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Dimensional Analysis

J.C. Gibbings

Dimensional Analysis

123

Emeritus Reader J.C. GibbingsUniversity of LiverpoolDepartment of EngineeringBrownlow HillLiverpoolL69 3GHUK

ISBN 978-1-84996-316-9 e-ISBN 978-1-84996-317-6DOI 10.1007/978-1-84996-317-6Springer London Dordrecht Heidelberg New York

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Preface

. . . – a University, taken in its bare idea – has this object –. Iteducates the intellect to reason well in all matters, to reach outtowards 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 honourablehistory. It was developed by the greatest of scientists and mathematicians; such in-cluded Newton, Fourier, D’Arcy Thompson, Vaschy, Rayleigh, Buckingham, Ri-abouchinsky, Einstein, Bridgman and Sedov. It is remarkable for its universality ofapplication, increasingly so in recent times.

This book is for engineers and scientists as a student’s learning volume, as a lec-turer’s text, as a graduate’s reference handbook and, as it contains much that is newfrom the author’s work, as a research publication. In serving students it is designedto give instruction both to undergraduates and, in its more advanced applications, itcould form the basis of postgraduate courses. A selection of the very elementary el-ements would be suitable for school pupils of science. Since the definitive FranklinInstitute conference of 1971, there have been significant developments in settingout the totality and ordering of the fundamental logic, in a resolution of outstand-ing problems basic to the analysis and in deriving a general and rigorous statementof the theorem upon which the analysis rests, as well as further contributions. Sothis volume is submitted as the first up-to-date comprehensive presentation of theseand 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 com-ment 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 manyexpositions makes it difficult to advance a defence against the commonly made claim thatdimensional analysis is only effective because the correct answer has previously been oth-erwise obtained. When the position has been reached of publication of an unnecessarilycomplicated derivation which in addition is not soundly based, arriving at an answer that isincorrect, then the time has come to reconsider –

Introductory presentations that give the impression to students that dimensionalanalysis 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 withoutambiguity and lacunae. It is a quality that is necessary in the use of dimensionalanalysis. Scientists practise logic when forming an analytical description of an ob-served phenomenon; mathematics for them is the language for doing this, prosehaving too limited a function. Engineers also practise this logic, together with op-timisation and balance in the process of design. Instilling well-ordered reasoningis a prime function of education which distinguishes it from training. Dimensionalanalysis produces relations between variables that are particularly useful when a‘formal’ analysis is not available. However, alone it does not produce answers in thesense 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 beshown to fail. Its highly valuable strength is to be the support of experiment, throughchecks on the validity of experimental design, in the ordering of the experimentalprocedure, by the enabling of a synthesis of empirical data and in making feasiblesome experimentation; in all these it is a very powerful tool. Not all forms of exper-iment allow the support of dimensional analysis. Where they do, not to employ it isto make the experimental design much the clumsier and to make cumbersome andlimited the resulting output of data. Correctly used, the analysis does not give wronganswers. The vital initial step to precision is the correct formulation of the physicsof the phenomenon being studied. This is stressed strongly here where guidancecomes from examples of errors and supported by exercises. Some of these could beuseful to tutors as topics for discussion. The ancient idea of the pupil learning at thefeet of the master remains an invaluable part of a university education. Resultingfrom teaching experience, Chapter 1 presents an elementary introduction to explainmathematical manoeuvring that can be novel in form to students and to bring themto an easy initial understanding of both the method and the great power of the anal-ysis. The full logic of the analysis in Chapters 2 and 3, is given in a book for the firsttime. Other new matters include, in logical order, the following principal items:

1. A listing of the full logic of the analysis.2. A listing of the basic primary quantities with original fundamental definitions.3. A rigorous procedure for the incorporation of universal constants.4. A general and rigorous proof of the pi-theorem.5. A complete resolution of the Rayleigh–Riabouchinsky controversy.6. An original criterion for a validity of experiment.7. The careful justification of approximations to the full physics of a phenomenon.8. The replacement of unjustified assumption in formal analysis.

Some matters such as the topic of partial modelling, would be of particular inter-est to engineers. Finally, attention is drawn to the wisdom of prior application ofdimensional 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 withProf. Nicholas Rott. I appreciate the kindness of Dr. R.J. Brook who provided a copyof Dr. E.J. Miller’s paper on his milk mixing experiments. The statement of Kelvin’sis given by permission of The Institution of Civil Engineers. The author is gratefulto Mr. Jeremy Benton for his selection of the three pictures related to and of theAirbus 380 aeroplane and these are copyright Airbus and reproduced by permissionof Airbus UK. The picture of shell buckling in Figure 9.5 was kindly provided byProf. Norman Jones and this and the redrawn version of Darcy Thompson’s anatom-ical 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 reproducedby 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 SciencePublishers B.V. The pictures of the helicopter and ship interaction in Figure 9.6are included by permission of Prof. G. Padfield. A few exercises are taken from theauthor’s past examination papers and are reproduced by permission of the Universityof Liverpool. Much assistance in computer preparation of the manuscript was givenby Mr. Roy Coates, by Mr. Michael Gibbings and by Mr. Kevin Rodgers whilstinvaluable support in preparation of all the diagrams was given by Mrs. SandraCollins. Originally I was encouraged by Prof. Sir John Horlock to review the subjectand then Prof. Norman Jones urged me to expand this aim into a book. I am gratefulfor their kindly encouragement.

