tables of values of k are available for many forms of
filter material. However, as k has dimensions then these values are of limited usage
as they refer only to the flow of water. This is despite the work of such as Yalin who
derived the relation of Equation 7.20 from dimensional analysis [4].
From Equation 7.20 and the definition of Darcy’s k,
CH D
g d2
:
k
Insertion of values for water in this equation enables Darcy’s k to be converted to
values of CH .
Equation 7.20 has a physical interpretation. For writing it as:
1 H
v
v2
Fr
D
D
D
:
2
CH w
gd
vd gd
Re
Then the right hand side is the ratio of the Froude number to the Reynolds number.
7.11 Effectiveness of Experimental Variables
Returning to the example of the test of a fan system, if interest is limited to a fan of
a single blade angle then Equation 7.4 for the efficiency reduces to:
V
Df
:
(7.21)
nd
Another marked usefulness of dimensional analysis is shown by this equation.
Again, only one of the three independent variables, which are V , n and d , needs
to be varied in experiment for a full determination of the influence of the other two
to be obtained. And again, adjustment of the velocity, V might much most readily
be achieved by an adjustable throttling of the flow through the containing duct.
Variables also need to be controlled so that an experiment is well designed for
the correlation of the output data. To continue with this example of a fan flow, if the
viscosity, is added to the list of independent variables in Equation 7.1 then the
166
7 Systematic Experiment
Compact Solution 7.7
P
V
n
d
ML2
T3
M
L3
M
LT
L
T
1
T
L
P
L5
T3
L2
T
P
n3
n
V
n
L5
L2
L
P
n3 d 5
nd 2
V
nd
1
1
1
Figure 7.9 Sketch of the
variation of a fan power coefficient against the Reynolds
number
drive power is given by:
P D f .; ; V; n; d / :
The pi-theorem then leads to Compact Solution 7.7.
This gives:
P
V nd 2
Df
;
:
n3 d 5
nd
(7.22)
(7.23)
In the studies of a fan and an airscrew, the dependent group is known as the power
coefficient, the first independent group is called the advance ratio and the second
independent group is the Reynolds number.
If an experiment is based upon Equation 7.23 then the desired results should be
plotted as shown in Figure 7.9. This would involve the following difficulty. By holding the value of n constant, varying the value of V and measuring the corresponding
values of P just one of the family of curves is obtained. This is because by holding
, n, d and constant then the Reynolds number, .nd 2 /= is being held constant.
So to get the family of curves, for each curve it is necessary to change one of the
7.12 The Validity Criterion
167
Figure 7.10 Sketch of the
variation of a fan power coefficient against the advance
ratio
Figure 7.11 Illustration of
the use of a transition rod
to trip the laminar boundary
layer
other variables, n, d , or . Clearly, the second and the third of these three are
the most inconvenient so the experimenter is faced with providing a variable-speed
drive motor. Then the results could be plotted as in Figure 7.10.
7.12 The Validity Criterion
The choice of variable to be changed during experiment is not always either the
most convenient experimentally or an arbitrary one. This matter is now illustrated
by three examples.
An experiment was set up to investigate the flow in the boundary layer at incompressible speeds over a flat plate. This is sketched in Figure 7.11. Interest was in
the transition from laminar, steady, flow in the boundary layer to a turbulent one at
transition between the two regimes; this was influenced by the presence of a circular
rod across the flow. Then with the notation of:
xT
xk
k
V
Distance to the transition start;
Distance to the rod position;
Transition rod diameter;
Flow velocity;
Fluid density;
Fluid viscosity.
168
7 Systematic Experiment
Compact Solution 7.8
xT
xk
k
V
L
L
L
L
T
M
L3
M
LT
T
L2
V
1
L
xT V
xk V
kV
1
1
1
there is:
xT D f .xk ; k; V; ; / :
The pi-theorem gives Compact Solution 7.8.
This shows that:
xT V
xk V kV
Df
;
:
(7.24)
(7.25)
It might be considered, on the basis of the physics of this phenomenon, that the flow
would be better related to the local conditions at the transition rod by introducing
the variable, ık which is the displacement thickness of the boundary layer at the rod
position but in the absence of the rod. Then this thickness is given by:
ık D f .xk ; V; ; / :
The solution becomes that of Compact Solution 7.9.
This gives the result that:
ık V
xk V
Df
:
Putting this equation into Equation 7.25 gives:
ık V kV
xT V
Df
;
:
(7.26)
(7.27)
(7.28)
7.12 The Validity Criterion
169
Compact Solution 7.9
ık
xk
V
L
L
L
T
M
L3
M
LT
T
L2
V
1
L
ık V
xk V
1
1
This is rearranged to:
xT V
Df
k kV
;
ık :
(7.29)
Following the prior discussion, with two independent non-dimensional groups there
is the requirement to vary just two of the independent variables in an experiment.
Correspondingly, one variable associated with the dependent group is thus measured. In the present case of a total of six variables, three can then be held constant
in the experiment.
Designing an experiment on the basis of Equation 7.29, it would be convenient
to fix the values of , and k. and then to vary the values of V and ık and to
measure values of xT . Doing this would be expected to vary the values of the three
non-dimensional groups in Equation 7.29 but on doing so these results would plot
as only a single curve rather than as a family of curves. The reason would be that the
value of xk was being held a constant so that a non-dimensional group was being
held constant; this would be the group of k/xk .
This is the validity criterion that has to be checked before the design of an experiment is finalised: it is that a non-dimensional group cannot be formed from amongst
those variables that are to be held a constant.
The second example comes from a study of the electro-hydrodynamic convection
of electric charge in a dielectric liquid which is set in motion by application of an
electric potential applied to an electrode bounding the fluid [5]. This is illustrated
by a flow-visualisation picture in Figure 7.12.
With the following notation:
170
7 Systematic Experiment
Compact Solution 7.10
ic
s
A
L
"
M
L3
A2 T4
ML3
0
'
M
LT
A2 T3
ML3
ML2
AT3
ic2
"
0
"
'2 "
ML3
T4
1
T
ML
T2
ic2
"
'2 "
L6
T4
L2
T
L4
T2
ic2 "3
40
"
0
' 2 "3
20
L6
L2
L4
ic2 "3
40 s 6
"
0 s2
' 2 "3
20 s4
1
1
1
ic
s
"
0
'
Convection current.
Electrode spacing.
Liquid density.
Dielectric coefficient.
Liquid viscosity.
Conductivity at zero charge density.
Electrode potential.
this is expressed as:
is D f .s; ; "; ; 0 ; '/ :
Analysis goes as in Compact Solution 7.10.
This results in:
ic2 "3
"
' 2 "3
Df
;
:
0 s 2 20 s 4
40 s 6
Figure 7.12 Picture of the
electro-hydrodynamic flow
pattern from a sharp point;
point at the top of picture,
'0 D 8 kV [5]
(7.30)
(7.31)
7.12 The Validity Criterion
171
Table 7.2
"
'
M
L3
A2 T4
ML3
M
LT
ML2
AT3
'2"
ML
T2
'2 "
T
L2
L2
T
' 2 "
2
1
Writing:
˘1 ic2 "3
;
40 s 6
"
;
0 s 2
˘2 ˘3 ' 2 "3
20 s 4
then Equation 7.31 can be rewritten as:
"
˘11=2
Df
1=2
1 ˘1
;
˘2 ˘3
#
or,
ic "3=2
Df
1=2 20 s 3
"
0 s 2 ic 1=2 s
;
" "3=2 ' 2
#
:
(7.32)
Suppose that an experiment is designed on the basis of Equation 7.32. In that experiment, the values of ", , and ' could be held at constant values so that those of 0
and s are the experimental variables with the dependent variable ic being measured.
It would be found that a family of curves would not result from this experimental design. For a non-dimensional group can be formed from those variables that are held
constant. This is seen by the following tabulation in Table 7.2. (It is to be noted that
this tabulation is not here an application of the pi-theorem but merely a convenient
means of forming a non-dimensional group.)
Thus the variables which were proposed to be held constant can be formed into
the non-dimensional group of:
' 2 "
:
2
172
7 Systematic Experiment
The proposed design of the experiment has been failed by this validity test. If the
experiment was designed on the basis of Equation 7.31 by holding constant the variables , s, " and whilst varying 0 and ', a family of curves could be constructed.
For it would not be found possible to construct a non-dimensional group from the
former four variables.
This can be seen from the following tabulation of Table 7.3.
It is seen from this tabulation that it is not possible to cancel the dimension in A
so that no non-dimensional group can be formed.
A third example is for the heating by natural convection from a vertical flat surface. This has been studied in Chapter 6. The variables involved there were:
P 0
Q=k
gˇ
Heat rate over thermal conductivity;
Combine acceleration due to gravity with the volume
coefficient of gas expansion;
Coefficient of specific heat;
Thermal conductivity over density;
Viscosity over density;
Scale size;
Temperature difference.
Cv
k0 =0
0 =0
`
T
That discussion showed that after making several approximations to the full set of
governing equations the result can be expressed by:
"
#
gˇ02 Cv2 `3 T gˇ02 `2 QP
QP
Df
;
:
(7.33)
k0 `T
k02
k0 20
This indicates that experimental values would be expected to plot as a family of
curves. The corresponding two variables to be varied in experiment would most
conveniently be the size variable, ` and the temperature difference, T . Then values
P being the dependent variable, would be measured. An experiment could be
of Q,
designed by varying the size and the temperature difference so that the quantity,
Table 7.3
s
"
M
L3
L
A2 T4
ML3
M
LT
"
T
L2
A2 T3
L4
"4
3
A2 L2
"4
3 s 2
A2
7.12 The Validity Criterion
173
Table 7.4
gˇ
Cv
k0
0
0
0
L
T2 L2
T2 L4
T3 L2
T
gˇ
Cv
k0
0 Cv
1
L
L
T
k0
0 Cv
1
`3 T is held constant thus holding the first independent non-dimensional group
in Equation 7.33 a constant and then enabling a curve between the first and third
groups to be traced out; this is illustrated in Figure 7.13. Changes to the value of
`3 T would be expected to provide the family of curves.
Applying the validity criterion, we seek a non-dimensional group from amongst
the four remaining variables which are:
gˇ; cv ;
k0 0
;
0 0
and which would have been held constant. Tabulation gives Table 7.4.
This shows that in the proposed experiment the non-dimensional group Cv 0 =k0
would have been held constant. This quantity is the Prandtl number and is a function
of the thermodynamic state of the gas. So an experiment would only be valid if at
least one of its thermodynamic properties is varied thus adding to the complexity
of the design. The reason why only one curve would be obtained is seen also by
P from the
noting that by using the dependent group to eliminate the quantity Q,
Figure 7.13 Sketch of the
results of an experiment on
natural heat convection
174
7 Systematic Experiment
second independent group means that that group contains the quantity `3 T which
is identical to that quantity in the first independent group.
These examples show the importance of a check against the validity criterion,
a criterion previously introduced [6].
7.13 Synthesis of Experimental Data
In all cases, experimental data can be enormously compacted when plotted or tabulated in terms of non-dimensional groups. The experimental range of validity of
data is expressed so simply in terms of the corresponding limiting values of the
non-dimensional groups as distinct from the experimental ranges of individual variables.
7.14 Concluding Comments
The benefits to experiment listed in Section 7.1 have now been explained in detail.
The discussion of this chapter shows that, when dimensional analysis can be
used, it has a most powerful effect upon the design of an experiment through a remarkable reduction of effort and a great benefit in the clarity of understanding of
the output data.
Also there is a corresponding extreme benefit in the way that expressing this data
in the form of non-dimensional groups leads to a great compactness of as well as an
increased clarity in the ordering of the experimental results.
Finally, the validity criterion has been introduced as a necessary check on the
choice of independent variables to be varied for valid experiment.
Exercises
7.1
7.2
In an experiment to evaluate Equations 7.12, or 7.13 show how it can be designed so that a family of curves is readily determined. The data is to be used
to determine a fan diameter when all the other variables are specified. Arrange a plot of the experimental results so that the required diameter can be
determined directly without any trial and error routine.
Show that for the flow of fluid of density and viscosity, at a mean velocity,
V through a pipe of diameter, d and with an internal surface roughness height
of ", the pressure gradient @p=@x is given by:
d @p
Vd V "
D
f
;
:
V 2 @x
A test is to be designed to determine the values of the pressure gradient by
varying the values of d and V . Check this experimental design.
7.14 Concluding Comments
175
Figure 7.14 Illustration of
a cooling fin heated at one end
7.3
7.4
7.5
Two parallel plates of area, a, and spaced a distance, z, apart, form an electrical capacitor. Putting the attractive force between these plates, fe , as a function of the charge on each plate, , and the dielectric coefficient, ", of the
intervening medium, show that,
2
"fe
z
Df
:
2a
a
Ignoring edge effects so that fe / a, show that this force is independent of
the value of z.
A hydraulic pump, when running at its maximum efficiency point is rotating at 30 rps and has a flow rate of 0.063 m3 s1 , with a pressure rise of
3:8 105 kg m1 s2 and requires a drive power of 32.5 kW. What would be
these values when running at the same efficiency and at 25 rps? (Exclude the
influence of viscosity)
Large water turbines, especially of the Pelton-wheel type, are very efficient
so that it is experimentally difficult to measure this efficiency accurately in
the usual way by measuring the output-shaft torque. An alternative way is to
measure the change in water temperature between inlet and outlet. With the
following variables:
Cv
m
P
W
7.6
Coefficient of specific heat
Mass-flow rate
Power output
Temperature change
Efficiency
set out a non-dimensional functional relation for the efficiency. What are the
effective dimensions in this solution?
Figure 7.14 illustrates a cooling fin having an input heating rate at one end.
Using the parameters shown and denoting the heat transfer coefficient from
the fin to atmosphere of h and a thermal conductivity of the material of the fin
176
7 Systematic Experiment
Figure 7.15 Sketch of
electric-field surface-wetting
of k, show that:
QP
h` t A
Df
; ; 2 :
.Tf T /`
k ` `
What are the effective dimensions? For a fixed shape and steady running conditions, then:
QP
h`
Df
:
.Tf T /`
k
7.7
What variables would have to be changed in an experiment to determine this
latter function?
An electric field when applied to a liquid of very low conductivity can influence the liquid-surface wetting. An experiment to study this is illustrated in
Figure 7.15. With the following notation:
d
g
y
"
'
Electrode spacing;
Gravity acceleration;
Liquid rise;
Surface tension;
Permittivity;
Density difference, liquid/gas;
Potential difference.
show that:
y
Df
d
gd 2 ' 2 "
;
:
d
What variables would be suitably changed in an experiment? How would the
above non-dimensional groups be re-arranged for the design of the experiment?
References
177
References
1. J C Gibbings. The dependence of conductivity of a weak electrolyte upon low solute concentration and charge density, J. Electroanal. Chem., Vol. 67, p.129, 1976.
2. J C Gibbings, G S Saluja. The electrostatic boundary layer in stationary liquids, J. Electrostatics,
Vol. 3, No. 4, pp. 335–370, 1977.
3. G S Saluja. Static electrification in motionless and moving liquids, Ph D Thesis, Mechanical
Engineering Department, University of Liverpool, October 1969.
4. M S Yalin. Theory of hydraulic models, MacMillan, London, 1971.
5. J C Gibbings, A M Mackey. Charge convection in electrically stressed, low-conductivity, liquids, Part 3: sharp electrodes, J. Electrostatics., Vol. 11, pp. 119–134, 1981.
6. J C Gibbings. The planning of experiments: part 3 – Application of dimensional analysis, The
systematic experiment (Ed. J C Gibbings), Cambridge Univ. Press, 1986
Chapter 8
Analytical Results
I have sought the principles of the resistance of fluids as if
analysis had not to enter therein, and only after having found
these principles have I tried to apply analysis to them.
D’Alembert
8.1 Analytical Results from Dimensional Analysis
As well as its prime use in the support of experiment, dimensional analysis has been
used to obtain analytical results where formal analysis either is not available or
is necessarily semi-empirical. Such results are derived from the information available from the composition of non-dimensional groups. All derived non-dimensional
groups give relations between the variables contained in those groups but sometimes
it is possible to go further. Four examples illustrating these features are now given.
8.2 Example I: Flow Turbulence
Notation for Example I
a
b
B
F
K
`
Pr
q
r
t
tB
u
U
uc
us
Coefficient; Equation 8.4
Jet width; coefficient Equation 8.4
Log-law constant
Œf ./1=2
Circulation
Scale length
Prandtl number
Velocity
Radial ordinate
Time
Burst time
Time mean velocity
Reference velocity
Centre-line velocity
Boundary velocity of viscous sub-layer
J.C. Gibbings, Dimensional Analysis. © Springer 2011
179
180
8 Analytical Results
u
u0
v0
y
Friction velocity
Streamwise turbulence velocity
Cross-stream turbulence velocity
Distance from wall
˛
ˇ
ˇ0
ıs
ısC
v
w
Coefficient; Equation 8.6, 8.12
Integration constant; Equation 8.13
Units-conversion factor; angle
Sub-layer thickness
Thickness Reynolds number
Phase angle between turbulence components
von Karman constant
Vortex spacing
Viscosity
Kinematic viscosity
Air density
Shear stress
Wall shear stress
8.3 The Complexity of Flow Turbulence
An outstanding example of an output from dimensional analysis has been in the application to turbulent flows. This is a major physical phenomenon, which is of considerable concern particularly to engineers and meteorologists. Yet despite its very
great practical importance, this flow characteristic is so complex and detailed in its
nature that practical solutions almost invariably involve a combination of dimensional analysis with one, or often more, empirical coefficients. A formal solution of
the governing equations without such coefficients is rare [1].
Because of its importance, a dimensional analysis for turbulent flows is given
now in detail and with some original derivations.
8.4 The Physics of Turbulence
This study provides a good illustration of the importance of a careful assessment of
the physics of a phenomenon prior to the application of the pi-theorem.
The governing equations are known. They have been quoted in full in Chapter 6
in the discussion of convective heat transfer. So the variables are known. For incompressible flow past a wall and for the local velocity and its fluctuating component,
they are:
u D f .U; ; ; `; t; y/
(8.1)
8.4 The Physics of Turbulence
181
with,
u0 D f .U; ; ; `; t; y/ :
(8.2)
Turbulence in a flow is of small amplitude in the velocities compared with typical
mean velocities, being three-dimensional in form, of random in occurrence at any
position and of high frequency. The structure is an intensely packed one. This explains why formal solution of the governing equations presents such a formidable
challenge. But there are some characteristics of the physics which can be deduced
from experiment on different occurrences of turbulence. Several of these flows are
now listed.
a)
b)
Figure 8.1 illustrates the flow in the turbulent shear layer between two uniform
streams of differing velocity. The velocity distribution plotted is the time mean
value so that the turbulent fluctuations are not shown.
In this flow the width of the layer increases with this distance and so the
Reynolds number of the turbulent portion of the flow, based upon this width
and the constant mean velocity at the mid-point of the shear layer, increases
along the flow. Yet the normalised velocity distribution is constant along the
flow and so there is no Reynolds number effect and so no influence of the numerical value of the viscosity upon this profile.
Figure 8.2 sketches the flow past a flat plate normal to the oncoming uniform
stream.
Figure 8.1 Velocity distribution across a shear layer
flow: velocity, u, versus the
distance, z
Figure 8.2 Flow past a flat
plate set normal to the oncoming stream
182
c)
d)
e)
8 Analytical Results
At a sufficiently high Reynolds number, the turbulent separation is fixed at the
edges of the plate and the drag coefficient, as defined in Chapter 1, is a constant
independent of the numerical value of the Reynolds number. So again this flow
is independent of the numerical value of the viscosity.
Again for the flow of Figure 8.2, there can be a level of turbulence in the oncoming free-stream. Then the measured values of the fluctuating component
along the stagnation streamline up to the front stagnation point were found to
retain a constant value while the mean flow velocity dropped from the freestream value down to zero [2]. This meant that by Kelvin’s theorem [3, 4], the
overall flow pattern was an irrotational one, the vorticity being conserved.
The diagram of Figure 8.3 is of the two-dimensional developed turbulent jet
flow.
As for case (a), the normalised velocity distribution in the turbulent region of
a developed jet flow, that is downstream of section ’A A’ as illustrated, is
independent of the value of the increasing Reynolds number along the flow
and so of the numerical value of the viscosity.
The developed turbulent wake flow downstream of a two-dimensional solid
body is sketched in Figure 8.4.
Again as for case (a), from a short distance downstream of the trailing edge of
the solid body, the varying value of the Reynolds number along the developed
wake flow has no effect upon the normalised velocity distribution and so again
the numerical value of the viscosity has no effect upon this flow.
Figure 8.3 Velocity distribution across a two-dimensional jet flow
Figure 8.4 Velocity distribution across a two-dimensional wake flow
8.4 The Physics of Turbulence
f)
g)
h)
i)
183
The temperature recovery factor.
The temperature recovery factor is the ratio of the temperature change across
a boundary layer to the corresponding isentropic change. For a laminar, and
hence fully viscous, boundary layer it has the value of Pr1=2 which for air is
0.85 whereas for a turbulent layer experiment gives Pr 1 [5]. This indicates
that the turbulent boundary layer is largely independent of the value of the
viscosity being then an irrotational flow except for the very thin viscous sublayer that adjoins the surface.
The boundary layer in a zero streamwise pressure gradient.
The normalised velocity profile of a turbulent boundary layer, outside of the
viscous sub-layer, and in a zero streamwise pressure gradient, is universal in
shape and so is independent of the value of the Reynolds number. The further
influence of a rough surface upon this profile is merely to shift away from the
wall that portion of the profile that is outside the viscous sub-layer. Thus this
profile, if normalised to the velocity at the edge of the viscous sub-layer, is also
a universal one again independent of the value of the Reynolds number [6].
The turbulent flow in a pipe.
The fully developed, normalised velocity distribution across a turbulent flow in
a straight pipe of constant diameter and outside of the viscous sub-layer also is
independent of the Reynolds number of this flow.
The charging of particles in a turbulent flow.
The electrostatic charge induced upon a particle in a turbulent flow has a value
that is significantly less than the Pauthenier limit. This has been explained as
being due to the absence of rotation of the particle in the turbulent flow [7].
Also study of the particle charging and tracing in the highly turbulent flow
through an electrostatic precipitator has assumed the absence of rotation of the
particles with a successful result ( [8], p. 133). This again implies an irrotational
flow and hence one independent of the numerical value of the viscosity.
All these examples indicate that turbulence is composed of small vortices contained
within a flow which in the main is irrotational with viscous effects limited to the
small core regions of each vortex ( [9], p. 163). This was Prandtl’s opinion when,
in his classic dissertation on turbulence, he continually referred to the entities of
turbulence as being vortices ( [9], pp. 162–163).
It is relevant to note that an irrotational flow is not necessarily an inviscid one.
The outer portion of a vortex is an example [10]. There is further evidence that
the vortices are generated by viscous stresses at the wall and then convect across
the flow [1]. Further there is evidence that initially all the vortices are of the same
strength [11]. Finally to complete the physics description, Kelvin’s theorem would
state that each vortex is convected without change in strength in the potential flow
velocity field generated by all the others [3]; this point was made by Prandtl ( [9],
p. 163). All this, by Kelvin’s law [3], is consistent with vortices of turbulence being
embedded within a largely irrotational flow.
This detailed consideration of the physics of turbulent flow now enables progress
to be made using dimensional analysis. The first step follows from the forego-
184
8 Analytical Results
ing flow examples which enables the viscosity to be excluded from Equations 8.1
and 8.2.
8.5 The Turbulent-Power Law
The nature of turbulence is illustrated in Figure 8.5 showing a typical plot of velocity
at one point in the flow of a turbulent boundary layer as it varies with time [11]].
This shows the turbulence appearing in bursts. The length of each burst is seen to
be closely constant, the mean velocity remaining so at this position. Also the height
of the fluctuation in velocity is also closely constant. The circulation, K, which
measures the angular velocity of a vortex and hence the velocity fluctuation, is thus
constant. The intermittency, I , which is the ratio of that part of the signal occupied
by the vortices is then related to the vortex spacing by:
D
utb
:
I
(8.3)
A typical variation of the intermittency across a turbulent boundary layer is sketched
in Figure 8.6. Considering Equation 8.3 this diagram shows that:
as y ! 1;
I ! 0;
u ! U;
!1
Figure 8.5 Typical velocity-time traces of turbulent bursts in a partially turbulent flow: traces are
shown for two values of the intermittency, Figure 8.6 Variation of intermittency of turbulence across
a boundary-layer flow
8.5 The Turbulent-Power Law
185
Figure 8.7 Variation of vortex spacing across a fully
turbulent boundary-layer flow
and,
as y ! 0;
I ! 1;
u ! 0;
!0
A simple representation of this variation of the vortex spacing satisfying these
boundary conditions is given by:
D ay b
(8.4)
with b < 1. This is sketched in Figure 8.7. The validity of this assumption can be
further justified later.
The convection velocity field is formed from the sum of the convection velocities
of the vortices. The velocity around a vortex falls as the reciprocal of the distance
from its centre [10]. Thus locally within a flow the velocity is mostly induced from
those vortices which are fairly adjacent together with their own velocities [3]. It
follows that in considering the velocity distribution locally away from a surface,
the distance from the surface does not influence the local velocity distribution. The
small change in the mean convection velocity generated by the turbulent vortices,
ıu over the distance from the surface, ıy is now given by:
ıu D f .ıy; K; /
The pi-theorem solution is in Compact Solution 8.1.
This gives:
ıu
ıy
Df
K
which rearranges to:
ıu2
Df
ıyK
ıy
(8.5)
186
8 Analytical Results
Compact Solution 8.1
ıu
L
T
ıy
K
L
L2
L
T
ıu
K
1
L
ıu
K
ıy
1
1
As ıy ! 0 then:
2 @u
D constant ˛
K @y
(8.6)
Substitution from Equation 8.4, and noting that K is a constant, and integrating
gives:
uD
˛K
:y .12b/
2b/
a2 .1
(8.7)
This is the power law which is quite a good approximation over all the boundary layer except for within the viscous sub-layer. This applicability has been found
also for the Prandtl solution giving this power law which was based upon the wall
shear in a pipe flow and upon an assumption about the velocity near to the wall:
so, as Piercy commented, ‘This law — is (rather surprisingly) found to hold closely
throughout the greater part–’ [3]. The demonstration here gives validity of the application across this greater part of the flow.
With the present demonstration of Equation 8.7 it is now possible to reverse
Prandtl’s analysis so as to derive the corresponding power law for the surface friction
which Prandtl had to adopt from experiment.
8.6 Prandtl’s Mixing Length
Reynolds set out time-mean values of the governing equations for turbulence and
showed that a predominant effect of the turbulence in the momentum equations was
from terms such as the mean of the cross-products u0 v 0 . Despite this term being one
in momentum flux it has come to be referred to as a Reynolds stress. Reynolds’
expression for this shear stress is [4]:
D u0 v 0
(8.8)
8.6 Prandtl’s Mixing Length
187
Compact Solution 8.2
K
ˇ0
L2
T2
L
L2
T
˛
1
˛
K 2
1
L2
2
K 2
1
ˇ0 1
In the boundary layer it comes from the component of the vorticity vector that is
perpendicular to both the y and the flow directions. Its value depends upon both
that of u0 and of v 0 and also the phase angle, , between these two components. So
following the previous arguments, this localised stress can be expressed as:
D f .; K; ; ˇ0 / :
(8.9)
The pi-theorem solution is in Compact Solution 8.2.
This gives the relation:
u0 v 0 2
D f .ˇ0 / :
K2
Substituting from Equation 8.6 gives:
u0 v 0
2 .@u=@y/2
D f .ˇ0 / :
(8.10)
Then, from Equation 8.8,
D 2
@u
@y
2
f .ˇ0 / :
(8.11)
This is Prandtl’s relation ( [9], p. 130) with his mixing length identified with
Œf .ˇ0 /1=2 and with the sign and scaling factor given by the value of f .ˇ0 /.
Taylor used a different form of derivation for this relation but in doing so assumed
a constant value of , a limitation not needed in the present derivation ( [4], p. 163).
Using Prandtl’s relation, values of his mixing length have been calculated from
experimental results ( [4], Art. 160). These can now be identified with the assumed
variation for the vortex spacing proposed in Equation 8.4. Both show closely sim-
188
8 Analytical Results
ilar distributions over most of the outer region of the flow thus giving the further
justification for the latter equation.
Goldstein pointed out that using Prandtl’s relation gives acceptable results for the
velocity profile [4]. This arises from differing values of the mixing length leading to
values of the velocity distribution that are very nearly the same. It is also influenced
by the velocity being obtained from an integration.
8.7 The Log-law
Close to the wall in a flat-plate turbulent boundary layer the turbulence is still mainly
isentropic and so the viscosity is excluded as a variable. The vortex spacing could
be influenced by the presence of the wall and so by the distance y. Should there
now be momentum effects in the flow then the density becomes a variable but not
the velocity u as it is an alternative dependent variable. As the mean rate of shear is
high here then the wall value might be significant. Thus for the vortex spacing here
we have that:
D f .; w ; y/ :
This leads to Compact Solution 8.3.
Because the dimension in T cannot be cancelled, this gives:
D ˛y
(8.12)
with ˛ a coefficient. This result excludes the variables w and .
Measurements of Prandtl’s mixing length show that close to the wall it is proportional to the distance from the wall with a universal value of ˛ [4]. Identifying this
with the vortex spacing confirms Equation 8.12.
Compact Solution 8.3
w
y
M
L3
M
LT2
L
L
w
L2
T2
w
y 2
y
1
T2
1
8.7 The Log-law
189
In this region close to the wall the shear stress is a constant. Substituting this and
Equation 8.12 into Equation 8.11 gives:
1=2
@u
1
1
D
@y
˛y Œf .ˇ0 /1=2
or, writing:
F Œf .ˇ0 /1=2 :
Then:
F u
1=2
.=/
D
1
ln y C ˇ
˛
(8.13)
which is the log-law for the velocity u, in the near-wall region, with ˛F now identified with , the von Karman constant [6]. Further, identifying Prandtl’s mixing
length with in Equation 8.12 gives a relation that is satisfied by experiment in the
near-wall region for which Equation 8.13 is valid [4].
Detailed experiment shows that is a constant that is independent of the Reynolds
number [6, 12]. Then F / 1=˛. As experiment related to Equation 8.12 confirms
that ˛ is not Reynolds number dependent, then neither would be F .1
Near the wall where the log-law is valid, ¤ f .y/. Thus:
w
D
u2 :
(8.14)
For a viscous sub-layer of thickness, ıs , the boundary velocity, us is given by:
@u
us D ıs
:
@y yD0
As,
@u
w D @y yD0
then,
ıs D
1
us u2
The results in [4] that show a Reynolds number dependency are for low values of the Reynolds
number which accord with the flow at the end of transition where the turbulence has not fully
developed.
190
8 Analytical Results
giving:
ısC ıs u
us
D
:
u
(8.15)
In Equation 8.13, putting u D us at y D ıs determines the value of ˇ. Then this
equation with Equation 8.15 becomes:
u
1 yu 1
D ln
ln ısC C ısC :
u
(8.16)
The constant in this equation is written as:
1
B ln ısC C ısC :
(8.17)
An accepted value of the thickness of the viscous sub layer is ısC D 11. With D
0:405 [6, 12], this gives B D 5:08 a value that agrees well with the experimental
value that Coles derived after extensive review of the existing data [13].
Equation 8.17 also explains the effect of surface roughness because that increases
the value of ıs and hence the value of B [12].
Comparison of Equations 8.13 and 8.16 shows that the viscosity enters into the
log-law only from the imposition of the inner boundary conditions for it is not
present in the former equation. This is consistent with the physics of the flow as
presented here which is one of an isentropic one. It would have been quite inconsistent to have entered earlier the viscosity as a variable governing this almost entirelyisentropic turbulent flow.
8.8 Jet Flow
A jet flow is illustrated in Figure 8.3. Beyond the section A-A the flow is said to
be fully developed and from then on all the profiles of u and of u0 are of the same
shape; that is all profiles are identical when non-dimensionalised in term of the jet
width, b, and of the centre-line values of u and of u0 . This is expressed by:
y u
Df
(8.18)
uc
b
and by:
y u0
Df
:
uc 0
b
(8.19)
The flow is not adjacent to a surface so that there is no further generation of vorticity.
Thus the existing vortices, being convected in the flow, also diffuse outwards [1].
The latter contribution to the motion is smaller than the former and so it might be
8.10 Example II: Particle Abrasion in Flows
191
expected that is uniform across the jet. So, unlike Equations 8.4 and 8.12, with the
outward convection, there would be:
/b:
(8.20)
Identifying with Prandtl’s mixing length as before, this is the basic assumption
made by Tollmien ( [4], Art. 255) which gives good agreement with the experimental
velocity profiles across the jet. The measured velocity towards the outer edge of
the jet is lower than values calculated because diffusion of the vorticity becomes
significant [1].
The same assumption of Equation 8.20 was made by Prandtl for the turbulent
wake flow as solved by Schlichting with equally good results ( [4], Art. 252).
8.9 General Comments
This detailed discussion of turbulence has introduced several valuable lessons. First,
it emphasises the value of care in the preliminary assessment of the physics of a phenomenon by developing argument for the largely isentropic nature of turbulence
once it has been created at surfaces by viscous stresses. Consequently, it is possible
to exclude initially the viscosity as a variable. By postulating that bursts of turbulence are in the form of vortices, it is possible to invoke Kelvin’s theorem for their
movement with the flow and to make use of the expression for the velocity distribution about a vortex. Secondly, it illustrates a case where formal analysis alone without the introduction of empirical coefficients, has barely started so that dimensional
analysis has proved of great value. Thirdly, this example shows how dimensional
analysis can intertwine with standard analysis making the whole possible.
8.10 Example II: Particle Abrasion in Flows
Notation for Example II
A
C
D
Fe
Fp
g
L
`
m
m
P
Surface area
Number concentration of particles
Drag force
Electrostatic force
Pressure force
Gravitational acceleration
Length size of the flow
Length size of particles
Mass of particle
Wear rate
192
8 Analytical Results
p
q
qp
s
U
Pressure
Flow velocity
Particle velocity
Stress
Reference flow velocity
W
"
p
Weight force
Dielectric coefficient
Fluid viscosity
Fluid density
Particle density
Charge density
Another phenomenon that is of considerable concern to engineers is that of the
wear by dust impact within fluid machines [14, 15]. Because of its association with
complex fluid flows, this problem is amenable to solution by dimensional analysis [16, 17].
8.11 The Forces
The forces acting upon a particle as it travels with the flow are four in kind.
First, there is a drag force, D, due to a relative motion between the particle and
the fluid. If this relative velocity is sufficiently small so that the associated Reynolds
number is small then from the discussion of Chapter 1, there is the result that:
D D f qp q ; `; :
(8.21)
Again following the discussion in Chapter 1, this gives from application of the pitheorem that:
D
D constant :
qp q `
(8.22)
Secondly, the weight force is given by:
W / gp `3 :
(8.23)
Thirdly, the force from the pressure gradients is given by:
Fp / `3 p
(8.24)
and this might include a significant buoyancy force. Within the flow,
p D f .; q; ; L/ :
(8.25)
8.11 The Forces
193
Compact Solution 8.4
p
q
L
M
L2 T2
M
L3
L
T
M
LT
L
p
L
T2
L2
T
p
q 2
q
1
L
L
Lp
q 2
qL
1
1
Solution of the pi-theorem takes the form of Compact Solution 8.4.
This gives that:
Lp
qL
Df
:
q 2
(8.26)
The flow through a fluid machine will usually be highly turbulent so that from the
previous discussion in this chapter viscous effects are not significant so that Equation 8.26 reduces to:
Lp
D constant :
q 2
(8.27)
Substitution from Equation 8.24 gives that:
Fp L
D constant :
q 2 `3
(8.28)
Fourthly, the electrostatic force, Fe , is given by:
Fe D
Q2 CL
:
"
(8.29)
A comparison of possible numerical values of D, W and Fe each with Fp shows
that for an air flow both D and Fe are small and W is comparable. However, the
electrostatic force would act across the flow and so could be significant, a point
returned to later. In the case of the weight force, experiment does not distinguish
between the leading edge deposit of particles on the upper and the lower surfaces at
the leading edge of an aerofoil [18]. This suggests that it can be excluded from the
discussion.
194
8 Analytical Results
Compact Solution 8.5
qp q
q
p
L
T
L
T
M
L3
M
L3
`
p
1
L2
T
L
L
L
M
LT
qp q
q
q
1
L
`
L
qL
1
1
Then the variables are:
qp q D f q; p ; ; `; L; :
Application of the pi-theorem gives Compact Solution 8.5.
This shows that:
qp q
p ` Df
; ;
:
q
L qL
This can be rearranged as:
qp q
p p q` qL
Df
;
;
:
q
(8.30)
(8.31)
(8.32)
With the viscous force on a particle being neglected then the second independent
group, which is the associated Reynolds number, can be excluded. Also, with little
influence of the machine Reynolds number upon the overall flow pattern then the
third independent group can also be omitted.
Thus Equation 8.32 reduces to,
qp q
p
Df
:
(8.33)
q
For a flow pattern under these conditions the local velocity, q is given by:
q
D constant
U
(8.34)
8.12 The Wear Rate
195
As qp q q so that qp q 0. Then qp q so that from Equation 8.34, to
a good approximation,
qp / U :
(8.35)
8.12 The Wear Rate
It follows that for the rate of wear, m,
P there is:
m
P D f qp ; m; `; C; A; s
where the stress s, is associated with the material under impact. At the moment it is
assumed that the material of the particles is so much harder and stronger than that of
the surface being abraded so that the numerical value of its material characteristics
will not affect m.
P
The pi-theorem solution is in Compact Solution 8.6.
This gives the result that:
"
#
m`
P
s`3 3 A
Df
; ` C; 2 :
(8.36)
mqp
mqp2
`
If conditions are supposed uniform across A, then m
P will be proportional to A so
that Equation 8.36 reduces to:
"
#
m`
P 3
s`3 3
Df
;` C :
(8.37)
mqp A
mqp2
Compact Solution 8.6
m
P
M
T
qp
L
T
m
M
`
L
C
A
s
1
L3
L2
M
LT2
m
P
m
s
m
1
T
1
LT2
m
P
mqp
s
mqp2
1
L
1
L3
m`
P
mqp
`3 C
A
`2
s`3
mqp2
1
1
1
1
196
8 Analytical Results
If each impact is an isolated occurrence and the contribution to m
P is the same for
each impact, then m
P will be proportional to the number of impacts which is in turn
proportional to C . Thus Equation 8.37 reduces further to:
"
#
m
P
s`3
Df
:
(8.38)
mqp AC
mqp2
Suppose that the particles cause damage by knocking off protrusions upon impacting
the surface. If the relevant breaking stress is halved and the particle exerts the same
force, then the area of fracture doubles. This implies a doubling of the number of
protrusions and so doubling m.
P Thus m
P / 1=s and so Equation 8.38 reduces to
ms`
P 3
D constant :
m2 qp3 A C
(8.39)
Combining Equation 8.35 with Equation 8.39 gives
ms`
P 3
D constant :
m2 U 3 AC
(8.40)
ms
P
D constant :
p2 `3 U 3 AC
(8.41)
This can be rewritten as
If Equation 8.33 is taken into account then,
ms
P
Df
p2 `3 U 3 A C
p
:
(8.42)
It is common practice to define a non-dimensional mass rate by the relation [15],
e
m
P
:
p UCA`3
Inserting this equation into Equation 8.42 gives,
p
es
D
f
:
p U 2
(8.43)
(8.44)
Equation 8.42 shows that the wear rate is proportional to the cube of the velocity
a result given by Truscott as being the most common value quoted by various authors
from the results of experiment and of analysis [19]. Another result from the same
source that the wear is proportional to the particle volume is also shown by this
equation. Further Truscott also quotes the finding by several authors that the wear
rate is proportional to the particle concentration for low concentrations and this also
is shown by Equation 8.42.
8.13 Classes of Impact
197
Truscott’s review showed that several sources indicate that,
m
P / p :
Accepting this result then Equation 8.42 becomes,
ms
P
D constant :
p `3 U 3 AC
(8.45)
8.13 Classes of Impact
It was found by Goodwin et al. [14] that impact occurred in one of two regimes. In
one, for a smaller size of the particles, the erosion coefficient, e, was proportional to
the square of qp . In the other, for the larger particles, e was independent of particle
size, `. These have been described as respectively Class I and Class II impacts [17].
Goodwin et al found that Class I impacts were associated with a low degree of
fragmentation of the particles whilst Class II accorded with a high amount. A criterion for the upper limit to Class I was found to be ` D constant. At the lower limit
of Class II qp is proportional to `. The impulse imparted by a particle per unit frontal
area is proportional to
p qp `3 =`2 D p qp ` :
The kinetic energy of the particle per unit volume is 12 p qp2 . The ratio of these
two quantities is then proportional to `=qp . A Class II impact thus corresponds to
a higher impulse in comparison with the kinetic energy.
A Class I impact corresponds to Equation 8.44 with Equation 8.35 giving
es
D constant :
p qp2
(8.46)
Whilst this equation gives the above mentioned experimental result that e / qp2 [14],
those experiments showed that e / `2 . This indicates that the parameter representing the material property of a surface is better chosen as a force, F rather than the
stress, s. Thus Equation 8.46 becomes
eF
D constant :
p qp2 `2
(8.47)
This gives the representation, shown by experiment, of both the velocity and the size
effects for Class I impacts.
198
8 Analytical Results
8.14 Particle Fragmentation
For Class II impacts we introduce a fractional degree of fragmentation, t. Thus
t D f p ; `; qp ; sp ;
where sp is a stress associated with particle fracture.
The solution of the pi-theorem is in Compact Solution 8.7.
This gives that:
"
#
sp
tDf
:
p qp2
Thus introducing this extra parameter, t, Equation 8.46 becomes,
"
#
sp
es
Df
:
p qp2
p qp2
(8.48)
The experimental results of Goodwin et al. show that for Class II impacts e / qp2:3 .
This is satisfied by Equation 8.48 if,
"
#1:15
es sp
D constant
sp p qp2
(8.49)
in which the independence of e with ` is consistent with experiment [14]. Both
Equations 8.47 and 8.49 satisfy the experimental result that the erosion rate is little
effected by concentration [14].
However, the previously mentioned electrostatic force can influence the paths of
the particles [20] and it has been suggested that this might have caused the 7 % scatter of the particles found by Goodwin et al. when fed into a vacuum chamber [14].
If this is significant then a further corresponding non-dimensional group is formed
Compact Solution 8.7
t
p
`
qp
sp
1
M
L3
L
L
T
M
LT2
sp
p
L2
T2
sp
p qp2
1
8.17 Example III: Electrostatic Fluid Charging
199
from,
Fe
Q2 CL
/
:
Fp
"q 2 `3
This introduces the concentration. Expansion of a space charge cloud [21] would
lower the local value of C and this would be consistent with an observed slight drop
of " with a large increase in the nominal value of C [14].
8.15 Particle Shape
Goodwin et al. found experimentally that, for Class II impacts e / .1 /2:3 where
is a shape factor of the particles. Introducing this into Equation 8.49 gives,
"
#1:15
sp
es
D constant :
sp p qp2 .1 /2
(8.50)
This then reproduces the experimental determination of the variation of e with qp , `
and and the independence from C .
8.16 Concluding Comments
Again, dimensional analysis has added to the understanding of a complicated phenomenon and enabled the derivation of useful relationships between variables which
had previously been obtained by several experimenters. It also has given insight into
the nature of the phenomenon and hence of the relevant independent variables.