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Contents

1 An Elementary Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Purpose of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Units and Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Synthesis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Re-ordered Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 Preliminary General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.9 Fluid-mechanic Force on a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 Benefits of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Concepts, Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Summary of Basic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 The Definition of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 The Definition of Primary Physical Concepts . . . . . . . . . . . . . . . . . . . 292.4 The Definition of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 The Definition of Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 The Definition of Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 Summary of Primary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Constant Relative Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Dimensional Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.10 Units-conversion factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.11 Products of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.12 Dimensional Equality in Functional Relations . . . . . . . . . . . . . . . . . . . 362.13 Limitation to Functional Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.14 The Complete Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.15 Derived Concepts and Their Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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2.16 Dimensions of Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . . . 442.17 The Inclusion of Units-conversion Factors . . . . . . . . . . . . . . . . . . . . . . 472.18 Formation of Dimensionless Groups from Units-conversion Factors 482.19 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 The Pi-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1 The Outline Form of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 The Basic Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 The Generalised Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Linear Mass Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 Non-linear Mass Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 Impact of a Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.4 Electromagnetic Field Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.5 Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Prior Proofs of the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.6 The Careful Choice of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.7 The Necessity for a Units-conversion Factor for Angle . . . . . . . . . . . 743.8 General Results from the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 763.9 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 The Development of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 83Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.1 The Case for the History of Dimensional Analysis . . . . . . . . . . . . . . . 834.2 The Onset of Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 The Onset of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 The Developing Use of the Pi-theorem . . . . . . . . . . . . . . . . . . . . . . . . . 854.5 The Place of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendix 4.1 The Reynolds Pipe-Flow Experiment . . . . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 The Choice of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Care in Choosing Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 The Number of Non-dimensional Groups . . . . . . . . . . . . . . . . . . . . . . 965.3 Mass and Force Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Mass and Volume Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.5 Temperature and Quantity Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 1025.6 Mass and Quantity Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7 The Angle Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.8 Electrical Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.9 Use of Vectorial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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5.10 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Supplementation of Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 Information from the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 The Bending of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4 Extrapolated Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Uncoupled Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.6 Forced Convection of Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6.1 Compressible-flow Energy Transfer . . . . . . . . . . . . . . . . . . . . . 1246.6.2 Incompressible-flow Energy Transfer . . . . . . . . . . . . . . . . . . . 131

6.7 The Rayleigh–Riabouchinsky Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1346.8 Natural Thermal Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.9 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7 Systematic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.1 The Benefits of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1507.2 Reduction of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.3 Further Reduction of Non-dimensional Groups . . . . . . . . . . . . . . . . . . 1537.4 Alternate Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.5 Parameter Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.6 Range of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.7 Superfluous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.8 Missing Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.9 Influence of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.10 Measurement Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.11 Effectiveness of Experimental Variables . . . . . . . . . . . . . . . . . . . . . . . . 1657.12 The Validity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.13 Synthesis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747.14 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.1 Analytical Results from Dimensional Analysis . . . . . . . . . . . . . . . . . . 1798.2 Example I: Flow Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.3 The Complexity of Flow Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.4 The Physics of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.5 The Turbulent-Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.6 Prandtl’s Mixing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.7 The Log-law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.8 Jet Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xii Contents

8.9 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.10 Example II: Particle Abrasion in Flows . . . . . . . . . . . . . . . . . . . . . . . . 1918.11 The Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.12 The Wear Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.13 Classes of Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1978.14 Particle Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.15 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.16 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.17 Example III: Electrostatic Fluid Charging . . . . . . . . . . . . . . . . . . . . . . 1998.18 The Physical Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008.19 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.20 The Variables and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.21 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.22 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2048.23 Example IV: Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . 2048.24 The Kinetic Theory of Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.25 Mean-free Path Length in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.26 The Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.27 The Pressure and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2098.28 The Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.29 The Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2128.30 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2138.31 Electrical Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148.32 The Einstein Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.33 Summarised Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.34 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