8.17 Example III: Electrostatic Fluid Charging
Notation for Example III
DC , D
E
F
is
k
`
t
u
Diffusion coefficients of ions
Electric field strength
Faraday constant
Streaming current
Ion mobility
Reference length
Time
Velocity
200
8 Analytical Results
U
z
Reference velocity
Ion valency
"
Dielectric coefficient
Electrical conductivity
Fluid viscosity
Fluid density
Charge density
8.18 The Physical Phenomenon
When a fluid, either liquid or gas, which contains electrical charges flows past a surface then an electrostatic charge exchange can occur at the surface so that a net
charge is left in the flowing fluid [22]. This has led to hazardous incidents going
back to those involved with the use of printing inks in the 19th century.
This forms another example where use can be made of uncoupled equations,
a detailed study of the physics of the flow showing that the equations governing the
fluid motion are uncoupled from those governing the electrostatics. Thus the former
can first be solved enabling that solution to be introduced into the latter.
Because of the complexity of fluid flows as already discussed, dimensional analysis has been found again to be of considerable help [20,23]. This is now described.
The phenomenon is sketched in Figure 8.8. Liquid is contained in a metal and
earthed reservoir. This liquid contains free charges equally positive and negative so
that the net overall charge, , is zero. The evidence from industrial practice is that
these ‘ions’ are very small and charged impurity particles [22]. On flowing into the
pipe, charges separate at the pipe wall so that a net current, is , usually negative,
flows to earth. This leaves a net positive charge flowing out of the pipe. The latter
charge can then form an industrial electrostatic hazard.
Figure 8.8 Diagram of the
flow system for the generation
of electrostatic streaming
current
8.20 The Variables and Groups
201
8.19 The Governing Equations
The governing equations have been set out [20]. They are:
d
˙
D D˙ r 2 ˙ r .˙ E/ ;
dt
D r ."E/ ;
D C C ;
˙ D jz˙ j F C˙ k˙ :
In addition to the variables in these equations there are those associated with the
flow pattern which are,
; U; ` and :
Further detailed study has been made of the interaction with the turbulence in the
flow [24].
This phenomenon is different to the previous two described in this chapter because now a careful consideration has to be made of the boundary conditions. Several different proposals have been made for this [22, 25].
8.20 The Variables and Groups
Because of the association of turbulent flows with this large number of variables,
this phenomenon is a most complicated one. But after detailed consideration of the
physics and chemistry involved it can be concluded that the variables can be reduced
to [20, 23],
is ; ; U; `; ; DC ; D ; "; :
Application of the pi-theorem is in Compact Solution 8.8.
Thus,
is2
"U DC D
Df
;
;
;
:
U 3 `3
` U ` U ` U `
This can be reformed as,
is2
Df
U 3 `3
" U ` DC DC
;
;
;
`2 U ` D
:
(8.51)
202
8 Analytical Results
Compact Solution 8.8
is
A
M
L3
U
"
L
T
A2 T4
ML3
`
L
A2 T3
ML3
DC
D
M
LT
L2
T
L2
T
is2
"
ML3
T
"U
U
DC
U
D
U
M
L2
L
L
T3
is2
U 3
M
L
is2
U 3
U
L3
L
is2
U 3 `3
"U
`
U `
DC
U`
D
U`
1
1
1
1
1
8.21 Experimental Verification
Using Equation 8.51 to correlate the results, an experiment was performed using
hydrocarbon liquids [26, 27]. The tests were performed for the turbulent flow in
a tube. For that programme it was intended to study phenomena that might accord
with industrial practice. So the evidence was that the ‘ions’ were charged impurity
particles. It was inferred that then the fourth independent group in Equation 8.51
would be a constant. Other calculations suggested that the influence of DC could be
excluded [22]. So these experiments were based on the reduction of Equation 8.51
to,
is2
" U `
Df
;
:
(8.52)
U 3 `3
`2 Writing,
0
I1
is2
U 3 `3
and with,
Re U `
:
Then a set of these experimental results is illustrated in Figure 8.9. This shows
a family of curves giving the relation between the three non-dimensional groups
of Equation 8.52. The value of the streaming current was corrected for an entry
8.21 Experimental Verification
203
Figure 8.9 Experimental
values of the electrostatic
streaming current: curves for
various values of "=d 2
from 0.039 to 0.51
length condition corresponding to the flow at entry to the pipe [26, 27]. It is seen
that the dependent group for the current covers a wide range of nearly three orders
of magnitude. The abscissa is that of the second independent group which is the
Reynolds number of the flow.
A similar plot of these results is given in Figure 8.10 where now the abscissa is
the first of the independent non-dimensional groups. In both of these figures there
is evidence of linearity on the logarithmic scales and so of the existence of simple
power relations. This enables the final correlation to be shown on the plot of Figure 8.11. There is seen to be a reasonable correlation of the results except for one
set: this is discussed further in the original paper [26]. However, the range of values
Figure 8.10 Nondimensional plot of Figure 8.9
204
8 Analytical Results
Figure 8.11 Non-dimensional correlation of the data of Figure 8.9
is reduced from that in Figure 8.9 of three orders of magnitude to one of a mean
value of about ˙10 %.
8.22 Concluding Comments
This is an example of a very complex phenomenon for which dimensional analysis
has been found to be of considerable help when a complete formal analysis is, so far,
quite impossible. It has shown relations between the various variables that would be
otherwise unknown.
8.23 Example IV: Kinetic Theory of Gases
Notation for Example IV
a
ai
Am
Cˇ
c
D
E
Fˇ
k
ki
kB
Molecular dimension
Ion dimension
Unit atomic mass
Force coefficient
Mean velocity
Diffusion coefficient
Electric field
Inter-molecular force
Radius of gyration
Ion mobility
Boltzmann constant
8.25 Mean-free Path Length in Gases
205
`m
m
M0
N
p
qi
r
R
Tv
T
u
U
z
Mean-free path length
Molecular mass
Molecular mass factor
Number density
Pressure
Ion charge
Reaction radius
Universal gas constant
Characteristic gas temperature
Temperature
Internal energy per mass-unit
Internal energy density
Ion valency
!
Thermal conductivity
Viscosity
Density
Angular velocity
8.24 The Kinetic Theory of Ideal Gases
The discussion now given for the elementary kinetic theory of gases completes the
previous discussion on the need to determine the units-conversion factors for both
temperature and quantity. Further it will show how dimensional analysis can correct an invalid assumption in formal analysis. The discussion is principally for the
characteristics of ideal gases [28].
8.25 Mean-free Path Length in Gases
The mean-free path length is expected to depend upon the interception characteristics and so would then be a function of the mean speed, c, the mean spacing of the
molecules measured as the number per volume unit, N , and a representative size of
the molecule, a. Also it is expected that the application of Newton’s laws require
the inclusion of the variables of the molecular mass, m and a force, Fˇ which is
a measure of the inter-molecular force.
Accepting that the force, Fˇ follows an inverse square law, we define a coefficient, Cˇ by,
Fˇ Cˇ
r2
so that Cˇ D constant for any particular gas.
(8.53)
206
8 Analytical Results
Compact Solution 8.9
`m
c
Na2
m
Cˇ
L
L
T
1
L
M
ML3
T2
Cˇ
m
L3
T2
Cˇ
mc 2
L
`m Na2
Cˇ Na2
mc 2
1
1
The problem is one of a molecule missing the collision area of the adjacent ones.
It is reasonable to suppose that the interception would be affected in the same way
by either doubling the number of particles by pairing each one or by doubling the
area of each particle. In other words, the effect of doubling N so that N2 D 2N1 can
be cancelled by making a22 D 12 a12 ; that is by holding Na2 D constant.
Then the variables are,
`m D f c; Na2 ; m; Cˇ
so that the pi-theorem solution is that of Compact Solution 8.9.
Thus:
Cˇ Na2
`m Na2 D f
:
mc 2
(8.54)
As the density tends to zero, the gas approaches the condition of a perfect gas [28].
Then N ! 0 and `m ! 1 so that Equation 8.54 reduces to:
`m Na2 D constant :
(8.55)
This is the result given by kinetic theory [29].
Effects of the vibrational and rotational motion of a molecule have been excluded
from this discussion. This is justified by their effective inclusion in the value of a.
Also the shape of a molecule, as distinct from its size, has been excluded. The same
argument applies for this.
8.26 The Internal Energy
207
8.26 The Internal Energy
The internal energy per volume unit, U , is expressed by,
U D f .c; N; m; a; k!/ :
(8.56)
The assumption is made that U is dependent upon the vibrational and rotational
motions of the molecules. This is represented by the parameter k! where k is a representative radius of gyration and ! is a representative angular velocity.
Then the pi-theorem gives Compact Solution 8.10.
This gives:
Ua3
3 k!
:
(8.57)
D
f
Na
;
c
mc 2
On a macroscopic scale, U / N so that,
U
Df
N mc 2
k!
c
:
The internal energy per unit mass, u, is given by, U D uAm M0 N so that Equation 8.57 becomes,
uM 0 Am N
k!
D
f
:
c
N mc 2
Noting that,
m D Am M0
Compact Solution 8.10
K
1
2
3
k
U
c
N
m
a
k!
M
LT2
L
T
1
L3
M
L
L
T
U
m
1
LT2
U
mc 2
k!
c
1
L3
1
Ua3
mc 2
1
3
Na3
1
1
1
208
8 Analytical Results
then,
u
c2
Df
k!
c
:
(8.58)
For an ideal gas from macroscopic thermodynamics [28],
u D CV T :
(8.59)
From Table 2.1 and using Equation 2.16 gives:
u
c2
/
CV T m
CV m
D
D constant :
kB T
kB
Then also from Equation 8.58 for this ideal gas approximation,
k!
D constant :
c
(8.60)
From macroscopic thermodynamics and the more general case [28],
uM0
D f .T / :
R
Therefore in Equation 8.58,
k!
Df
c
R
M0 c 2
f .T / :
(8.61)
Using the later Equation 8.71, then,
k!
Df
c
1
f .T / :
T
The right-hand side of this equation cannot satisfy the requirement of equality of
dimensions except for the case just considered of f .T / D T . A further variable
has to be added to the equation. This is the characteristic temperature of the gas, Tv .
Then the equation becomes,
k! D f .c; T; Tv / :
This is solved in Compact Solution 8.11.
Thus the equation becomes,
k!
Df
c
T
Tv
:
(8.62)
8.27 The Pressure and Temperature
209
Compact Solution 8.11
k!
c
T
Tv
L
T
L
T
k!
c
1
T
Tv
1
Substituting this into Equation 8.58 gives,
u
Df
c2
T
Tv
(8.63)
which is a result of kinetic theory.
8.27 The Pressure and Temperature
The pressure, p, on a solid surface is assumed to be a momentum effect of collisions.
On a continuum scale, within a gas the so-called pressure on an imaginary surface
drawn in the gas is from the equality of the interchange of molecules across this
surface to the incident and reflected motions at the solid surface: this implies that
this equality means that the solid surface collisions are perfectly elastic. This is an
important assumption.
Then we have as before,
p D f .c; N; m; a; k!/
(8.64)
it being noted that `m can be excluded by Equation 8.55 as N and a are included.
The pi-theorem solution is in Compact Solution 8.12.
Then for this example:
#
"
1=3
p
1=3 k! Cˇ N
;
:
(8.65)
D f aN ;
c
mc 2
mc 2 N
As mN is the density, , then,
p
c 2
"
Df
aN
1=3
k! Cˇ N 1=3
;
;
c
mc 2
#
:
(8.66)
210
8 Analytical Results
Compact Solution 8.12
K
1
2
3
k
p
c
N
m
a
k!
Cˇ
M
LT2
L
T
1
L3
M
L
L
T
ML3
T2
p
m
Cˇ
m
1
LT2
L3
T2
p
mc 2
k!
c
1
L3
1
p
mc 2 N
1
1
1
p
c 2
Df
Cˇ
mc 2 a
aN 1=3
1
3
Then as N ! 0,
Cˇ
mc 2
L
k!
c
1
3
:
(8.67)
The macroscopic equation of state for an ideal gas defining the temperature T is,
p
R
D
T
M0
(8.68)
so that comparison of Equations 8.67 and 8.68 gives,
M0 c 2
T D
f
R
k!
C
:
(8.69)
:
(8.70)
From Equation 8.62,
M0 c 2
T D
f
R
T
Tv
To retain the definition of temperature for a perfect gas by Equation 8.68 the last
term of Equation 8.70 cannot be effective and so,
M0 c 2
D constant
RT
(8.71)
p
D constant
c2
(8.72)
or, Equation 8.68 becomes,
8.28 The Viscosity
211
which again is a result of kinetic theory.
Also noting that,
m D M0 Am with kB D RAm :
where the dimensional constant RAm is written as kB , the Boltzmann constant. Then
from Equation 8.71,
kB T / mc 2
which is now a deduction of the form of Equation 2.16 and gives another result of
kinetic theory which identifies the temperature with the energy of translation.
8.28 The Viscosity
The viscosity is a measure of the momentum change so that it would depend upon
the values of m, c and N . Also, the motion concerned is related to that along the
mean free path, `m , which has a component perpendicular to the continuum shear
so that `m is of greater significance than a. The viscosity, , is then written as:
D f .N; m; c; `m / :
Solution by the pi-theorem is in Compact Solution 8.13.
This gives:
`2m
D f N `3m :
mc
(8.73)
Compact Solution 8.13
K
1
2
3
k
c
N
m
`m
M
LT
L
T
1
L3
M
L
m
1
LT
mc
1
L2
`2m
mc
N `3m
1
1
3
1
212
8 Analytical Results
The viscosity is a measure of the momentum flux per unit area and so is proportional
to N . Thus Equation 8.73 reduces to:
D constant
N mc`m
or,
D constant
c`m
(8.74)
the result given by kinetic theory.
For a fixed continuum shear rate, is proportional to the rate of momentum
interchange. In this uniform shear the momentum interchange would be proportional
to the distance travelled by a molecule in a direction perpendicular to the shear and
between collisions. This distance is proportional to `m and this would accord with
/ `m as is shown by Equation 8.74.
8.29 The Thermal Conductivity
The thermal conductivity, , is related, in its definition, to temperature. Its dimensions include one in temperature, so the dimensional constant, kB , must be introduced as a units conversion factor. The argument for preferring the variable, `m , on
physical grounds are as for the case of viscosity.
Thus we put,
D f ŒkB ; N; `m ; c; m :
Application of the pi-theorem gives Compact Solution 8.14.
It is noted that the variable m has been excluded by this application of dimensional analysis.
Thus:
`2m
D f N `3m :
kB c
(8.75)
The thermal conductivity, , is a measure of an energy interchange on a continuum
scale along a temperature gradient. It is a measure of the transfer of internal energy
and so of molecular kinetic energy through collisions. Thus on a macroscopic scale
it would be proportional to the total interchange and so to the value of N . Thus
Equation 8.75 becomes:
D constant
N kB c`m :
which is the result given by kinetic theory.
8.30 Diffusion
213
Compact Solution 8.14
K
1
2
3
k
kB
N
`m
c
m
ML
T3 ML2
T2 1
L3
L
L
T
M
kB
m
kB
1
LT
T2 L2
kB c
mc 2
kB
1
L2
`2m
kB c
N `3m
1
1
3
1
8.30 Diffusion
Using the previous consideration of the physics, the diffusion coefficient, D, is written as,
D D f .m; c; N; `m / :
Solution takes the form of Compact Solution 8.15 which gives,
D
D f N `3m :
c`m
(8.76)
The diffusion coefficient is defined as a measure of the ratio of the rate of transport
of N particles to the value of N . From this thermodynamic definition, D ¤ f .N /.
Compact Solution 8.15
D
m
c
N
`m
L2
T
M
L
T
1
L3
L
D
c
L
D
c`m
N `3m
1
1
214
8 Analytical Results
Thus Equation 8.76 reduces to,
D
D constant :
c`m
(8.77)
Again the result of kinetic theory.
8.31 Electrical Mobility
The electrical mobility, ki , is a measure of the velocity which results from a balance
between the electrostatic force on a particle of charge, qi , and the momentum interactions with the uncharged molecules. If Ohm’s law is invoked then this velocity is
defined as equal to ki E so that ki ¤ f .E/. Thus the mobility is expressed by,
ki D f .qi ; m; N; c; a; ai / :
The solution by the pi-theorem gives Compact Solution 8.16.
This results in,
h
ki cm
ai i
D f Na3 ;
:
qi a
a
(8.78)
As with pressure, on a continuum scale the phenomenon is a measure of a summation of momentum effects. Then m and N are combined in the product mN so that
Compact Solution 8.16
K
1
2
3
4
k
ki
qi
m
N
c
a
ai
AT2
M
AT
M
1
L3
L
T
L
L
ki m
AT2
ki m
qi
T
ki mc
qi
L
ki mc
qi a
Na3
1
1
4
1
ai
a
1
1
8.31 Electrical Mobility
215
Equation 8.78 becomes:
ha i
ki cmNa2
i
Df
jzje
a
(8.79)
where the charge qi D jzje. This relation shows the result that ki / 1=N / 1=.
From Equation 8.55, Equation 8.79 becomes,
ha i
ki cm
i
Df
jzje`m
a
(8.80)
ki cm
D constant
jzje`m
(8.81)
and when ai / a this becomes,
which is the result given by kinetic theory.
Further substitution from Equation 8.74 into Equation 8.80 gives,
ha i
ki i
D
f
jzjeN `2m
a
or from Equation 8.55,
ha i
ki Na4
i
Df
:
jzje
a
(8.82)
In liquids, `m is replaced as a significant parameter by the mean molecular spacing
sm [30] given by,
sm D 1=N 1=3 :
Thus Equation 8.82 becomes,
ha i
ki a4
i
D
f
:
3
jzjesm
a
If sm =a D constant, this further reduces to,
ha i
ki ai
i
Df
:
jzje
a
(8.83)
This is known as Walden’s rule for the mobility in liquids as it is related to the
viscosity.
The standard derivation given in the literature invokes the use of Stokes’ law
for the motion of a sphere at very low Reynolds number. However that law is for
a continuum flow and so does not stand examination for this microscopic motion;
216
8 Analytical Results
thus that standard form of derivation is fundamentally flawed. Here it is derived
quite satisfactorily from a rigorous use of dimensional analysis.
Some experimental evidence [31] shows that for ions above a certain size then
the right hand side of Equation 8.83 is a constant so that,
ki ai
D constant :
jzje
(8.84)
For ki ¤ f .z/ then from this equation,
ai / jzj :
There is a degree of support for this from experiment [32].
8.32 The Einstein Relation
The ionic velocity, v, being given by ki E and the ionic force F being given by ejzjE
then,
v
ki
D
F
ejzj
which from Equations 2.16, 8.77, and 8.81 gives,
v
`m
/
;
F
mc
D
/
;
mc 2
D
/
:
kB T
This result is the Einstein relation.
8.33 Summarised Results
The present discussion provides a combination of the rigorous derivations of dimensional analysis with some plausible assumptions of the physical character of
the various phenomena. These derivations set out the physical difference between
pressure and mobility on the one hand where the quantity .N m/ is a significant
parameter and the viscosity, diffusion and conductivity where `m is the significant
parameter. In all cases the standard results of elementary kinetic theory are readily
obtained.
References
217
It is also of interest in that dimensional analysis produces the relation for mobility in a liquid which so far has required the unjustified use for this microscopic
phenomenon of Stokes’ relation which is valid for only the low Reynolds number
continuum or macroscopic flow.
8.34 Concluding Comments
The first three phenomena discussed in this Chapter demonstrate cases of dimensional analysis succeeding where formal analysis is still not successful. It is noted
that all three phenomena involve flows that are turbulent. Further, for the second and
third examples there remain some doubts as to the physics of the events while in the
third case there are further doubts on the electrical-chemical behaviour. The fourth
case demonstrates the importance of clarifying the physics for a correct setting out
of the independent variables and the dimensions to be needed. It also corrects an
existing basic fault in a derivation.
Exercises
8.1
An electrical charge exchange can occur when a liquid of very low conductivity flows past a metal surface. This exchange is measured as a streaming
current. For a laminar flow in a pipe and with the notation of:
is
l
Re
U
"
0
Streaming current
Representative size
Reynolds number; .U l/=
Reference velocity
Dielectric coefficient of liquid
Zero-charge conductivity of liquid
Viscosity of liquid
Density of liquid
obtain a relation for the streaming current. Then show that as the conductivity
value tends to zero:
is / 0 R e :
References
1. J C Gibbings. Diffusion of the intermittency across the boundary layer in transition, J. Mech.
Eng. Sci., Proc. I. Mech. E., Vol. 217, Pt. C, pp. 1339–1344, 2003.
2. P W Bearman. An investigation of the forces on flat plates in turbulent flow, National Physical
Laboratory, Aero Rep. No. 1296, April 1969.
218
8 Analytical Results
3. N A V Piercy. Aerodynamics, English Univ. Press, 1937.
4. S Goldstein (Ed.). Modern developments in fluid dynamics, p. 97, Clarendon Press, Oxford,
(Dover, 1965), 1938.
5. J E A John. Gas Dynamics, Art. 17.3, Allyn & Bacon, Boston USA, 1969.
6. J C Gibbings. On the measurement of skin friction from the turbulent velocity profile, Flow
Meas. Instrum., Vol. 7, No. 2, pp. 99–107, June 1996.
7. S Masuda, M Washizu. Ionic charging of a very high resistivity spherical particle, J. Electrost.,
Vol. 6, pp. 57–67, 1979.
8. H Lei, L-Z Wang, Z-N Wu. EHD turbulent flow and Monte-Carlo simulation for particle
charging and tracing in a wire-plate electrostatic precipitator, J. Electrost., Vol. 66, No. 3,
4, pp. 130–141, March 2008.
9. L Prandtl. The mechanics of viscous fluids, Aerodynamic Theory (Ed. W F Durand), Vol. 3,
Div. G, pp. 34–208 & plates I to V, Springer, Berlin, 1935 (see p 163).
10. J C Gibbings. Thermomechanics, Art. 11.11, Pergamon Press, Oxford, 1970.
11. J Madadnia. Experimental study of stability and transition of boundary layer flow, PhD. Thesis, University of Liverpool, September 1989. (See Figs. 6.16c, 7.2a).
12. J C Gibbings, S M Al-Shukri. Effects of sandpaper roughness and stream turbulence on the
turbulent boundary layer, Jour. Mech. Eng. Sci., Proc. Inst. Mech. Engrs., Vol. 213, Part C,
pp. 507–515, 1999.
13. Proc. Computation of turbulent boundary layers, 1968 AFOSR-IFP-Stanford Conference,
Vol. 1 and 2, Ed. S J Kline et al., Stanford University, 1969.
14. J E Goodwin, W Sage, G P Tilly. Study of erosion by solid particles, Inst. Mech. Eng., Proc.,
Vol. 184, Pt. 1, No. 15, pp. 279–292, 1969–1970.
15. G P Tilly, W Sage. (Communication of Ref. 16), J. Mech. Eng. Sci., Vol. 14, No. 3, p. 227,
1972.
16. J C Gibbings. Dimensional analysis of wear by particle impact in fluid flows, J. Mech. Eng.
Sci., Vol. 13, No. 4, pp. 234–236, 1971.
17. J C Gibbings. Dimensional analysis of wear by particle impact in fluid flows, J. Mech. Eng.
Sci., Vol. 14, No. 3, pp. 227–228, 1972.
18. V H Gray, U H von Glahn, Effect of ice and frost formations on drag of NACA 651 -212 airfoil
for various modes of thermal ice protection, NACA Tech Note 2962, June 1953.
19. G F Truscott. A literature survey on abrasive wear in hydraulic machinery, Br. Hydromech.
Res. Assoc., Tech. Note 1079, October 1970.
20. J C Gibbings. Non-dimensional groups describing electrostatic charging in moving fluids,
Electrochim. Acta, Vol. 12, pp. 106–110, 1967.
21. Schon G, Masuda S. Expansion of a space-charge cloud, J. Appl. Phys. Vol. 2, Ser. 2, p. 115,
1969.
22. J C Gibbings. Interaction of electrostatics and fluid motion. Electrostatics 1979, Conference
Ser. No. 48, Inst. Phys., pp. 145–160, London, 1979.
23. J C Gibbings, E T Hignett, Dimensional analysis of electrostatic streaming current, Electrochim. Acta, Vol. 11, pp. 815–826, 1966.
24. J C Gibbings. Electrostatic transport equation for turbulent flow, J. Electrostatics, Vol. 17,
pp. 29–45, 1985.
25. J C Gibbings. On the charging current and conductivity of dielectric liquids, J. Electrost.,
Vol. 19, pp. 115–119, 1987.
26. E T Hignett, J C Gibbings. Electrostatic streaming current developed in the turbulent flow
through a pipe, J. Electroanal. Chem., Vol. 16, pp. 239–249, 1968.
27. E T Hignett, J C Gibbings. The entry correction in the electrostatic charging of fluids flowing
through pipes, J. Electroanal. Chem., Vol. 9, pp. 260–266, 1965.
28. J C Gibbings. Thermomechanics, Sec. 9.9, Pergamon, 1970.
29. I Estermann. Gases at low densities, (in) Thermodynamics and physics of matter, (Ed. F D
Rossini), High Speed Aerodynamics and Jet Propulsion, Vol. 1, Oxford, P. 738, 1955.
30. R A Robinson, R H Stokes. Electrolyte solutions, p. 125, Fig. 6.1, Butterworths, London,
1970.
31. loc. cit., p. 126, Table 6.3.
32. loc. cit., p. 125, Fig. 6.1.
Chapter 9
Model Testing
– fifty Lockheed Hudson patrol bombers — were lost at sea
without trace. – thanks to that model ditching-technique worked
out by D C McPhail – there was a rapid increase in the number
of Hudson crews rescued.
R. Turnhill, A. Reed
Notation
A
c
cw
Cd
Cf
CF
d
D
e
E
F
Fr
g
h
H
I
j
k
L
`
m
Ma
n
p
pa
p0
pv
P
Pm
Cross-sectional area
Wetted circumference
Wave speed
Drag coefficient
Friction coefficient
Force coefficient
Diameter; depth; grain size
Diffusion coefficient; drag
Spring elasticity
Young’s modulus
Force
Froude number
Acceleration from gravity
Water depth
Cylinder length
Second moment of area
Current density
Ratio of extensions; torsional radius of gyration
Wave length
Representative size
Mass; mass per unit length
Mach number
Rotational velocity; size ratio; Manning factor
Pressure
Atmospheric pressure
Free-stream pressure; Poisson ratio
Vapour pressure
Power
End load
J.C. Gibbings, Dimensional Analysis. © Springer 2011
219
220
9 Model Testing
Q
r
rv
r
R
Re
R0
t
V
w
wP
W
y
y0
Torque
Coordinate; scale ratio
Vertical scale factor
Slope factor
Ship resistance; cylinder radius
Reynolds number
Equivalent radius
Time; thickness
Velocity
Weight per unit length; stream width
Sediment mass-flow rate per unit width
Weight
Deflexion
Asymmetrical offset
ˇ0
ı
f
"
"f
"t
˘
B
g
0
w
'
Units-conversion factor, angle
Ratio of specific heats
Deflexion
Torsional oscillation frequency
Dielectric coefficient; roughness height
Flexural elastic modulus
Torsional elastic modulus
Channel slope
Electrical conductivity
Viscosity
Non-dimensional group
Density
Bridge ‘density’
Grain density
Surface tension coefficient
Stress
Wall shear stress
Electrical potential
9.1 The Application of Model Testing
Chapter 4 describes how the earliest applications of dimensional analysis were to
the testing of reduced size models of full-scale systems. The first two practical uses
were to the flow in pipes and the aerodynamic forces on ropes in an airflow. From
then on tests of scale models of various components of aeroplanes became common
practice to be followed by the testing of models of complete aeroplanes.
This application to model testing, where a test of the full-size system would not
be possible, has developed extensively since those early days so that models of such
9.2 The Essence of Model Testing
221
as ships, rivers, sea defences, road vehicles, windmills, buildings, artistic features
and bridges invariably are now the subject of testing.
Yet still there have been cases where dimensional analysis has not been applied so
causing considerable inconvenience. Once it was found that a large chemical plant
was constructed based upon a pilot plant which in turn was based on laboratory
experiments. It was found that at full-scale a chemical reaction occurred which was
quite different from the one required and which latter had been obtained both in the
laboratory tests and then in the pilot plant.
Another example occurred when an international competition was held by a city
for the design of an artistic feature. Without it having been the subject of model tests,
the winning artist was given his valuable prize before the committee of artists and
politicians was told by engineers that the shape was possibly the most favourable
one to be the subject of serious aerodynamic oscillation in a wind: it was never
built.
A contrasting example was reported by Chichester [1,3]. He had a yacht specially
built for his epic around the world solo voyage. This famous boat, called Gipsy Moth
IV, was found to have two most serious and highly inconvenient faults. First, the keel
was of an inadequate size and weight and secondly, there was not a suitable balance
between the mizzen sail and the headsails. After his triumphant return he found
that a prototype of a commercial model of the yacht had revealed these two faults
which were readily rectified by design changes to both model and full-scale boat.
Understandably, Chichester reported his dismay that these model tests had not been
performed before the full-scale design was determined [3]. As he wrote:
‘– what a pity that the designers of Gipsy Moth IV did not have time to make a model to sail
in the Round Pond before the boat was built! What an immense amount of trouble, worry
and effort this would have saved me, by discovering Gipsy Moth’s vicious faults and curing
them before the voyage!’
A similar occasion occurred over the design of the Skylon artistic feature and the
very large Exhibition Dome building, both erected for the London 1951 Festival.
Wind tunnel tests were made at a late stage resulting in extensive modifications to
these two structures that quite spoilt the designers’ planned appearances [2].
Model testing is a requisite of careful design being based upon this important use
of dimensional analysis [3]. This is now discussed.
9.2 The Essence of Model Testing
Suppose that some phenomenon can be expressed by the pi-theorem through a relation, for example, between the non-dimensional groups such as:
˘1 D f Œ˘2 ; ˘3 :
(9.1)
This function being determined, then graphically it could be illustrated as in Figure 9.1. When the full-scale phenomenon is represented by the point ‘D’ on this
222
9 Model Testing
Figure 9.1 Illustration of a relation between three pi-groups
family of curves then, as denoted on this diagram, the ˘ groups will have the following values:
˘1 D A ;
˘2 D B ;
(9.2)
˘3 D C :
This gives the conditions for a model test. For it is required that the ˘ groups have
values in the test that are equal to these full-scale values at ‘D’. Then, in this test, by
meeting the last two relations of Equation 9.2, it will follow from Equation 9.1 that
also the first of the relations of Equation 9.2 will be satisfied.
In practice it will often happen that the model test will be performed for a range
of values of the pi-groups so covering a range of the full size behaviour. Some
examples are now described.
9.3 The Windmill
A piece of equipment, used especially in agricultural areas in North America, is the
windmill used to pump water. It is constructed with windmill blades formed from
sheet metal and with a complex supporting structure.
Following the discussion in Chapter 1 on the nature of the airflow past sharp
edges, the numerical value of the air viscosity would be expected to have little or no
effect upon the performance of a windmill of this form. So the shaft-power output,
P , would depend upon the diameter, d , the wind-speed, V , the rotational speed n
9.3 The Windmill
223
Compact Solution 9.1
P
d
V
n
ML2
T3
L
L
T
1
T
M
L3
P
L5
T3
P
n3
V
n
L5
L
P
n3 d 5
V
nd
1
1
and the air density, . Or,
P D f .d; V; n; / :
(9.3)
The solution for the application of the pi-theorem is in Compact Solution 9.1.
This gives that:
P
V
D
f
:
(9.4)
n3 d 5
nd
Using a suffix ‘m’ to indicate the conditions of the model test, then from Equation 9.4 the requirement is that,
It follows that:
Vm
V
D
:
nm dm
nd
(9.5)
Pm
P
D
:
3
5
m nm dm
n3 d 5
(9.6)
This test might be set up in a wind tunnel. For a model of 1/10 th of the full-size
windmill, then from Equation 9.5,
Vm n
dm
D
D 0:1 :
V nm
d
(9.7)
Running the wind tunnel so that, Vm D 3 V then,
nm
D 3:10 D 30 :
n
(9.8)
With the torque on the model windmill loaded so that it runs at 6020 rpm, which
is nm D 630:4 rad s1 , then the full-size windmill would run at 630:4=30 D
21:0 rad s1 . With the corresponding torque on the model windmill so loaded as
to be measured at 0.95 N m, the model power would be 0:95 630:4 D 599 W. With
224
9 Model Testing
the test air density the same as the atmospheric value then from Equation 9.6 the
full size power would be given by,
P D Pm
n
nm
3 d
dm
5
so that:
P D 599 303 105 D 2219 W :
There is one further important point to be made about such a test which will be
returned to later. It is the matter of the stress in the model windmill. With the torque,
Q, we have that,
Stress /
Q 1
Q
P
2 / 3 /
:
d d
d
nd 3
For this example under load the ratio of the stresses is given by,
3
model stress
Pm n
d
0:27:103
D
D
D 9:0
full-size stress
P nm
dm
30
This immediately reveals a problem in the use of suitable material to be used for the
construction of the model windmill.
9.4 The Oil-insulated Transformer
Very large and very high voltage transformers rely on the use of extremely pure oil
as an insulator. Because of their high degree of purity, these oils have an extremely
low electrical conductivity. Then problems arise over leakage current, over electrical field induced breakdown and over fires and explosions started by electrostatic
discharging especially during the process of filling. The use of small-scale model
tests are a clear requirement for these very large plants. Suppose a model is to be
constructed to investigate the steady state voltage potential in the oil. Then the potential, ', could be regarded [4] as a function of a boundary reference potential, '0 ,
a coordinate, r, a representative size, `, the conductivity, , the dielectric coefficient,
" and the coefficient of diffusion of the conducting ions, D. So that,
' D f .'0 ; r; `; ; "; D/ :
Solving for application of the pi-theorem gives Compact Solution 9.2.
This gives the result that,
'
r `2
Df
;
:
'0
` "D
(9.9)
(9.10)
9.4 The Oil-insulated Transformer
225
Compact Solution 9.2
'
'0
ML2
AT3
ML2
AT3
'2 "
r
`
"
D
L
A2 T3
ML3
A2 T4
ML3
L2
T
'02 "
"
ML
T2
ML
T2
1
T
'
'0
1
L
"D
1
L2
r
`
`2
"D
1
1
The non-dimensional group, r=` implies that measurements of ' on a model of the
transformer are to be made at relative positions corresponding to the full-scale ones.
From the other independent group there is,
2
2
`
`
D
:
(9.11)
"D m
"D
Because the representative length appears in this group then in the model test one
or more of the electrical parameters would have to have a different value from the
full-scale one. Because the conductivity at full-scale is so extremely low it is readily
possible to increase considerably the value for the model test by adding a suitable
electrolyte. On the other hand, having the same values of " and D at model scale as
at full scale makes the experimental conditions easy to set.
Supposing, for example, that it is arranged that m D 100, then from Equation 9.11,
1=2
`m
D
D 0:1 :
`
m
That is, a 1/10 th scale model becomes feasible. The reference voltage, '0 , in the
model test is at choice, a reduced value making experiment more convenient. By
Equation 9.10, the measured values of ' are then scaled down in proportion.
The current density, j , through the insulating electrolyte varies with the time, t ,
after the initial application of the reference potential, '0 . Then,
j D f .'0 ; `; ; "; D; t / :
226
9 Model Testing
Compact Solution 9.3
j
'0
A
L2
ML2
AT3
`
"
D
t
L
A2 T3
ML3
A2 T 4
L2
T
ML3
T
j2
'02 "
M
LT3
ML
T3
T
j2
'02 2
1
L2
"D
t
"
L2
1
j 2 `2
'02 2
"D
`2
1
1
The solution is in Compact Solution 9.3.
It thus follows that,
j`
"D t
Df
;
:
'0 `2 "
(9.12)
From the second independent non-dimensional group of Equation 9.12, for the
model test,
t
t
D
:
" m
"
For the present numerical example it follows that,
tm
"m
D
t
m "
so that tm D t=100; a reduced time-scale that could be convenient experimentally.
The applied electric field will be proportional to '0 =`. If it is desired to keep this the
same in the model test as at full-scale to give equivalent field breakdown conditions,
then the applied potential in the model test, '0 , will be 1/10 th of the full-scale
potential, again easing experimental conditions. It follows from Equation 9.12 that,
j`
j`
D
:
'0 m
'0 Thus for this example, jm =j D 100. Finally, this leakage current, which is proportional to j `2 , will have the same value in the model test as at full-scale again
assisting measurement in the model experiment.
9.6 Inapplicability of Hooke’s Law of Elasticity
227
Figure 9.2 Spring loaded
restraint under impact
Compact Solution 9.4
m
M
V0
e
ı
L
T
T2
L
me
T2
meV02
L2
M
meV02
ı2
1
9.5 Collision Against a Spring Restraint
A spring restraint is used for protection against damage from an impact. The applications range over such as packaging to the restraint of trains arriving at a terminus.
The case is illustrated in Figure 9.2.
With a mass, m, travelling at an initial velocity, V0 , and impacting against a linear
spring of elasticity, e; then the initial deflexion of the spring to a rest position, ı, that
is, before any rebound, is given as:
ı D f Œm; V0 ; e :
Application of the pi-theorem leads to Compact Solution 9.4.
Thus we have:
ı2
D constant :
meV 20
(9.13)
(9.14)
A model test only requires a single set of measurements so as to determine the value
of the constant in Equation 9.14.
9.6 Inapplicability of Hooke’s Law of Elasticity
There are phenomena in which there is a non-compliance with a proportionality between load and deflexion at the point of application of the load. This is because there
are many cases of purely elastic structures which contain elements that individually
228
9 Model Testing
Figure 9.3 Suspension by
elastic strings
obey Hooke’s law but for which this does not apply for the structure as a whole. Thus
care must be taken when enhancing an answer from use of dimensional analysis by
introducing that proportionality between load and deflexion.
A simple example is that of an elastic string, initially straight, and suspended horizontally at each end. It is then loaded centrally with a weight of W giving a vertical
deflexion of ı. This is illustrated in Figure 9.3 and was described by Southwell [5].
Southwell derived the result that:
"
#
2
`
F D ı 1
;
(9.15)
k
.`2 C ı 2 /1=2
where k is the ratio of the extension in each half of the string to the tension in it.
Then for ı=` 1 this becomes:
F D
1 ı3
C
k `2
(9.16)
so that, to first order and for small deflexions, the force is proportional to the cube
of the deflexion.
9.7 Limitation to Elastic Deformation
Southwell went on to show that Hooke’s law also failed to apply to the case of elastic
bending of struts. The bending of a single strut is now considered. A real strut will
have some degree of eccentricity in both the shape of the cross-section and of the
asymmetry of the end load: this is sketched in Figure 9.4. Denoting this off-set by
the amount of y0 and with a maximum sideways deflexion y then the governing
differential equation is,
d2 y
y
EI 2
1 CP
D 0:
dx y0
y0
This shows that these two variables appear in only the combination of y=y0 [5].
Then, again within the elastic limit, the variables are:
.y=y0 / D f ŒP; `; I; E :
(9.17)
9.7 Limitation to Elastic Deformation
229
Figure 9.4 End loading of
a strut
Compact Solution 9.5
P
E
I
`
y
y0
ML
T2
M
LT2
L4
L
1
P
E
2
L
P
E`2
I
`4
1
1
The pi-theorem solution is in Compact Solution 9.5.
This application of the pi-theorem involves the two effective dimensions L and
M=T2 . The result is:
y
P I
Df
;
:
(9.18)
y0
E`2 `4
For small deflexions, there is elastic bending which is governed by the parameter
EI . Thus Equation 9.18 reduces to:
2
y
P`
Df
:
(9.19)
y0
EI
230
9 Model Testing
This is the correct answer for, as Southwell showed [5], in this case the analytical
solution gives:
" #
y
` P 1=2
D sec
(9.20)
y0
2 EI
so that y is not proportional to the load. However, when the load is small then
a series expansion leads to:
y y0 EI
constant
y0 P `2
(9.21)
so that the net deflexion is proportional to the load. This however is only valid for
this approximation.1 Also, from Southwell’s solution of Equation 9.20,
y ! 1 at
` P 1=2
D
2 EI
4
indicating a complete collapse before this value of P .
In a model test then from Equation 9.19,
2
2
P`
P`
D
EI m
EI
and
y
y0
m
y
D
y0
(9.22)
:
(9.23)
With,
I ˛w4 :
Then,
I
w4
D 4 :
Im
wm
Using the same material gives E D Em . Then from Equation 9.22 there is,
Pm
D
P
`
`m
2
Im
D
I
`
`m
2 wm 4
:
w
The artificial case for which y0 is taken as zero gives the analytical result that P `2 =E I D
constant. This solution then gives no relation between the load and the deflexion. It is an example
of unrealistic initial assessment of the physics.
1
9.8 Impossibility of Scale Structural Modelling
231
If the model strut is of the full-size shape proportions, then,
w
wm
D
`
`m
so that,
2
Pm
`m
D
:
P
`
(9.24)
In a model test, there is a need to determine both y0 and the value of the constant in
Equation 9.21. From this equation:
y D y0 C
`2 K
y0 P :
EI
Where K is the constant in Equation 9.21. A linear plot of experimental values of
y against P gives the value of y0 as the intercept at P D 0 and then the value of K
from the slope of this line.
9.8 Impossibility of Scale Structural Modelling
The impact restrained by an elastic spring and the elastic deflection of a strut have
now been modelled in Sections 9.5 and 9.7. There are commonly occurring cases
where either impact or excess deformation results in plastic yield of structures. This
plastic buckling is of considerable importance. Examples are for the collisions of
motor cars and ships.
The plastic flow of metals under these deflexions beyond the elastic limit is complex. It is somewhat akin to the flow of non-Newtonian fluids. Three similarities are
from:
a)
b)
c)
The stress-strain relationship is of a complex form.
There is a velocity effect upon the stress-strain relation.
There is a temperature effect upon the stress-strain relation.
Currently reliance is upon empirically determined data which is used to determine
full-scale behaviour.
The work done in a deformation under load results in an increase, according
to the First Law of Thermodynamic, in the internal energy of the plastic region.
There is an important distinction to be made. In steel within the elastic limit the
Poisson ratio, P0 D 0:29 and the internal energy is almost completely in the form
of strain energy. In the plastic region behaviour is like that of rubber in the elastic
region where P0 D 0:5 and so the change in internal energy is all thermal causing
a temperature rise [6]. This rise reduces the friction shear within the plastic region
so reducing the shear strain. For metal structures the thermal conductivity would be
mostly through the metal adjacent to the plastic deformation.
232
9 Model Testing
Such is this complexity of behaviour that Jones lists twenty two independent
variables leading to eleven non-dimensional groups. The form of these groups prevents scale modelling and so forces a limitation to severely partial scale modelling:
as Jones puts it, ‘–dimensionless retardation – is impossible to satisfy for geometrically similar scaling –’ [7].
As a result, for the impact of motor cars reliance is still placed upon full-size
tests. For ship collisions partial modelling of elements of structure has to be used.
Jones describes a comparatively simple model of the axial buckling compression
of a cylinder as is illustrated in Figure 9.5 [7]. This is reduced to four variables. This
gives:
Pm D f .0 ; H; R/ :
Where the notation is:
Pm
0
H
R
End load
Stress
Length of cylinder
Radius of cylinder.
The solution is in Compact Solution 9.6.
The solution is thus:
Pm
Df
0 R2
Figure 9.5 Axial plastic buckling of a cylinder [7]
H
R
:
9.9 Limitations to Partial Modelling
233
Compact Solution 9.6
Pm
0
H
R
ML
T2
M
LT2
L
L
Pm
0
L2
Pm
0 R2
H
R
1
1
Jones gives the analytical solution as:
Pm
/
0 R2
H
R
3=2
:
The result shown in Figure 9.5 shows how nominally identical cylinders can reveal
different patterns of buckling. Jones describes this as a switching effect that can
occur during compression [7]. The phenomenon of plastic deformation then is so
complex that any scale modelling is subject to severe simplification of the representation of the full-scale behaviour.
9.9 Limitations to Partial Modelling
A straightforward, but intractable, problem often arises when a model test is proposed. Experimental limitations can prevent some or all of the independent nondimensional pi-groups being set to the same value in the model test as in the fullscale phenomenon. The engineer who requires data from experiment to enable competent design then has a limit set to this information.
Where this requirement of equality of the independent groups has, in part, to be
relaxed, resort is made to partial modelling. Then only some of the non-dimensional
pi-groups can have identical numerical values between model and full scale.
This relaxation can sometimes be justified because a group represents the negligible effect of a weak variable. Sometimes use can be made of analytical results
to supplement the partial modelling and sometimes the limitation can be overcome
by extrapolation of data to the full scale criterion. Extrapolation is necessarily of
doubtful accuracy when unsupported by extra information: the examples of Chapter 8 make this very clear. It is in the skill of using supplementary knowledge that
reliable data is obtained.
Examples illustrating these procedures are now given.
234
9 Model Testing
9.10 Full-scale Comparison Method
An early example of overcoming partial modelling was given by Wilbur and Orville
Wright [8]. When they started on the problem of aerial flight they assiduously surveyed all the then current aerodynamic data but came to realise its very doubtful
accuracy and eventually discarded it all and started their own experimental programme.
They built a small wind tunnel and carried out a remarkably intensive test programme on aerofoils and other components [9]. They recognised the small scale of
their tests which we now assess in terms of the low values of the Reynolds number.
Following the discussion of Chapter 1, an aerodynamic force, F , has a corresponding non-dimensional coefficient, CF defined by:
. 1
2 2
CF D F
V `
(9.25)
2
with,
CF D f ŒRe :
(9.26)
For their model tests the Reynolds number based on the wing chord was of the order of 104 whilst their full-size ‘Flier’ powered aircraft flew at a Reynolds number
of over 106 . Having at that time no suitably extensive background of aerodynamic
data, a simple extrapolation of their wind-tunnel results would have been of highly
doubtful accuracy: in the light of present knowledge the error would have been
considerably greater than they could reasonably have estimated [10]. So they tethered one of their large man-carrying gliders to support itself in a steady wind and
measured the forces on it that were transmitted through the supporting cables. The
correction factor, to account for the change in the value of the Reynolds number,
thus derived was of limited validity yet they applied it to other wind-tunnel results
with the obvious success of their final achievement of powered flight. As they said
with justifiable pride; “– all our calculations were shown to have worked out with
absolute exactness so far as we can see –” [8].
9.11 Non-effectiveness of a Single Group
There are many other cases where the model test cannot reproduce the full-scale
value of the Reynolds number but where extrapolation is reliable because the numerical value of the Reynolds number has no effect. A case of this was mentioned
and illustrated in Chapter 1. It makes possible perfectly valid tests in wind tunnels
of such things as very large buildings and suspension bridges both having sharp
corners.
In the case of suspension bridges, the phenomenon of particular concern is the
oscillation in a wind though this can also occur with very large buildings of slender
proportions. This oscillation is of two forms. In one the bluff shape of the bridge
9.11 Non-effectiveness of a Single Group
235
decking induces a cyclic aerodynamic force which can cause the bridge deck to
oscillate up and down against the bending flexibility of the structure. The other is of
far more serious consequence as it involves this bending combined with a torsional
oscillation. When these two modes are suitably coupled in phase an aerodynamic
flutter can occur. This has been long understood in the design of aeroplane wings
and gives a particularly strong and oscillating forcing-load from the aerodynamic
lift.
The latter bridge oscillation at a torsional angular frequency, f with the corresponding units conversion factor of ˇ0 , is a function of the bridge shape and size, `,
the wind speed, V , the air density, , the flexural elastic modulus, "f as a force per
unit area, and the torsional one, "t , the weight of the bridge per unit length, w with
a corresponding mass, m, and the radius of torsional gyration, k.
Tabulating for application of the pi-theorem gives Compact Solution 9.7.
Then the frequency is represented by:
2
ˇ0 f `
` "f `2 "t `2 w` k
Df
;
;
;
;
;
(9.27)
V
m mV 2 mV 2 mV 2 `
where the value of the viscosity is excluded because of the negligible effect of the
Reynolds number.
Whilst the weight of the bridge affects its steady state deflexion and hence the
total stress levels, the oscillation depends upon the mass as discussed in the example
of Section 1.7. Thus excluding w as a variable and rearranging the groups reduces
Equation 9.27 to:
2
ˇ0 f `
`
"f "t k
Df
;
; ;
:
(9.28)
V
m V 2 "f `
Compact Solution 9.7
f
ˇ0
`
V
"f
"t
w
m
k
˛
T
1
˛
L
L
T
M
L3
M
LT2
M
LT2
M
T2
M
L
L
m
"f
m
"t
m
w
m
1
L2
1
T2
1
T2
L
T2
f
V
"f
mV 2
"t
mV 2
w
mV 2
˛
L
1
L2
1
L2
1
L
f ˇ0
V
1
L
f ˇ0 `
V
`2
m
"f `2
mV 2
"t `2
mV 2
w`
mV 2
k
`
1
1
1
1
1
1
236
9 Model Testing
Writing m D B `2 so that B is a bridge ‘density’ then:
ˇ0 f `
"f "t k
Df
;
;
;
:
V
B V 2 "f `
(9.29)
In a model test of scale n, and with the density at atmospheric value so that from the
first independent non-dimensional group of Equation 9.29, the model ‘density’ has
to be the same as the full scale value or:
Bm D B :
(9.30)
"tm
"fm
V2
D
D m2 :
"t
"f
V
(9.31)
km
`m
1
D
D :
k
`
n
(9.32)
fm
Vm `
Vm
D
Dn
:
f
V `m
V
(9.33)
From the next two groups:
From the last group:
Then the dependent group gives:
From Equation 9.33, to avoid too high a frequency in the model test and to avoid too
extreme a loading on the model, the test can be run so that Vm < V . This reduces
the aerodynamic loading because the aerodynamic pressure is proportional to m Vm2 .
Then, from Equation 9.31, "fm < "f which can be achieved by using plastic for the
construction of the model. This can require extra masses being attached to the model
and at suitable locations to satisfy Equations 9.30 and 9.32.
Thus there is considerable flexibility in the experimental design to enable achievement of equality of all the above non-dimensional groups with the sole exception
of a Reynolds number. Here a limitation has been found. In one test series it was
found that a 1/100th scale model did not reproduce the expected behaviour whereas
a 1/20th model gave acceptable results. The Reynolds number discontinuity was as
seen in Figure 1.5.
The numerical value of the Reynolds number can be of no influence in other
cases. These can be associated with flows past bluff bodies particularly having sharp
corners. Such a case is illustrated in Figure 9.6 (a) and (b). The figure (a) is a computer drawing of the super-critical flow, cavitation, hydraulic flume and water tunnel
at the University of Liverpool. This figure is of the test-section viewed from above.
The figure (b) shows a model of a ship having a rear landing platform with a model
of an approaching helicopter. This rig was being used to ascertain the wind patterns
in the region of the platform which have an important impact upon the piloting during landing. A laser system was used to map these flow patterns and the associated
wind velocities. Because of the bluff shapes of the ship there again is no marked
influence of the Reynolds number in these tests upon the flow patterns. Obtaining
9.11 Non-effectiveness of a Single Group
237
Figure 9.6 Hydraulic flume tests to determine wind flows at a ship helicopter landing platform; (a)
view of flume test section from above, (b) view of model ship and helicopter rig; rig is suspended
upside down from test section roof
238
9 Model Testing
this information has the important safety result that the data could be fed into the
helicopter flight characteristics which then would be entered into a full-scale flight
simulator so enabling pilots to practice a landing procedure in the difficult atmospheric conditions before making a real flight.
9.12 Analytical Input Method
The flow past an aerofoil creates a drag force D, which, in non-dimensional form,
equates to a drag coefficient, CD , defined in Section 1.6 by:
. 1
2 2
CD D
; V `
(9.34)
2
where is the fluid density, V the stream velocity and ` a measure of the size of
the aerofoil. With no heat transfer present, then following the discussion of Section 6.6.1,
CD D f ŒRe ; Ma ; :
In the subsonic flight of aeroplanes, that is when the Mach number, Ma < 1 0, and
when historically the effects of increasing Mach number first became of concern,
the only test facilities available to investigate this were of small size so that the
Reynolds number achieved was quite unrepresentative of the aeroplane values.
The problem was overcome at that time by use of an analytical result derived long
before by Prandtl and Glauert and whose analysis had been corrected by Gothert [11,
12]. As illustrated in Figure 9.7, this related the incompressible flow past an aerofoil
to that in subsonic compressible flow past an aerofoil differing only in the thickness
coordinate normal to the oncoming stream. The analysis then showed that, with
increase in the stream Mach number, the aerofoil pressure distribution corresponds
to that of a thicker aerofoil in incompressible flow, and by a factor that depends on
only this stream Mach number. The analysis is not valid in the region of the rounded
leading edge so the solution is limited in accuracy. Tests in incompressible flow on
a series of aerofoils of varying values of this thickness gave data for high values
of the Reynolds number which could then be used to determine the compressibility
effect [13]. The typical result obtained is shown in Figure 9.8.
The validity of this approach is limited by two factors. First, the analysis is for
a non-viscous potential flow whereas the drag force is dictated by viscous effects.
Figure 9.7 The shape transform of an aerofoil between
incompressible flow and compressible subsonic flow
9.13 Partial Extrapolation Method
239
Figure 9.8 The transformed values of drag coefficient as functions of Mach number
Secondly, the effect sought was that of the viscous boundary layer upon the pressure
distribution around the aerofoil surface as well as that due to viscous stresses there,
and the former comes from an interaction between the external potential flow and
the viscous boundary layer. At the time, and in the absence of further knowledge,
the results of such a partial modelling had to be adopted and in the event proved to
be successful.
9.13 Partial Extrapolation Method
Partial modelling arises from the need to determine, at the design stage, the resistance of a large ship. The importance of accurate data has been shown in a recent
study revealing estimates of the provision of engine power of up to 30 % high with
variation of 40 % in several predictions [14]. The flow past the ship which gives this
drag involves several effects. The pressure stresses arise from inertia terms which
involves the density and the velocity V and also from the work done against gravity to create the wave motion and so involves the ratio of the weight to the mass g.
Total forces bring in the size `. The development of spray involves the surface tension coefficient whilst cavitation can occur when the local pressure falls in value
from the free stream pressure, p0 to that of the vapour pressure of the water, pv .
Following the discussion of Section 6.6.2, if cavitation occurs it would initiate at
the minimum pressure point which occurs at the surface [15]. So the vapour pressure, pv forms the single variable .pv p0 /. Then, with the resistance of the ship,
R, as the dependent variable, there is:
R D f Œ; V; `; g; ; ; .pv p0 / :
The pi-theorem solution is in Compact Solution 9.8.
240
9 Model Testing
Compact Solution 9.8
R
V
`
g
p v p0
ML
T2
M
L3
L
T
L
L
T2
M
LT
M
T2
M
LT2
R
pv p0
L4
T2
L2
T
L3
T2
L2
T2
R
V 2
g
V2
V
V 2
pv p0
V 2
L2
1
L
L
L
1
R
V 2 `2
g`
V2
V `
V 2 `
1
1
1
1
This then gives that:
R
Df
V 2 `2
V 2 V ` pv p0
;
;
;
`g V 2 ` V 2
:
(9.35)
The first independent group is a Froude number, Fr , the second is a Reynolds number, Re .
If tests are proposed on a model of size 1=n of the full-size ship, then from the
first independent group, and with g D gm :
Vm
D
V
`m
`
1=2
p
D 1= n
(9.36)
and the test speed is to be correspondingly reduced from the full-size one which is
experimentally convenient.
From the last non-dimensional group in Equation 9.35:
pv p0m
V2
1
D m2 D :
pv p0
V
n
(9.37)
This relation can be satisfied by use of a low value of the model stream pressure
using a super-critical cavitation flume.
From the second independent group, and obeying Equation 9.36,
p
.m =m /
V`
D
D n n D n3=2 :
.=/
Vm `m
For a ship of 150 m in length to be represented by a model of 1 5 m length then:
.m =m /
D 1000 :
.=/
9.13 Partial Extrapolation Method
241
This presents an impossible experimental condition. Even running a test replacing
water at 15 ı C with mercury at 100 ı C gives this ratio as only 12 6. A precise
representation is not possible. Looking at the third independent group shows the
requirement of:
.m =m /
V 2 `m
11
D m2 D
D 104
.=/
V `
nn
and again an impossible requirement is set. Faced with this situation, resort is made
to partial modelling.
First, experience shows that the capillarity has no significant effect upon the value
of R and so the third independent group can be neglected. It is only effective with the
very small toy boats that were propelled by a crystal of camphor thereby changing
locally the surface tension. It is also important for the propulsion of some floating
insects.
Secondly, the resistance is composed of pressure and viscous stresses upon the
hull. The former are dominated by the influence of the first independent group of
Equation 9.35: the latter by the second. An example of this division of forces is
shown in Figure 9.9 [16]. The upper curve is for the total resistance as can be measured by a force balance. The component due to friction and separation at the stern
can be obtained by traversing the wake flow behind a model hull. The difference
Figure 9.9 Separation of the wave and the form drag on a ship hull model [16]
242
9 Model Testing
shown in Figure 9.9 is the wave drag component. Alternatively, the wave drag was
also derived from analysis of the measured wave patterns.
At high values of the Reynolds number, this viscous component of the resistance
is a ‘well behaved’ function of the second independent group and so lends itself to
reasonably accurate extrapolation based upon supplementary information.
On a model the breakdown of the flow in the surface boundary layer from laminar to turbulent conditions can occur well back along the model hull whereas at
full scale this occurs at a comparably equal downstream length which places it right
at the bow. Thus the full scale friction force is virtually all due to the turbulent
boundary layer whereas the model would be a mixture of laminar and turbulent.
This would make extrapolation of the resistance from model to full scale somewhat
problematical. A testing technique to avoid this is to attach carefully assessed roughness at the nose of the model to trip the laminar layer to a turbulent one [17] so that
the model pressure and viscous drags also are virtually all due to turbulent flow in
the boundary layer. The effective model shape is changed in this partial modelling
both by the attachment of the roughness trips and by the different relative thickness
of the boundary layer.
The combination of tests over the experimentally attainable range of the Reynolds
number together with a wealth of experience gained from both model testing and
full-scale trials gives reasonable confidence in extrapolations of model tests to the
full-scale values of the second independent group.
Further refinements to the experimental technique are required giving success
to this partial modelling method. Where precision of design estimate can be less
precise is, for example, in the case of very-large crude-carriers whose viscous resistance extrapolated to very high values of the Reynolds number is the greater portion
of the whole.
The same problem arises in the case of compressible flow where the drag due
to the presence of shock waves in regions of supersonic flow add to the pressure
drag. Experimental separation of the wave drag component is rather rare. It has
been achieved by working from measurements of the shape of the shock wave [18].
Whereas the ship wave drag is a function of the Froude number, in the case of
the wave drag in a supersonic flow the controlling parameter is the Mach number.
9.14 The Range Limitation Method
A limitation to partial modelling can occur with the compressible flow of gases,
Such flows are considered in greater detail in Chapter 3. As in the discussion of
Section 6.6.1, a force F exerted by the gas stream is represented by:
F
D f ŒRe ; ; Ma :
V 2 `2
(9.38)
9.14 The Range Limitation Method
243
It is readily shown that where viscous effects are absent in the flow, the density
change can be written, for Ma2 2=. 1/, in the series [19]:
s 1 2
D
Ma C (9.39)
s
2
Here the suffix s refers to stagnation values. The pressure differences are given in
terms of the pressure coefficient, Cp , by:
ps p
1
Cp 1
D 1 C Ma2 C (9.40)
4
V 2
2
Up to Ma D 0 14 there is only a 1 % error in neglecting the density change and
a 0 5 % error in determination of the pressure coefficient: then the flow can be
considered as an incompressible one. At Ma D 0 32 the respective errors are 5 %
and 2 5 %.
For a valid model test for low-speed incompressible flow the full-scale value of
the Reynolds number needs to be reproduced. With a ratio of full size dimension to
model value of `=`m D n this requires:
m Vm D n:
V m
(9.41)
The viscosity is a function of the particular gas and, except for great extremes of
pressure, only the gas temperature. To vary this property in practice involves undue complexity in the test rig and instrumentation with only limited control of the
Reynolds number.
The left-hand side of Equation 9.41 can be suitably adjusted by running the
model test at n times the full-scale velocity. This procedure can be limited by the
need to avoid the above mentioned compressibility effect as well as by the large
powers required of the test rig which, in general terms, increases as the cube of
the test velocity. Alternatively, the test channel can be pressurised, a density of
twenty-five times that of the atmospheric value having been used in wind-tunnel
tests. This leads to large stresses on the model as an alternative dependent variable
is .p p0 / =V 2 and so for equality of this .pm p0 / = .p p0 / D n.
When the flow is compressible, and particularly for Ma > 1 0, then partial modelling is the rule, experiment being limited to reproduction of the full-scale Mach
number. As with ship model testing the influence of Reynolds number is extrapolated using background knowledge and an extensive body of prior experimental
data. This still has its hazards when high accuracy is required. It has been said that at
the onset of the use of jet engines to propel civil airliners one of the great companies
won the market against a rival company largely because the formers aerodynamicists made the more precise estimate of the extrapolation of wind-tunnel model tests
to the full-scale Reynolds number.
244
9 Model Testing
Compact Solution 9.9
v
R0
d
g
t
L
T
M
L3
L
M
LT
L
L
T2
T
L2
T
v
g
v2
vt
L
1
L
L
vR0
d
R0
gR0
v2
vt
R0
1
1
1
1
9.15 The Distortion Method
Model reproduction of the flow in rivers and estuaries requires partial modelling of
several shape factors and several compromises in balancing conflicting requirements
in the model.
The flow is one under gravity and against friction, and can be unsteady as for
example when floods occur or when waves travel along the stream. The mean velocity, v is a function of the cross-section area of the river, A, the mean depth, d ,
the water density, , the viscosity, , the acceleration due to gravity, g, and a time,
t. The area can be represented by an equivalent radius R0 from 2A D R02 . The pi
theorem solution is in Compact Solution 9.9.
The result is:
vR0
d v 2 t
Df
;
;
:
(9.42)
R0 R0 g R02
The first and dependent group is a Reynolds number, the second is a shape factor
on the river cross-section, the third is a Froude number, and the last one is a nondimensional time group.
Starting with the steady flow in a straight parallel channel, of constant crosssectional area and shape, inclined at the angle to the horizontal, the mean velocity,
v, is expressed by [20]:
2g sin A
v2 D
:
(9.43)
Cf
c
Here A is the cross-sectional area, c is the wetted circumference and Cf is the friction
coefficient defined by:
1 2
Cf w =
v
(9.44)
2
with w being the wall shear stress.
9.15 The Distortion Method
245
Figure 9.10 Scale factors for a model test on a channel flow
The speed of a water wave, cw , of length, L, in water of depth, h, is given by [21]:
cw2 D
gL
2
h
tan h
:
2
L
When h L then:
cw2 D gh :
Two requirements are set for the flow through a model. First, the full-scale flow
will be turbulent, and over a rough surface and well into the region where, for the
random irregularity of the roughness, the friction coefficient does not vary greatly
with the Reynolds number if at all. Secondly, for smooth river flow away from such
as weirs and waterfalls the flow velocity will be sub-critical, that is below the above
wave speed corresponding to the Froude number:
v2
< 1:0 :
gh
(9.45)
The first criterion presents the greater difficulty for two reasons. First, the Reynolds
number has to be greater than a limiting value, which is not precisely known for the
particular nature of the surface roughness, in order that the flow is in the fully-rough
region [22]. Secondly, practicality requires the surface roughness to be modelled by
arbitrarily chosen roughness forms and locations. To achieve the necessary value
of the Reynolds number the cross-section of the real flow is distorted in the model
by an extension of the depth scale in order that the flow can attain the fully-rough
condition. Further, the slope of the flow, , is increased in the model by distorting
the vertical scale of the water level.
Thus for a model reduced in plan view by the factor 1=n, the vertical depth scale
of the cross-section is extended by the factor rv and the slope is extended by the
factor r . This is illustrated in Figure 9.10. Then:
wm =w D 1=nI
dm =d D rv =nI
m = D r
(9.46)
246
9 Model Testing
so that:
dm w
rv
D
n D rv :
d wm
n
(9.47)
For a rectangular cross-sectional shape of the stream of width w and depth A=w,
then the wetted circumferences would be given by:
c D 2d C w
with,
cm D 2dm C wm
so that the ratio of these is,
cm
1 1 C 2rv .d=w/
D
c
n 1 C 2.d=w/
and this relation is used in the following example. Also for this illustration, the following relations from Nikuradse’s experiments upon the flow in sand-rough circular
pipes will be adopted [22]. That is, the minimum Reynolds number for ‘fully-rough’
flow is given by:
Re D 4:64 103 .R0 ="/5=6
(9.48)
and the corresponding friction coefficient by:
Cf D 0:0339 .R0 ="/0:315 :
(9.49)
Here " is the surface-roughness height. The Reynolds number is then:
Re 2R0 v
:
To meet the requirements of Equation 9.45 and 9.48 requires in the model that:
2
vm
1
dm g
and:
2vm 2Am 1=2
4:64 103 .R0 ="/5=6
m :
Combining these gives:
dm g 2
vm
4:64 103 2
!2
5=3
R0
:
2Am " m
Then using the scaling relations, Equations 9.46 and 9.47 leads to:
!2 rv2
w 4:64 103 R0 5=3
:
n3
A2
2
2g " m
9.16 Complexity of Modelling
247
With = D 1:14 106 m2 s1 for water and g D 9:81 m s2 then the limiting
criterion is given by:
5=3
rv2
R0
6 w
1:12 10
:
3
2
n
A
" m
There are several parameters that can be set for the model tests. With full-scale
values of A and w being set, the choice of n gives R0 and with an upper limit set to
rv for experimental convenience, the criterion has to be met by the model value of ".
Refined relations for other roughness forms are available but still have to be applied
with care for real river flows [23].
The reproduction of the full-scale roughness effect presents a particular difficulty. Equation 9.49 was developed from extensive testing upon pipe flows and, as
representing satisfactorily the roughness effect, has been generally accepted as being applicable also to channel flows. In comparison, Manning gave an empirically
derived relation which was [24]:
vD
1 A 2=3
:sin1=2 n c
(9.50)
In this relation, n, is Manning’s friction factor, v, A and c are measured in SI units
and is measured in radians. As,
n
T 2=3
T
L D 1=3
L
L
it is seen that Manning’s factor has the fundamental disadvantage of not being nondimensional. It was introduced well before dimensional analysis came to be used
in engineering. Yet it is still used universally and large tabulations exist of values
for various types of surface without any reference being made to the corresponding
values of g and of A=c. This is seen when from Equations 9.43 and 9.50 there is:
n2 D
Cf R0 1=3
:
2g 2
9.16 Complexity of Modelling
The flow of sediment in river and estuary flows is of considerable importance. Because of the complexity of these flows partial modelling is necessary.
A considerable skill has been developed for model testing as has been shown,
for example, by Yalin [25]. Problems are severe as is illustrated by the following
sources of these difficulties.
a)
Even after the grain size and the distribution of this size together with the
turbulence character of this flow have been reproduced, at least four nondimensional groups remain to be satisfied.
248
b)
9 Model Testing
It can be difficult to reproduce accurately small-scale versions of the weights,
the shapes and the sizes of the grains of sediment especially when a range of
sizes exist in the full-scale flows.
There is a complexity of the full-scale flow patterns especially when the flow
is both channel flow combined with wave motion as in estuaries.
There is enormous complexity of the full-scale sedimentation movements [26].
Because of (a) above, some non-dimensional groups cannot be reproduced in
value at model scale so quite considerable partial modelling is necessarily resorted to.
There are difficulties in measuring sediment movement whilst the flow is running so that much data is limited to the overall change from the beginning to
the end of a test run.
c)
d)
e)
f)
With the following notation:
d
g
`
wP
V
g
Grain size
Acceleration due to gravity
Scale size
Sediment weight-flow rate per unit width
Reference flow velocity
Fluid viscosity
Fluid density
Grain density
Then the functional relation is:
wP D f d; g; `; V; ; ; g :
(9.51)
The pi-theorem solution is as in Compact Solution 9.10. The result is that:
wP
gd `
Df
; ;
;
g V 3
V 2 d g Vd g
Compact Solution 9.10
wP
d
g
`
V
g
M
T3
L
L
T2
L
L
T
M
LT
M
L3
M
L3
w
P
g
g
g
L3
T3
L2
T
1
w
P
g V 3
g
V2
g V
1
1
L
L
gd
V2
`
d
g Vd
1
1
1
9.17 Model Testing in Engineering Design
249
or,
wP
Df
g V 3
V 2 d V ` ; ;
;
g` ` g
:
In this equation the first independent group is the Froude number and the third is the
Reynolds number.
It can now be seen why there are problems in gaining equality in a model test.
Whilst the Froude number can be reproduced, the Reynolds number cannot. This
latter is important in that it would control the development of the vortices of turbulence which spring from the boundary and which can lift sediment into the flow: this
is the behaviour seen with atmospheric whirlwinds. However progress can be made
by the use of surface roughness to develop the necessary level of turbulence. When
the last independent group is satisfied then there is a problem in obtaining a model
grain size sufficiently small.
Yet despite these difficulties some most useful agreements of data between model
and full-scale results have been achieved [25]. Extensive discussion is given by Yalin
in the work just quoted.
9.17 Model Testing in Engineering Design
Model testing is but one aspect of experimental practice [27]. The way in which it
fits in to the whole design process in engineering is well illustrated in Figures 9.11.
This shows three aspects of the aerodynamic design of the Airbus A380 aeroplane.
Figure 9.11(a) illustrates the output of a highly advanced computer study to determine the aerodynamic behaviour of the flow around the proposed aeroplane at
the landing configuration together with the corresponding calculation of the aerodynamic loading. This would form the first stage of the design. Such computations
have become markedly improved in accuracy in recent times from the rapid growth
in the operating capacity and speed of computers. Yet, as is illustrated in the discussion of turbulence in flows given in Chapter 8, such calculations in detail still
require the input of empirical coefficients to enable modelling of the flow.
To provide that information and as a check on the calculations, Figure 9.11(b)
illustrates a model test in a wind-tunnel reproducing the same landing condition.
The results from this testing then enables comparison with the computer output so
that the latter can be refined in its accuracy of prediction of the full-scale condition.
Finally, Figure 9.11(c) shows the full-size aeroplane in the same landing configuration as it would be flight tested with extensive on-board instrumentation. Comparison of flight-test results with the computer output and the model tests enables
the latter two to be further refined.
In this way engineers hone their design skills and build up experience and expertise so essential to successful design developments.
250
9 Model Testing
Figure 9.11 Airbus 380 aeroplane, landing configuration (a) Computer output, (b) Wind tunnel
test, (c) Full-size aeroplane
9.18 Assessment of the Physics
These examples show that a good understanding of the physics is required before
undertaking the use of dimensional analysis to order an experiment: a considerable
understanding is needed when partial modelling is undertaken both for the design
of the experiment and the interpretation of the resulting data.
This latter is the engineering approach in which some answer must be obtained
for design or operational needs even though partial modelling may be somewhat
approximate in its output. The skill in engineering is in using scientific knowledge
and background experience to minimise the approximation. But this is no more than
the need to recognise that an analytical model of a real event necessarily has an
9.18 Assessment of the Physics
251
inherent approximation as mentioned in Chapter 2. It is in assessing the degree of
that wherein lies the skill.
Exercises
9.1
9.2
9.3
9.4
The spin of an aeroplane can be studied in a spinning tunnel which has a test
section containing a stream of air flowing vertically upward. A model aeroplane can have the controls set so that it spins steadily within this stream, the
air velocity being adjusted so that the vertical aerodynamic force balances
the weight of the model. Obtain the requirements of a model test. Consider
both the case of steady spinning and that of recovery from the spin when the
control surfaces are operated by radio control.
Tests are to be made on a model of the pressurised cabin of an airliner to
find the effect of sudden failure of a window upon the subsequent trajectory
of the passengers. Demonstrate that for dynamical similarity the absolute
temperature of the air within the model must be proportional to `; where
` is a representative length. Also show that if the external pressure is held
constant then the density of the dummy passengers must be inversely proportional to `. [Univ. Liverpool., 1958.].
Extend the discussion of Section 9.7 to take account of the masses contained
within the full-scale shell structure such as, for a car the engine and other
masses.
A safety valve, which is illustrated in Figure 9.12, operates in the following
manner. A piston of mass m rests on a seal A and is restrained vertically by
a pre-tensioned spring. When the release pressure is reached, the seal at A
is released and the piston rises allowing air to flow into the chamber B. Initially the outlet from this chamber at C is constricted and so the full release
pressure now acts upon the whole piston area. The piston rises rapidly until it
clears the constriction at C rising to a maximum height where it is held and
so releasing the air to the atmosphere. Tests are to be performed upon a small
scale model of this valve to determine this vertical movement. Air at the fullscale pressure is used in the container and this exhausts to the atmosphere.
Neglecting viscous and heat conduction effects in the flow, demonstrate that
full dynamical similarity can only be obtained if the absolute temperature
of the enclosed gas is proportional to a typical length `. Also show that the
Figure 9.12 Design of
a safety release pressure
piston
252
9.5
9.6
9.7
9.8
9.9
9 Model Testing
weight of the piston and the pre-tension force are to be proportional to the
square of the size; and that the spring rate (in units of force per length) must
be proportional to size. Show also the importance of retaining the full-size
overall pressure ratio. (Univ. Liverpool)
A very large vacuum vessel is to be set up out of doors. Set in the bottom
is to be a circular safety bursting disc. There is concern for the safety of
persons in the vicinity of this disc should it fail and a person be sucked
into the vessel. A scaled down model test, in which the external pressure
is equal to the full-scale atmospheric pressure, is to be set up to determine
a safe approach distance. Show that the density of a dummy figure should be
inversely proportional to the size and that the pressure in the vessel must be
in a fixed proportion to the outside pressure. Show that this latter condition
can be relaxed when the inside pressure is sufficiently low so that then the
outside temperature must be proportional to the model size. Show that also
the velocity of the dummy would be proportional to the square root of the
size. [Univ. Liverpool. 1988].
An aerofoil wing is tested in a wind tunnel at a Mach number of 1:35 and
a Reynolds number of 7:8 106 giving a drag coefficient, based on the wing
plan area, of 0.0087. Calculate the friction drag coefficient, using the relation
given in Exercise 9.9 below, at a full-scale Reynolds number of 8:2108 . Then
calculate the full scale total drag.
A design proposal has been made for a container to hold survival equipment
which is to be dropped into the sea from an aeroplane and be just buoyant
when stationary. It is to have a small drogue parachute to hold the orientation
at impact. Cavitation, which occurs when the local fluid pressure in the flow
drops to the local vapour pressure, is possible at entry. Model tests are to be
made to determine the maximum depth of immersion and the impact load on
the nose of the container. Set out the requirements of a model test to a scale
of 1/20 in which the atmospheric pressure can be varied to suit.
A proposal for a ship is for a length of 180 m with a propeller of 2.55 m
diameter rotating at 85 rpm just below the sea surface. The design is to be
tested using a model of scale 1 : 120 in test flume using fresh water. Calculate
the diameter and the rotational speed of the model propeller. If the ship is
sailing in sea water of density 1.035 that of fresh water, calculate the ratio
of the power to that determined in the model test. Determine also the size of
the model ship. Exclude viscous and cavitation effects.
A ship is planned having a length, `, of 165 m and a wetted area, S , of
5445 m2 . An estimate is required of the total resistance at a speed of 22 knots
(UK) in salt water of density, , of 1.03 kg m3 and viscosity, of 1:14 103 kg m1 s1 . A test on a model of length 1:2 m at the appropriate speed
in fresh water gave a total resistance of 0:94 N. The frictional drag, is given
by
1
D D CD V 2 S
2
References
253
Figure 9.13 Flow down a spillway
and the drag coefficient can be estimated from
CD D h
0 455
i258
log10
9.10
9.11
9.12
_V `
Prepare the estimate. Calculate the drag due to friction as a percentage of the
whole resistance at both model and full scale and consider the implications
for the precision of your estimate.
The sketch of Figure 9.13 shows a spillway for the flow from a reservoir.
Derive the non-dimensional groups for the flow rate of the water of density,
, surface tension, , and viscosity, . Decide on a form of partial modelling.
A full-scale spillway is of 75 m in width and is to be represented by a model
of 2.0 m width. The vertical scale of the model is 1 : 9.3 and the model flow
rate is measured to be 0.47 m3 s1 . What is an estimate of the full-scale flow
rate?
Model tests are required on a harbour for which the outer breakwater wall
is subject to waves of 1.5 m height travelling at 10.2 m s1 . A scale model
of 1/300 full-size is chosen. Deduce the size and speed of the waves in the
model. Tides occur at intervals of 12 h. What is this tidal period to be in the
model test?
A section of a river has a width of 100 m and a mean depth of 7 m. The mean
slope of the river is 2 104 rad. Determine the dimensions of a model when
the horizontal scale is 1 : 2000 with a vertical enhancement of 40. Take the
model and full-scale mean flow velocities to be equal.
References
1. F. Chichester. Gipsy Moth circles the world, Ch. 8, p. 106, Hodder and Stoughton, London,
1967.
2. J.C. Gibbings. (Obituary supplement), The Times, No. 67769, p. 39, Thursday May 22 2003,
London.
3. J.C. Gibbings. The systematic experiment (Ed J.C. Gibbings), Ch. 3, The planning of experiments,: Part 3 – application of dimensional analysis, Cambridge, 1986.
254
9 Model Testing
4. J.C. Gibbings. Non-dimensional groups describing electrostatic charging in moving fluids,
Electrochim. Acta, Vol. 12, pp. 106–110, 1967.
5. R.V. Southwell. Theory of elasticity, 2nd Ed., pp. 19, 425, 426, Oxford Univ. Press, Oxford,
1946.
6. J.C. Gibbings. Thermomechanics, Sects. 8.10, 8.11, Pergamon, Oxford, 1970.
7. N. Jones Structural impact, Cambridge University Press, (Corr. Ed.) 1997.
8. J.C. Gibbings. Achievement of aerial flight: an engineering assessment, Aer. J., Vol. 85,
No. 846, pp. 257–265, July/Aug. 1981.
9. M.W. McFarland (Ed.). The papers of Wilbur and Orville Wright, McGraw-Hill, New York,
1953.
10. F.W. Schmitz. Aerodynamik des Flugmodells, Tragflügelmessungen I und II bei kleinen
Geschwindigkeiten, Luftfahrtverlag Walter Zuerl, 3rd Ed., 1975.
11. F. Cheers. Elements of compressible flow, John Wiley & Sons, London, 1963.
12. A.H. Shapiro. The dynamics and thermodynamics of compressible fluid flow, Vol. 1,2, Ronald,
New York, 1953.
13. Th von Karman. Compressibility effects in aerodynamics, J. Aer. Sci., Vol. 8, July 1941.
14. [Anon]. Are vessels overpowered? Mar. Eng. Rev., The Institute of Marine Engineers, Dec.
1996, p.14.
15. H. Lamb. Hydrodynamics, (6th Edn.), Cambridge University Press, 1932.
16. M. Insel, A.F. Molland. An investigation into the resistance components of high speed displacement catamarans, Trans. Royal Inst. Naval Archit., Vol. 134, pp. 1 – 20, 1992.
17. J.C. Gibbings, O.T. Goksel, D.J. Hall. The influence of roughness trips upon boundary layer
transition, Parts 1,2 and 3, Aeronaut. J., Vol. 90, pp. 289–301, 357–367, 393–398, 1986.
18. J.C. Gibbings. Pressure measurements on three open nose air intakes at transonic and supersonic speeds with an analysis of their drag characteristics, Br. Aer. Res. Council, Current Paper
No. 544, 1960.
19. H.W. Liepmann, A.E.Puckett. Aerodynamics of a compressible fluid, Wiley, New York, 1947.
20. A. Mironer. Engineering fluid mechanics, McGraw-Hill, 1979.
21. R.H. Sabersky, A.J.Acosta. Fluid flow, Macmillan, New York, p. 281, 1964.
22. S. Goldstein. Modern developments in fluid dynamics, Dover, New York, 1965.
23. V.T. Chow. Open-channel hydraulics, McGraw-Hill, New York, 1959.
24. R. Manning. Flow in channels, Trans. Inst. Civil Engineers, Ireland, Vol. 20, p. 161, 1890.
25. M.S. Yalin. Theory of hydraulic models, MacMillan, London, 1971.
26. K.R. Dyer. Coastal and estuarine sediment dynamics, Wiley, Chichester, 1986,
27. J.C. Gibbings. (Ed.) The systematic experiment, Cambridge Univ. Press, Cambridge, 1986.
Chapter 10
Assessing Experimental Correlations
– an experimental reading is not to be trusted until its
background has been thoroughly investigated.
F. Drabble
Notation
a, b, c
A, B
C
c1 , , c6
d
D
D12
DC , D
e
E1 , E2
F
g
H
i0 , i1
j0 , j1
j
K1
`
n, m
n
P
QP
Re
t
w
x
xt
Constants; Equation 10.19
Dimensional products
Dimensional constant
Regression coefficients
Paddle, fan diameter
Container, pump diameter
Diffusion coefficient
Ion diffusivity
Paddle clearance
Errors
Fraction of cream to milk
Acceleration due to gravity
Head of liquid
Electrical current, zero, infinite time
Electrical current density, zero, Infinite time
Electrical current density
Parameter
Scale size
Non-dimensional indices
Rotational speed
Power
Volume flow-rate
Reynolds number; .!d 2 /=
Time
Paddle width
Variable, control setting
True value
ˇ0
Units-conversion factor for angle
J.C. Gibbings, Dimensional Analysis. © Springer 2011
255
256
"
0
˘
'0
!
.!t/c
10 Assessing Experimental Correlations
Permittivity, error
Electrical conductivity
Liquid viscosity
Non-dimensional group
Liquid density
Efficiency
Torque
Electrical potential
Angular velocity of paddle
Computed regression value
10.1 Interpretation of Dimensionless Correlations
The power of dimensional analysis in enabling greatly simplified correlation of experimental data has been demonstrated in Chapter 1. Care is needed on occasions in
interpreting the precision of these correlations. Again, a clear understanding of both
the physics of the phenomenon and of the physical significance and composition of
each non-dimensional group is required.
10.2 Interpretation of Experimental Error
In a phenomenon involving a functional relation between only two non-dimensional
groups ˘1 and ˘2 , these are then related by:
˘1 D f .˘2 / :
(10.1)
When each group contains the same variable, x, then, as the groups are in the form
of products, we can write,
˘1 D Ax n ;
˘2 D Bx m :
(10.2)
It is common practice to judge the level of the random error in experimental results
by measuring the scatter about the mean correlating curve whether the latter is derived in an analytical form or as a hand-drawn curve on a plot of the results showing
the nature of Equation 10.1. Furthermore, a rather restrictive, but very common,
practice is, when x D f .y/, to assess the scatter on only the values of x.
With the measured value, x; having a fractional random error " from the true
value, xt , then,
"D
x xt
:
xt
10.2 Interpretation of Experimental Error
257
Typically in most good quality experiments, errors range from 1 to 10 %. So, for "
small,
x n D xtn .1 C "/n
D xtn .1 C n" C 12 n.n 1/"2 C /
so that,
x n xtn .1 C n"/ :
(10.3)
Then, using Equations 10.2 with 10.3, the corresponding error on ˘1 denoted by E1
is given by:
˘1 ˘1t
˘1t
Ax n Axtn
D
Axtn
x n .1 C n"/ xtn
D t
xtn
E1 so that,
Similarly, the error on ˘2 is,
E1 D n" :
(10.4)
E2 D m" :
(10.5)
A plot of ˘1 versus ˘2 can be misleading in estimating the error as is illustrated in
the sketch of Figure 10.1. With the true value at point ‘A’ in this diagram and with
n > 0 and m > 0 then the corresponding experimental value would plot at point
‘B’. If n > 0 and m < 0 the experimental value would plot at point ‘C’. Thus in
the first case a cursory glance at the experimental plot would mislead as to a high
accuracy and the second case would equally mislead as to a poor one. Other cases
for a negative value of the slope of the correlation curve and for other values of n
and m are readily sketched.
There are two important lessons. First, from the above example a correlation
curve, for which the values of the non-dimensional groups are equally straddled
about the curve, could be quite misleading depending upon the slope of the curve.
The second is that care must be taken in interpreting experimental accuracy from
plots of non-dimensional groups and from statistical analyses. Clearly discrepancies
are enhanced by high values of the indices n and m. This can often happen. For
example, the non-dimensional group for the power of a propeller is,
P
n3 d 5
so that the error in measuring n is trebled and that in d is increased five-fold.
258
10 Assessing Experimental Correlations
Figure 10.1 Illustration of
assessment of experimental
error
Massey has called this matter a ‘spurious correlation’ [1] but this can be a limiting expression. The more appropriate use for this term would be as used later in
this chapter. A suitable term for the present difficulty would be ‘misleading errorestimation’.
Again referring to Figure 10.1, the ‘true’ error arising from error in only x, is
readily assessed, once the correlating curve is acceptable, by drawing the line from
the plotted point ‘B’ at the slope of given by:
˘1 ˘1t
˘1t n"
n ˘1
tan D
D
˘2 ˘2t
˘2t m"
m ˘2
to intercept the correlated curve giving the ‘true’ data point at ‘A’. This is the equivalent of measuring the vertical discrepancy between an experimental point and the
correlation curve when, for example there is a plot between two variables with an
error assigned to only one.
It is also common practice when computing a regression curve to assume the
error to be only in the ordinate values which are conventionally chosen as measuring
the dependent variable. In the present example, if there is error only in x then there
are errors in both the ordinate and the abscissa and which can be of different degree.
So this should be accounted for in deriving the regression curve [2].
10.3 Deduction of Physical Results
Case A. The Hydraulic Turbine
A Pelton wheel is a form of hydraulic turbine. Figure 10.2 shows results of a test
on such a machine to determine the power output. This is represented by an output
10.3 Deduction of Physical Results
259
Figure 10.2 Pelton wheel experimental data: torque group versus flow-rate group: curves are for
a range of values of the Reynolds number
torque, , from a shaft rotating at n rps on a machine of reference diameter, D. The
water of density, and viscosity, is flowing through the turbine nozzle at a volume
P under a head of water of height, H; so that the applied pressure difference
rate, Q,
is gH . The flow nozzle for the water jet has a needle valve to adjust the nozzle
cross-sectional area with a value of the control setting, x.
It is noted that:
QP D f .gH; x; D/
(10.6)
so that QP and gH are alternative variables.
Then with the function of:
P ; ; x; D; n; ˇ0 :
D f Q;
The pi-theorem solution is in Compact Solution 10.1.
Thus the pi theorem leads to:
D
D x ˇ0 nD 3
Df
; ;
QP 2
QP D
QP
or:
Df
2 2 5
ˇ0 n D
"
ˇ0 nD 2 x
QP
; ;
D ˇ0 nD 3
#
:
(10.7)
A convenient procedure is to fix H , x and hence QP and then vary by varying
the output load on the turbine and measuring n. This gave the curves shown in
260
10 Assessing Experimental Correlations
Compact Solution 10.1
P
Q
x
D
n
ˇ0
ML2
T2
L3
T
M
L3
M
LT
L
L
˛
T
1
˛
ˇ0 n
1
T
L5
T2
L2
T
P2
Q
1
L
P
Q
1
L
D
P2
Q
D
P
Q
x
D
1
1
1
ˇ0 n
P
Q
1
L3
ˇ0 nD3
P
Q
1
Figure 10.2. This would seem unsatisfactory as the value of both the second and the
fourth groups in Equation 10.7 vary along each curve. The groups can be rearranged
so that:
"
#
P
x
QP
Q
Df
;
;
(10.8)
d ˇ0 nD 3 D
ˇ02 n2 D 5
and now the final group has a fixed value for each curve of Figure 10.2.
The experiment was repeated for three more values of x giving the set of four
curves for a range of Reynolds number in Figure 10.2. There is now a problem of
interpretation. The question is, by bearing in mind Equation 10.8, does the distinction between the four curves give a measure of changes in the second or in the fourth
groups of Equation 10.8 or of both?
An alternative formulation is obtained by replacing the variable, QP with thehydraulic head term, gH so that the third group in Equation 10.8 becomes .gH /= ˇ02
n2 D 2 . This represents the square of the ratio of the velocity of the water jet to
the peripheral speed of the deflectingbuckets of the turbine. Also, the fourth group
can correspondingly be replaced by 2 gHD 2 =2 . Further, the efficiency of the
turbine, , which is given by:
ˇ0 n
gH QP
can be alternatively used as the dependant variable so that:
x
gH
2 gHD 2
; 2 2 2;
Df
:
D ˇ0 n D
2
(10.9)
10.3 Deduction of Physical Results
261
Figure 10.3 Pelton wheel experimental data: efficiency versus pressure-head group: codes as in
Figure 10.2
It is known that a significant effect of Reynolds number, represented in Equation 10.9 by the last group, will show up particularly in values of the maximum
efficiency. Plots of the experimental data are shown in Figure 10.3. There is a common curve for the sets of data at the three highest values of the Reynolds number
with an indication of a common maximum value of the efficiency for all four flow
rates. This suggests that the divergence between the first three and the lowest fourth
flow rate, at the lower value of the second independent group of Equation 10.9, is
an indication of the effect of x=D and that there may well be no Reynolds number
effect for all the results.
To confirm this latter conclusion, experiment would have to be done with variation of three independent variables. Such might be, n, x and gH .
Case B. Conductivity of Highly-Resistive Liquids
Processing a liquid of very low conductivity in industry can give rise to electrostatic
problems. The evidence is that the charge carriers are impurity particles of undetermined composition because of their extremely small size [3]. With these liquids
there is an interaction with electrodes that gives rise to the electrical boundary layer
previously described in Section 7.8. Thus a problem arises in the determination of
the very low conductivity because of the absence of a single value of the field. Further, above a limiting applied potential difference across the electrodes convection
262
10 Assessing Experimental Correlations
Compact Solution 10.2
j
'0
t
0
"
DC
D
`
A
L2
ML2
AT3
T
A2 T 3
L2
L2
L
ML3
A2 T4
ML3
T
T
'0 0
"
0
A
L
T
j
'0 0
1
L
"
0 t
DC t
D t
1
L2
L2
j`
'0 0
DC t
`2
D t
`2
1
1
1
of the liquid can take place and so this effect has to be excluded by limiting the
upper value of the applied potential [3].
The current density, j , in a conductivity cell is taken as a function of the time, t,
from application of the externally applied potential, '0 , the liquid conductivity at
zero charge density, 0 , the liquid permittivity, ", the diffusion coefficients of the
positive and negative ions, DC , and D , and the size of the conductivity cell, `.
Thus the functional relation is:
j D f Œ'0 ; t; 0 ; "; DC ; D ; `
Solution for the pi-theorem then is in Compact Solution 10.2.
This solution gives that:
j`
" DC t D t
Df
;
; 2
:
0 ' 0
0 t `2
`
This is rewritten as:
j`
Df
0 ' 0
0 t DC " DC
;
;
" 0 `2 D
(10.10)
:
(10.11)
Before application of the potential there is a charging current [4]. This current is very
small compared with those to be measured in a conductivity cell so the distortion of
the charge distribution and hence of the field distribution within the liquid is taken
as being negligible. Thus, at the instant of application of the potential, the field is
10.3 Deduction of Physical Results
263
taken as being uniform in a plane electrode cell. At this zero time we have that:
j0 / .'0 0 / =` :
From then on the current falls in value with time to an asymptotic value corresponding to infinite time. From Equation 10.11 and for a fixed liquid at t D 1 then the
ratio of these two current values, i1 = i0 is given by:
i1
j1
j1 `
D
D
:
i0
j0
'0 0
It then follows from Equation 10.11. and for a fixed electrolyte that,
i1
DC "
Df
:
i0
0 `2
(10.12)
(10.13)
This physical modelling of the phenomenon is confirmed by the experimental correlation shown in Figure 10.4 [3,5]. These experiments were carried out after a careful
check confirmed that there was no electrically generated convection of this liquid
electrolyte. The value of DC and of " were not determined in this experiment. As
each would have a constant value then the values of 0 `2 are representative of the
independent group of Equation 10.13. This correlation in terms of non-dimensional
groups shows clearly the importance of the diffusion in the measurement of conductivity of these liquids, an effect that increases with increase in the diffusion coefficient.
Figure 10.4 Experimental
values of electrical current
versus the diffusion group [3]
264
10 Assessing Experimental Correlations
This example again illustrates how much understanding can be gained when inspection of the form of the non-dimensional groups is combined with inspection
of the experimentally derived form of the functional relation between the nondimensional groups and with knowledge of the physics of the phenomenon.
10.4 Dimensional Analysis with Statistical Regression
The prior stress placed upon the application of dimensional analysis to the analysis of experimental data will now be demonstrated in detail through a particular
example.
The statistical analysis of experimental data is common practice by scientists and
engineers. In industry and research establishments, and when these units are very
large, the group doing the statistical analysis can be separate from those who have
performed the experiment. The value of incorporating dimensional analysis with the
statistical analysis has been remarked upon elsewhere, including advocacy by statisticians [6–8]. Not to do this can result in a false correlation especially when neither
the experimental group nor the statistics group has applied dimensional analysis
before the regression has been started [9, 10].
It is illustrated here by the following example that was reported by Miller [11].
It has been considered in detail by Brook and Arnold who made a detailed study of
the numerical accuracy of their statistical regression analysis of these results [12].
10.5 A Mixing Experiment
The aim of Miller’s experiment was to determine the duration-time of rotation of
a stirring paddle needed to just disperse a layer of cream that had been separated up
to the top of a container of milk by gravity forces. The experimental rig is shown
sketched in Figure 10.5. This shows the various shape variables which are; D, the
vessel diameter; d , the paddle diameter; w, the paddle width; e, the paddle clearance; H , the liquid depth and the paddle angular velocity, !. The flow would be
one of viscous rotating motion within the vessel and of turbulent separation from
the sharp edges of the paddle. This motion would contain vortices in combination
with diffusion. The liquid variables would be the proportion by volume of cream to
milk, F ; the viscosity, ; the density, and the diffusivity, D12 where the diffusion
coefficient is common between the cream and the milk [13].
Two other variables that Miller considered were the difference of the density
between cream and milk and the acceleration due to gravity. These would enter into
a buoyancy effect which would be significant in the original settling out of the cream
This would take a time that would be several orders of magnitude greater than the
stirring time and so could be neglected in the latter operation.
10.5 A Mixing Experiment
265
Figure 10.5 Sketch illustrating the milk-mixing rig
Thus we have that:
t D f .D; d; w; e; H; !; ; ; D12 ; F / :
(10.14)
Solving as usual gives Compact Solution 10.3.
This gives that:
d w e H !D 2 D12
!t D f
; ; ; ;
;
;F :
D D D D
!D 2
Compact Solution 10.3
t
T
!t
1
D
L
d
L
w
L
e
L
H
L
!
D12
F
M
LT
M
L3
L2
1
T
L2
!
D12
!
1
L2
L2
1
T
T
d
D
w
D
e
D
H
D
!D2
D12
!D 2
1
1
1
1
1
1
266
10 Assessing Experimental Correlations
It is convenient for the analysis of the experimental results to rewrite this as:
d w e H !d 2 D12
!t D f
; ; ; ;
;
;F :
(10.15)
D D D D
Miller excluded the variables and D12 . But if the density is excluded in Equation 10.14 then, as seen in the above tabulation of dimensions in Compact Solution 10.3, this equation cannot be made dimensionally homogeneous because there
would be only the variable containing the dimension of mass.
In liquids containing a small concentration of the solute, by Walden’s rule
D12 / and so the penultimate group in Equation 10.15 would be constant in
value. Also, in the experiment the third and the fifth group were held constant. Thus
Equation 10.15 reduces to:
d e !d 2
!t D f
; ;
;F :
(10.16)
D D
10.6 The Regression Function
The third independent group in Equation 10.16 is the Reynolds number. Experience
suggests that for this type of flow the two variables, Re and d=D might be the
principle ones. An appeal to the physics of this fluid flow indicates that it would
comprise a combination of separated flow from the sharp edges of the paddles and
a viscous mixing flow from both the vortices generated by this separation and the
general rotating motion in the container. The former might not show a Reynolds
number dependence and the latter might give a simple power-law one.
This is seen to be so in the plot of the results given in Figure 10.6. As expected,
this confirms the power-law variation and also shows that the power index is independent of the shape parameter d=D. This plot gives a first estimate of the value of
this index as 1:35. Thus Equation 10.16 becomes:
d e
.!t/Re1:35 D f
; ;F :
(10.17)
D D
There are indications in Figure 10.6 that the parameter d=D has a greater influence
than that of e=D. Consequently Figure 10.7 is a plot of .!t/ Re1:35 against d=D.
This indicates a further power law relation which, because of the constant slopes of
about 2.0 in Figure 10.7, is a simple multiplying one so that Equation 10.17 reduces
to:
he
i
K1 .!t/Re1:35 =.d=D/2 D f
;F :
(10.18)
D
Figure 10.6 indicates that there is a better correlation for d=D D 0:228 and 0.380
than there is for d=D D 0:294: Figure 10.7 confirms this.
10.6 The Regression Function
267
Figure 10.6 Experimental data for time group versus Reynolds number
Figure 10.7 Correlation of
group product versus diameter
ratio
Equation 10.18 leads to the further plot shown in Figure 10.8. It is seen that as
the correlation is being built up, so that the non-dimensional variables successively
taken into account are becoming weaker in their influence, then doubtful experimental points become revealed. There are now six of these marked in the plot of
268
10 Assessing Experimental Correlations
Figure 10.8 Correlation of K1 versus paddle clearance ratio
Figure 10.9 Correlation of
coefficient of Equation 10.18
versus the cream ratio
Figure 10.8. This figure suggests a correlation in the form of a set of parabolas of
the form:
a C b.e=D/ C c.e=D/2
(10.19)
with only the constant term, a, varying with the nondimensional variable F .
The further cross-plot shown in Figure 10.9 indicates that the coefficient, a, of
Equation 10.19 is a simple linear function of F . In this plot the six doubtful points
marked in Figure 10.8 are now excluded. Thus the final form of the function of
Equation 10.16 is obtained as:
.!t/Rec1
D c3 C c4 F C c5 .e=D/ C c6 .e=D/2 :
.d=D/c2
(10.20)
10.7 Statistical Analysis on the Non-dimensional Groups
269
10.7 Statistical Analysis on the Non-dimensional Groups
The form of Equation 10.20 requires the use of a generalised non-linear, multiple
variable, regression technique [14]. Using that quoted technique, a least-squares fit
to Equation 10.20 was obtained, excluding the points marked in Figure 10.8. The
regression was on the actual values of each variable. The values of the coefficients
were determined as follows:
c1 D 1:36 ;
c2 D 2:30 ;
c3 D 1:67:1010 ;
c4 D 4:48:1010 ;
c5 D 1:53:1012 ;
c6 D 5:61:1012 :
The mean error on !t was computed to be ˙8:3 %.
The plot of Figure 10.10 shows the good fit of this regression formula where
the ˙10 % bands show accordance with this error calculation. The points that were
excluded from the regression curve are now plotted in this figure. They reveal two
admirable fits to straight lines which straddle the presently accepted curve. This
raises a possibility that somewhere in the experiments some fixed variable was erroneously reported.
Figure 10.10 Final correlation for the time group
270
10 Assessing Experimental Correlations
Inspection of the original data and of Figure 10.6 does not reveal the cause of
these divergences in what is, for this type of experiment, an otherwise pleasingly
accurate one. In terms of the variables in Equation 10.14 only the variables w and
H were excluded. Inspection of Figure 10.10 suggests that the divergences came
from two systematic causes.
The statisticians who analyzed Miller’s results adopted the Rayleigh assumption
of a simple power product for the form of the function. They used their study to
remove some of the variables in Equation 10.14 resulting in the simple power law
relation of:
t D C !ad b :
(10.21)
Further, they did a simplified logarithmic linear regression curve fit. These approximations are now shown not to be valid on several counts. First, there were excluded,
as determined by the statistical analysis, several variables which are now seen to be
important. One noteworthy omission is the viscosity. Secondly, the simple powerlaw assumption is seen to be not acceptable being unacceptably restrictive. Thirdly,
the present analysis has revealed what appears to have been a systematic divergence
in reporting some of the data.
This case study shows that Miller’s experiment was greatly more precise than he
was given to understand. It has revealed the great advantages of both a study of the
physics of a phenomenon and the application of dimensional analysis before using
statistical analysis to determine the form of the regression function that represents
experimental data. These matters have been considered in detail elsewhere ( [2],
Ch. 9).
10.8 Summarising Comments
The great value of dimensional analysis in synthesizing experimental data has been
demonstrated both here and in Chapters 1 and 7. The discussion here shows also
how care has to be taken in interpreting results in both their significance and their
accuracy.
Exercises
10.1
10.2
Tests are to be made upon the trajectory of artillery shells by firing identical
shells at a set of values of the muzzle velocity and the muzzle inclination
to the horizontal. Consider how the test results may be expressed in terms
of non-dimensional groups and how some of these groups may be assigned
fixed values in the test.
Derive Equation 10.9.
References
271
References
1. B.S. Massey. Units, dimensional analysis and physical similarity, Van Nostrand Reinhold,
London, 1971, pp. 87-88.
2. J.C. Gibbings (Ed.). The systematic experiment, Cambridge University Press, 1986.
3. J.C. Gibbings, G.S. Saluja, A.M.Mackey. Current decay and fluid convection in a conductivity
cell, Inst. Phys. Conf. Ser., No.27, Chapter 1, pp. 16–33, 1975.
4. J.C. Gibbings, G.S. Saluja, A.M. Mackey. Electrostatic charging current in stationary liquids,
Static Electrification 1971, Conf. Ser. No.11, Inst. Phys., pp. 93–110, 1971.
5. G.S. Saluja. Static electrification in motionless and moving liquids, Ph.D. Thesis, Univ. Liverpool, October 1969.
6. A.G. Baker. (Letter), R. Stat. Soc.; News and Notes, Vol. 18, No. 10, p. 2, June 1992.
7. D.J. Finney. Dimensions of statistics, J. App. Stat., Vol. 26, pp. 285–289, 1977.
8. P.T. Davies. Dimensions of statistics and physical quantities, J. App. Stat., Vol. 29, No. 1,
pp. 96–97, 1980.
9. P.N. Rowe. The correlation of engineering data, The Chemical Engineer, p. CE69, March
1963.
10. D. Wilkie. The correlation of engineering data reconsidered, Int. J. Heat & Fluid Flow, Vol. 6,
No. 2, pp. 99 - 103, June 1985.
11. E.J. Miller. Preliminary design investigation into milk agitation, N. Z. J. Dairy Sci. Technol.,
Vol. 14, pp. 265-272, 1979.
12. E.J. Brook, G.C. Arnold. Applied regression analysis and experimental design, Marcel
Dekker, New York, 1988.
13. J.C. Gibbings. Thermomechanics, Sect. 12.9, Pergamon, 1970.
14. J.C. Gibbings. A generalised iteration method for deriving multiple regression curves of nonlinear functions, J. App. Stat., Vol. 20, No. 1, pp. 57–67, 1993.
Chapter 11
Similar Systems
So scientists have reproached him for having sometimes
lavished his calculus on physical hypotheses, or even on
metaphysical principles, of which he had not sufficiently
examined the likelihood and solidity. [Eloge de M. Euler, 1783]
R. Giacomelli, E. Pistolesi
Notation
c
C
D
E
fa
F
g
h
is
k1 , k2
L
`
n
p
QP
Re
T
U
V
x
y
Mean wing chord
Bulk compressibility modulous; ion concentration
Diffusion coefficient; leg width
Electric field
Aerodynamic frequency parameter
Force; Faraday constant
Gravity acceleration
Heat transfer coefficient
Electrostatic streaming current
Coefficients
Leg length
Scale size
Beat frequency
Pressure
Heat rate
Reynolds number
Temperature
Velocity
Velocity
Joint location measure
Bone lengths
˛
ˇ
"
Coefficient
Coefficient of volume expansion
Dielectric coefficient
Thermal conductivity
Viscosity
Density
Surface tension coefficient; charge density
J.C. Gibbings, Dimensional Analysis. © Springer 2011
273
274
11 Similar Systems
11.1 The Concept of Similitude
The concept of similitude and its applications does not appear to be universally
described. Palacios [1] refers to it by writing “– the Principle of Similitude on which
are based the experiments with reduced models –”. Thus his idea is the use of the
results of dimensional analysis for designing model tests as has been described here
in Chapter 9. Duncan had a similar approach for he wrote that: “The determination
of the quantitative conditions for similarity of behaviour is an essential part of the
study of physical similarity and these are most conveniently found by the technique
of dimensional analysis.” [2].
Again, Isaacson and Isaacson link similarity to the relationships derived from
dimensional analysis that are used to validate model testing and which can enable
scaling parameters to be determined [3]. Massey adopts the same idea of the application of the idea of similarity to model testing [4].
In contrast, Kline advocates its use in place of dimensional analysis by considering instead certain physical quantities that govern a phenomenon [5]. However,
initially he limits these to mechanical forces and length ratios. He specifically excludes generalised forces “– such as in irreversible thermodynamics.”. He continues
his discussion by introducing forms of energy and also factors defining the thermodynamic properties of a system. As mentioned earlier, this idea of similar systems
based upon kinematics and dynamics was first used by Newton. This was eventually
overtaken by the growing use by Rayleigh of dimensional analysis.
Here the idea of similitude is used for two purposes. First, to gain physical insight
into a solution that has been obtained using dimensional analysis. This is because
the latter form of solution, as presented here, is both rigorous and straightforward
especially for complicated phenomena. Also, as shown in Sections 5.6 and 2.10,
dimensional analysis can resolve real difficulties that have arisen in the past.
Secondly, it is used, with reservations to be discussed here, when a phenomenon
cannot be stated in terms of functional relations so that the very basis of dimensional
analysis is lacking. This, for example, provides a different application to anatomical
studies some of which will be described later.
11.2 Physical Significance of Non-dimensional Groups
It is often, but by no means always, possible to ascribe a physical significance to
a non-dimensional group in the form of a product of two or more physical quantities.
This can aid understanding of the physics of a phenomenon. The following are just
a few examples.
11.2 Physical Significance of Non-dimensional Groups
275
11.2.1 The Physical Significance of Reynolds Number
The non-dimensional product of .V `/=, which was introduced in Section 1.7,
has a physical connotation. An inertial force in a fluid flow is represented by the
product of a mass and an acceleration; the former is represented by `3 , the latter
by u du=ds which is represented by u.u=`/. The inertial force is then represented
by `3 u2 =` D u2 `2 . A viscous force is represented by the product of an area and
a shear stress or by `2 .u=`/. The ratio of these two forces is .V `/= which is
called the Reynolds number.
This non-dimensional product has been named after Osborne Reynolds even
though Stokes first demonstrated the dependence of a flow on this parameter [6]
some 35 years before Reynolds famous paper [7]. This Reynolds number is given
the symbol Re – Duncan says that Re is an etymological abomination as that should
mean .R e/! [2]. This practice of naming non-dimensional groups after persons,
eminent or otherwise, has long got completely out of control. Massey as far back as
nearly four decades ago, lists 282 of these [4]; others abound. The result is duplication, repetition and other aids to confusion. Here few such names are used.
11.2.2 The Physical Significance of Further Groups
The ratio of the force on a body immersed in a uniform stream to the above representation of the inertia force is,
Body force
F
D
:
Inertia force
U 2 `2
This is the non-dimensional group introduced in Section 1.6. It has been called the
Euler number or the Newton number.
The gravity force can be represented by the product of g and a volume or by
g`3 . Dividing this into the inertia force gives the non-dimensional group of:
Inertia force
U 2 `2
U2
D
D
:
3
Gravity force
g`
g`
This ratio is named the Froude number which then represents this ratio of these two
forces in a fluid flow.
A surface tension force can be represented by `; that is the surface tension times
a length. Dividing this into the inertia force above gives:
Inertia force
U 2 `2
U 2 `
D
D
:
Surface tension force
`
This ratio of forces is called the Weber number.
276
11 Similar Systems
The buoyancy force in a liquid resulting from a temperature difference is represented by g`3 ı. With the coefficient of volume expansion given by:
ˇ
1 @
:
@T
Then the buoyancy force is represented by:
g`3 ˇT :
Multiplying this force by the representation of the inertial one and then dividing by
the square of that of the viscous force from above gives:
Inertia force Buoyancy force
U 2 `2 gˇT `3
2 gˇ`3 T
D
D
.Viscous force/2
.U `/2
2
which is named the Grashof number. This non-dimensional group is now seen to
represent a combination of three forces.
In a compressible flow of a gas the stress is represented by ıp and the strain
by ı= which latter is non-dimensional. The bulk compressibility modulus, C , is
defined as:
C 1 d
dp
so that:
Stress
1
D
:
Strain
C
The stress force is then represented by:
Stressforce D Strain
`2
C:
Then we have that:
Inertia force
U 2 `2 C
d
D
D U2
2
Stress force=Strain
`
dp
which is the Mach number.
Considering now the phenomenon of heat transfer we have that the convection
heat rate is given by:
QP conv D h`2 T
11.2 Physical Significance of Non-dimensional Groups
277
and the conductivity heat rate is given by
QP cond D `2 :.T =`/
The ratio of these two heat rates is then:
QP conv
h`2 T
h`
D 2
D
` .T =`/
QP cond
which is called the Nusselt number.
We now turn to the phenomenon in electro-chemistry where there is an electrostatic streaming current in a fluid flow [8]. The result of the application of dimensional analysis is given in Section 8.16. It is as follows:
is2
"U D" DC U `
D
f
;
;
;
:
(11.1)
U 4 "`2
0 ` 0 `2 D The convection current is the charge per unit time which can be written as:
Charge density Volume
Unit time
3
`
D
`=U
is D
D `2 U :
The electrical force is given by:
Field force D Field Charge
D E`3 :
The Poisson relation is represented by:
E
D
:
"
`
Thus:
2 `4
"
is2
D 2 :
U "
Field force D
Dividing this by the inertia force gives:
iS2
Field force
D
Inertia force
U 4 "`2
(11.2)
278
11 Similar Systems
which is the first non-dimensional group in Equation 11.1.
The conduction current is represented by:
Conduction current D `2 E
D
`3
"
so that:
Convection current
"U
D
Conduction current
`
which is the second non-dimensional group in Equation 11.1.
The diffusion current is represented by:
Diffusion current D F .DC rCC D rC / `2
D FD` .CC C /
D D` :
Thus:
Diffusion current
D"
D 2:
Conduction current
`
This is the third non-dimensional group in Equation 11.1. The fourth group is just
a measure of the ratio of the mobility of the positive and negative ions whilst the
fifth group is the Reynolds number.
11.3 Numerical Value of a Group
It is important to recognise that whilst these non-dimensional groups have a physical
significance they do not give a direct numerical measure or even of the order of the
ratios that they represent as Kline pointed out [5]. For example, as seen in Figure 1.5,
the value of the Reynolds number that indicates the boundary of a flow for which
the viscous force dominates is Re 4 whilst that for the case where the flow is
independent of the numerical value of the viscosity is about 104 . Whilst this gives
a physical meaning to this non-dimensional product it is nowhere near a measure
of the order of magnitude of this ratio. Again, for example, on a slender aerofoil
at a low angle to an oncoming flow, when the value of this non-dimensional group
is of the order of 108 , the force due to the pressures can be of the order of only
one twentieth or less of that due to viscous shear. Again, values of the Grashof
11.5 Similarity in Anatomy
279
number are typically of the order of 1012 and some of the groups of Equation 11.1
are typically of the order of 1014 .1
11.4 The Use of Similarity
When the physics of a phenomenon is known there have been attempts to replace
dimensional analysis by adopting concepts of similarity. This is done by specifying the existence of a group through its physical representation as, for example, by
groups such as considered in Section 11.2. above. This procedure can be acceptable
in problems in simple kinematics and Newtonian mechanics.
It does, however, raise considerable difficulties in more complex phenomenon
such as with compressible flow. Whilst groups such as the Reynolds, Mach, and
Grashof numbers can be deduced in this way there is no similar argument that will
bring in the ratio of the specific heats. There is an even greater difficulty in that there
is no argument in terms of such non-dimensional ratios which will deduce the need
for inclusion of any units-conversion factors.
Thus such an approach based upon similarity is very limited in these sorts of
application.
11.5 Similarity in Anatomy
D’Arcy Thompson, starting in the nineteenth century, made great use of similarity
in anatomy [9, 10].
One interesting example that he reported was on the measurements that he made
of the locations of the joints along the front legs of the ox, the sheep and the giraffe.
The diagram that he gave is reproduced as Figure 11.1. The numerical values that
he reported were as given in Table 11.1.
Here values of x are numerical counters for the joints on a scale of 0–1. Values
of y are measured lengths on a scale of 0–100. Values from this Table are shown
plotted in Figure 11.2. Here the separate curves become a function of L=D. For
correlation, these curves suggest a power equation of the form:
.1 x/ D k1 .1 y/k2 :
(11.3)
With x D 0 at y D 0 then k1 D 1.
To obtain a first estimate of k2 , Equation 11.3 is differentiated to give:
dx D k1 k2 .1/.1 y/.k2 1/ dy :
1
This demonstrates the remark made in the Preface that dimensional analysis cannot give numerical values. Yet there have been proposals that all non-dimensional groups have a numerical value
of the order of unity.
280
11 Similar Systems
Figure 11.1 Reproduction of
leg proportions measured by
D’Arcy Thompson (see [9])
Table 11.1 (See [9])
Joint indicators (Figure 11.1)
Numerical locations; x
Length values; y
Ox;
0
Sheep; 0
Giraffe; 0
Animal;
L=D
Ox
2.5
Figure 11.2 Plot of the proportions of the leg elements:
symbol codes; ı, Ox; ,
Sheep; , Giraffe
a
0.25
b
0.5
c
0.75
d
1.00
18
10
5
27
19
10
42
36
24
100
100
100
Sheep
3.7
Giraffe
6.4
11.5 Similarity in Anatomy
281
Figure 11.3 Correlation of curve slopes
At y D 0:
dy
dx
D
0
1
k1 k2
so that from Equation 11.3,
k2 D 1=.dy=dx/0 :
(11.4)
Values are shown plotted in Figure 11.3. These indicate a proportional relation so
that from Equation 11.4 we have that:
k2 D ˛
L
:
D
Thus Equation 11.3 becomes:
ln.1 x/ D ˛
L
ln.1 y/ :
D
(11.5)
This correlation is confirmed in Figure 11.4. The regression straight line gives ˛ D
0:923. In Figure 11.5 the plot is of values computed from Equation 11.5, denoted
by yc and the values in Table 11.1. This Figure shows the final correlation accuracy
where values of y are compared with those calculated from Equation 11.5. The
calculated mean error was 8.5 %; the band width drawn is for ˙10 %.
This correlation is very good for what might initially be thought of as three very
disparate animals: it is much better than D’Arcy Thompson realised.
Other studies that have been reported have been into the swimming of fish [11]
and into the flying of birds [12]. Lighthill, for example has demonstrated the impor-
282
11 Similar Systems
Figure 11.4 Correlation plot
Figure 11.5 Final correlation
plot
tance of the criterion set by the non-dimensional group of:
fa D
2 nc
U
for the beating of bird and insect wings. Here fa is the aerodynamic frequency
parameter, n is the beat frequency, c is the mean wing chord and U is the flight
speed [13].
A review of the use of non-dimensional groups in assessing biological phenomena has been given by Vogel [14]. This demonstrates the wide range of applications
to these phenomena. A further general contribution was made in 1977 [15].
References
283
11.6 Concluding Comments
It is seen that similitude is particularly useful when there is no way of starting dimensional analysis with the statement of an equation forming a functional relationship.
Thus it is particularly applicable to the study of animal structure and behaviour. Like
dimensional analysis it does not give a numerical answer but it enables the obtaining
of correlations between variables.
Exercises
11.1
11.2
11.3
It has been proposed that the comfortable walking speed of humans accords
with the legs swinging as if they were pendulums at their natural frequency
of swing. Show that then the walking speed is proportional to the square root
of a person’s height
The medical profession measures the level of obesity in human beings by
the value of a ‘Body-mass index’, B. This is defined by B D m= h2 where
m is the body mass in kilograms and h is the height in metres. Comment on
the validity of this criterion as used for this purpose. Derive a valid one from
the assumption that obesity is measured as a ratio of the mass of fat content
to the total mass. A criterion that can be readily measured is required.
For a member of a rowing eight, assume that the force exerted by the muscles
is proportional to the square of the rower’s size and the distance over which
this force is exerted is proportional to the size. Then assume that for such
a slender boat the wave resistance is negligible so that the resistance then
is proportional to the square of the velocity and the square of the boat size
related to the immersed volume which is proportional to the buoyancy force.
Then show that the heaviest crew should win.
References
1. J Palacios. Dimensional analysis (English Ed.), MacMillan, London, 1964.
2. W J Duncan. Physical similarity and dimensional analysis; an elementary treatise. Edward
Arnold, London, 1955.
3. E de St Q Isaacson, M de St Q Isaacson. Dimensional methods in engineering and physics,
Edward Arnold, London, 1975.
4. B S Massey. Units, dimensional analysis and physical similarity, Van Nostrand Reinhold,
London, 1971.
5. S J Kline. Similitude and approximation theory, McGraw-Hill, New York, 1965.
6. G G Stokes. On the effect of the internal friction of fluids on the motion of pendulums, Trans.
Camb. Philos. Soc., Vol. 9, Pt. 2, No. 10, pp. 8–106, 1856 (Read 9th Dec. 1850).
7. O Reynolds. An experimental investigation of the circumstances which determine whether
the motion of water shall be direct or sinuous, and the law of resistance in parallel channels,
Philos. Trans., R. Soc., Vol. 174, pp. 935–982, 1883.
284
11 Similar Systems
8. J C Gibbings, E T Hignett. Dimensional analysis of electrostatic streaming current, Electrochim. Acta, Vol. 11, pp. 815–826, 1966.
9. D’Arcy W Thonmpson. On growth and form, Cambridge University Press, 1917 (1948).
10. D’Arcy W Thompson. (Letter), Nature, Vol. 95, p. 2373, April 1915.
11. M J Lighthill. Hydromechanics of aquatic animal propulsion, An. Rev. Fluid Mech. Vol. 1,
pp. 413–446, 1969.
12. T Y-T Wu, C J Brokaw, C Brennen (Eds.). Swimming and flying in nature, Plenum Press,
New York, 1975.
13. M J Lighthill. Aerodynamic aspects of animal flight, In; T Y-T Wu, C J Brokaw, C Brennen
(Eds.). Swimming and flying in nature, Vol. 2, pp. 423–491, Plenum Press, New York, 1975.
14. S Vogel. Exposing life’s limits with dimensionless numbers, Phys. Today, November 1998,
pp. 22–27, American Institute of Physics, 1998.
15. T J Pedley (Ed.) Scale effects in animal locomotion, Academic Press, London UK., 1977.
Appendix A
Derivation of Dimensions of Quantities
Notation
A
B
c
C
CV
D
Di
E
F
h
H
I
j
k
L
m
n
nP i
P
q
QP
R
S
T
u
U
W
WP
Area
Magnetic flux density
Concentration
Electrical capacitance
Coefficient of specific heat
Electric displacement
Diffusion coefficient
Electric field; illumination
Force; luminous flux
Specific enthalpy
Magnetic field strength
Luminous intensity
Electric current density
Thermal conductivity
Electrical inductance; luminance
Mass
Normal to isotherms
Diffusion flux rate
Magnetic pole strength; pressure
Electrical charge
Heat rate
Electrical resistance
Specific entropy
Temperature
Specific internal energy
Internal energy
Work
Power
J.C. Gibbings, Dimensional Analysis. © Springer 2011
285
286
"
A Derivation of Dimensions of Quantities
Permittivity
Electrical conductivity
Permeability; viscosity
Density
Shear stress
Electrical potential
The dimensions of several electrical, magnetic, light, thermal and mechanical
quantities are now listed. In each case the derivation is outlined.
The dimensions of M, L, T, A, and C are used.
A.1 Electro-magnetic Units
1. The Maxwell equations can be written as:
D D "E ;
q
rD D 3 ;
L
(A.1)
(A.2)
@D
;
@t
B D H ;
r B D 0;
@B
r E D :
@t
rH Dj C
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
2. Permittivity:
F D
q2
I
"r 2
"
A2 T 2 T 2
A2 T 4
D
2
ML
L
ML3
3. Magnetic pole strength:
W D 4 ipI
p
ML2 1
ML2
D
T2 A
AT2
4. Permeability:
p2
F D
I
`2
ML2
AT2
2
1 T2
ML
D 2 2
2 ML
L
A T
A.2 Magnetic Units
287
5. Electrical potential:
i D W ML2
I
T3
ML2
AT3
6. Electrical field:
ED
@ ML
@n AT3
7. Electrical conductivity:
j D EI
A A2 T3
A2 T 3
D
L2 ML
ML3
8. Electrical capacitance:
C D
q AT AT3
A2 T 4
D
2
1 ML
ML2
9. Electrical resistance:
RD
ML2 1
ML2
3
D 2 3
i AT A
A T
10. Electrical inductance:
LD
ML2 T
ML2
3 D 2 2
di=dt AT A
AT
11. From Equation A.1:
D D "E
A2 T4 ML
AT
D 2
3
3
ML AT
L
12. From Equation A.2:
rD D
q
AT
D 3
3
`
L
A.2 Magnetic Units
13. Magnetic field strength:
H
D A
A
D j C 2 C 2I
`
t L
L
H
A
L
288
A Derivation of Dimensions of Quantities
14. Magnetic flux density.
B D H ML A
M
D
2 2 L
A T
AT2
A.3 Diffusion
15. Diffusivity.
nP i D cDi
@ ci 1
L2
I Di 2 L3 L D
@n c
T
LT
A.4 Illumination units
16. Luminous intensity, I , cd:
I C
17. Luminous flux, F , lm:
F I ˝C
18. Luminance, L:
L
I C
` 2 L2
19. Illumination, E, lux D lm m2 :
ED
dF C
dA L2
20. Exposure (camera), H , lux sec:
H D Et
CT
L2
21. Mechanical equivalent of light, P0 , watts lm1 :
1 dE
P0 dt
ML2 1
ML2
P0 3 D 3
C
T
T C
F D
A.6 Mechanical Units
289
A.5 Thermal Units
22. Specific internal energy, u.
uD
U ML2 1
L2
2
D 2
m
T M
T
23. Thermal conductivity, k.
QP
@T
Dk
I
A
@n
k
ML2 1 L
ML
D 3 I
3
2 T L
T 24. Coefficient of specific heat, CV :
CV u L2
T T2 25. Specific enthalpy, h:
h
p
M L3
L2
L2
Cu C
D 2
2
2
LT M
T
T
26. Specific entropy, s:
sD
1 Q 1 ML2 1
L2
D 2
2
mT M T T A.6 Mechanical Units
27. Viscosity, , kg m1 s1 :
du
dn
ML 1 L T
M
2 2
D
LT
T L 1L
D
Name Index
A
Crowe C T
Acosta A J 82, 254
Advisory Committee for Aeronautics 93
Al-Shukri S M 218
Annual Report of the Aeronautical Society of
Great Britain 93
Anon 254
Arnold G C 271
Augustine (Saint) 53
D
B
Bacon F (Lord Verulam) 53
Bacon R H 93
Bairstow L 93
Baker A G 271
Barr D I H 82
Bearman P W 217
Benjamin T B 82
Birkhoff G 82
Booth H 93
Brennen C 284
Bridgman P W 53, 82, 93, 114, 147, 149
Brokaw C J 284
Bronowski J 54, 92
Brook E J 271
Brooke Benjamin T 114
Buckingham E 23, 53, 81, 93
C
Cardwell D S L 92
Cheers F 254
Chichester F 253
Chow V T 254
Churchill S W 146
82
D’Alembert 179
Davies P T 271
Drabble F 255
Duncan W J 1, 283
Dunn J F 53
Dyer K R 254
E
Esnault-Pelterie R 53, 82
Estermann I 218
F
Finney D J 271
Focken G M 81, 114
Fourier J B J 53, 92
Fox L 54
G
Gessler J 114
Giacomelli R 93, 273
Gibbings J C 23, 53, 81, 93, 114, 147, 177,
217, 218, 254, 271, 284
Goksel O T 254
Goldstein S 23, 82, 92, 147, 218, 254
Goodwin J E 218
Gray V H 218
Green S L 54
H
Hall D J 254
Hawkes N 53
291
292
Name Index
Hawking S 54
Hignett E T 218, 284
Huntley H E 114
I
Inglis C (Sir) 55
Ipsen D C 82
Isaacson E de St Q 82, 283
Isaacson M de St Q 82, 283
Palacios J 283
Pankhurst R C 54, 147
Piercy N A V 23, 93, 218
Pistolesi E 93
Plotinus 53
Pomerantz M A 92
Prandtl L 93, 115, 146, 218
Puckett A E 147, 254
R
J
Jeffreys H 23, 53
John J E A 218
Jones N 254
Jones R V 95
K
Keenan J H 53, 81
Kelvin Lord 23
Kline S J 54, 218, 283
Kroon R P 54
Kuethe A M 147
L
Lamb H 254
Langhaar H L 82
Lei H 218
Liepmann H W 147, 254
Lighthill M J 284
M
Macagno E O 92
Mackey A M 177, 271
Madadnia J 218
Manning R 254
Massey B S 82, 114, 271, 283
Masuda S 218
Maxwell J C 23, 53, 93
McFarland M W 93, 254
Melville-Jones B 93
Miller E J 271
Mironer A 254
Morrison L V 53
Munk M M 114
N
Newton I (Sir)
P
92
Rayleigh J W S (Lord) 81, 93, 147
Reed A 219
Reynolds O 25, 93, 283
Riabouchinsky D 93, 147
Roberson J A 82
Robinson R A 218
Rott N 83, 92, 93
Rowe P N 271
Russell B 53, 114
S
Sabersky P H 82, 254
Sage W 218
Saluja G S 177, 271
Schetzer J D 147
Schmitz F W 254
Schon G 218
Sedov L I 53
Shapiro A H 254
Southwell R V 54, 254
Staicu C I 82
Stokes G G 92, 283
Stokes R H 218
Szucs E 92
T
Taylor E S 53, 82
Thompson D’Arcy W 92, 284
Thomson J 92
Thomson W (Sir Lord Kelvin) 23
Tilly G P 218
Truscott G F 218
Turnhill R 219
V
Van Driest E R 82
Vaschy A 93
Vogel S 284
Von Glahn U H 218
Von Karman, Th 254
Name Index
W
Waismann F 53
Wakefield G L 53
Wang L-Z 218
Washizu M 218
White M 146
Wilkie D 271
Williams W 114
Wilson W 53
293
Wu T Y-T 284
Wu Z-N 218
Y
Yalin M S
177, 254
Z
Zemansky M W
53
Subject Index
A
analytical results 179
anatomy, similarity 279
angle, units-conversion factor
dimensions 106
choice of 96
electrical 107
mass and force 97
mass and quantity 104
mass and volume 100
number reduction 45
physical quantities 6
symbolism 85
symbols 5
temperature and quantity 102
vectorial 109
dissipation function 126, 128
drag coefficient 12
drag, fluid 18
particle 192
wave 242
74
B
beam, bending 116
boundary layer, transition fixing
242
C
collisions, spring restraint 227
compact solutions 15
complete equations 28, 37
compressible flow, thermal convection
concepts, derived 28, 39
measure 27
nature 27
primary 27
types 43
consistent equation 5
124
D
data, synthesis 174
points 7
definition, concepts 28
design, model testing 249
dimensional analysis, benefits 19, 150
statistics 264
dimensional system 5
dimensionless groups, non effective 234
dimensions 3
dimensions, angle 106
E
Einstein relation 216
electrical conductivity, liquids 261
electrical dimensions 107
electromagnetic field energy 64
electrostatic charging, fluids 199
experimental comparison 202
physics 200
equality, dimensional 33
equations, complete 37
uncoupled 121
errors, dimensionless 256
Euler number 275
experiment, errors 256
number of readings 19
range of application 156
validity criterion 167
experimental data 6
experimental limits 10
295
296
Subject Index
experimental results, interpretation
extension 29
extrapolated solution 120
extrapolation, partial 239
270
28
27
M
F
Mach number 128, 276
magnitude, constant relative 33, 35
pi-groups 121
mass, oscillating 13, 62
non-linear 62
Maxwell 5
measurement 3
limitations 162
mixing length, turbulence 186
mixing processes 264
model testing, application 220
engineering design 249
essence of 221
modelling, analytical input 238
complexity 247
distorted 244
full-scale comparison 234
flow patterns, atmospheric 236
fluid mechanic force 16
force, definition 32
frictional resistance 87
Froude number 240, 275
function, general 11
functional relationship 11
operations, limit 36
G
gases, kinetic theory 204
Stokes law inapplicability
Grashof number 142, 276
gravitation law 119
linear scales
logical steps
215
H
N
heat exchanger 66
history, dimensional analysis 83
first stage 84
similitude 84
Hooke’s law
inapplicability 227
hydraulic turbine, deduction of results
non-dimensional groups, numerical significance 278
P
Kelvin (Lord) 3
kinetic theory of gases 204
diffusion 213
Einstein relation 216
electrical mobility 214
internal energy 207
mean-free path 205
pressure, temperature 209
thermal conductivity 212
viscosity 211
partial modelling, limits 233
particle, abrasion 191
fragmentation 198
impact classes 197
shape 199
physics, assessment 250
pi theorem 15, 57
general results 76
generalised 59
previous proofs 67
transformation 60
pi-groups 11
magnitude 121
pipe flow, liquid 72
Reynolds’ experiment 85, 90
planetary motion 118
Prandtl number 127
pressure coefficient 243
properties, thermodynamic 129
L
Q
J
jet flow 190
impact of 63
K
length
29
258
quantity, definition
32
Subject Index
297
R
Rayleigh–Riabouchinsky problem 134
regression analysis, non-dimensional 266
relative magnitude 28
results, experimental 258
Reynolds number 12, 127
physical significance 274
U
uncoupled equations 121
units 3
units, reference measure 27
units-conversion factors 3, 28, 34, 47
angle 74
dimensionless groups 48
dimensions 44
inclusion 47
universal constants 47
S
scale modelling, partial limit 233
scales, linear 28
similarity 273
anatomy 279
similitude 274
solutions, asymptotic 120
extrapolated 120
statistics, non-dimensional variables
structural frame 75
structures, scale model limits 231
struts, deformation 228
supplementation 116
symbolism, dimensions 85
synthesis of data 174
Systeme International d’Unites 5
jet flow 190
log-law 188
mixing length 186
physical nature 180
power law 184
264
T
thermal convection, compressible flow 124
incompressible flow 131
natural 139
thermodynamic properties 129
time, definition 30
transformer, electrical modelling 225
turbulence, complexity 180
V
validity criterion 167
variables, choice of 72
dependent, alternate 155
effectiveness 165
influence of 159
missing 157
reduction of 151
superfluous 157
vibration, stretched wire 111
W
wear rate, particle abrasion 195
Weber number 275
Windmill, model test 222