9 Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2199.1 The Application of Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.2 The Essence of Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.3 The Windmill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.4 The Oil-insulated Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2249.5 Collision Against a Spring Restraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279.6 Inapplicability of Hooke’s Law of Elasticity . . . . . . . . . . . . . . . . . . . . 2279.7 Limitation to Elastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.8 Impossibility of Scale Structural Modelling . . . . . . . . . . . . . . . . . . . . . 2319.9 Limitations to Partial Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.10 Full-scale Comparison Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2349.11 Non-effectiveness of a Single Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 2349.12 Analytical Input Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2389.13 Partial Extrapolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399.14 The Range Limitation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2429.15 The Distortion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Contents xiii

9.16 Complexity of Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.17 Model Testing in Engineering Design . . . . . . . . . . . . . . . . . . . . . . . . . . 2499.18 Assessment of the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

10 Assessing Experimental Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25510.1 Interpretation of Dimensionless Correlations . . . . . . . . . . . . . . . . . . . . 25610.2 Interpretation of Experimental Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.3 Deduction of Physical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25810.4 Dimensional Analysis with Statistical Regression . . . . . . . . . . . . . . . . 26410.5 A Mixing Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.6 The Regression Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26610.7 Statistical Analysis on the Non-dimensional Groups . . . . . . . . . . . . . 26910.8 Summarising Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

11 Similar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27311.1 The Concept of Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27411.2 Physical Significance of Non-dimensional Groups . . . . . . . . . . . . . . . 274

11.2.1 The Physical Significance of Reynolds Number . . . . . . . . . . . 27511.2.2 The Physical Significance of Further Groups . . . . . . . . . . . . . 275

11.3 Numerical Value of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.4 The Use of Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27911.5 Similarity in Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27911.6 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

A Derivation of Dimensions of Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285A.1 Electro-magnetic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286A.2 Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287A.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288A.4 Illumination units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288A.5 Thermal Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289A.6 Mechanical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Chapter 1An Elementary Introduction

It can be truly said that the very easiness of the processof dimensional analysis tends somewhat to encourageinexperienced investigators to plunge into the analysiswithout sufficient preliminary study of the problem.W.J. Duncan

Notation

a Area; oscillation amplitudea, b ConstantsA AreaA0 Units-conversion factorc Velocity of lightC Electric capacitanceCd Drag coefficientd Diameter; distanceD Diameter; drag forcee Spring elasticity; elementary chargef AccelerationF Forceg Gravity accelerationh Heighthp Planck constantH PowerkD Drag coefficientK1, K2, . . . CoefficientsL Inductance` Lengthm MassM0 Molecular molar massn, N Rotational speedp PressureP Propeller powerq VelocityQ Flow quantity rateR Gas constant; electric resistanceRe Reynolds number

J.C. Gibbings, Dimensional Analysis. © Springer 2011 1

2 1 An Elementary Introduction

s Distancet TimeT Temperatureu, v VelocitiesV Flow velocityx Variabley Variablef Function indicator

" Permittivity� Viscosity˘ Non-dimensional product� Density� Potential difference! Frequency

� “Is dimensionally equal”

A Electric current dimensionC Luminous intensity dimensionF Force dimensionH Area dimensionL Length dimensionM Mass dimensionn Quantity dimensionT Time dimension˛ Plane angle dimension� Temperature dimensionS 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 toenthuse the reader for the widespread usefulness of the subject. Thus it is written forthose first coming to it and they include both the student and the research worker inengineering 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 careneeded in application of the full logic. This fine detail is well realised in the contro-versy between two such great masters as Lord Rayleigh and Professor Riabouchin-sky 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 perfor-mance and the analysis of experiment and also the synthesis of the resulting data.The great majority of experiment requires methods of measurement that use numer-ical scales from both defined units and dimensions. Rare exceptions to this are, forexample, botany and anatomy where classification can be in terms of graphical de-scriptions of shape and colour though even here some measure of size is commonlyused. In this book measurement is used as a basis of science and engineering andhence of dimensional analysis.

In 1883 Kelvin stated this importance of measurement in an address to the Insti-tution of Civil Engineers when he said [1]:

“In physical science a first essential step in the direction of learning any subject, is to findprinciples of numerical reckoning, and methods for practicably measuring, some qualityconnected with it. I often say that when you can measure what you are speaking about, andexpress it in numbers, you know something about it; but when you cannot measure it, whenyou 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 tothe stage of science, whatever the matter may be.”

The first step in the present logic requires consideration of a basic feature ofmeasurement in science. There are two matters in measurement; one is that of thedimensions used, the other of the units used. To illustrate: if a pupil states that onecow approximately equals ten sheep then this statement is faulty on two counts. Twopossible correct statements are that the mass of ten sheep roughly equals the massof one cow or alternatively the cost of ten sheep approximately equals the cost ofone cow. Now there are equality of dimensions. In these cases the dimensions are ofmass or of cost respectively. But these statements have to be further qualified. Thefirst must refer to the same unit which could be the kilogramme; the second couldbe the unit of the United States dollar. The first part of these statements gets theequality of dimensions right; the second does so for the units.

The fundamental basis of this is that it is quite meaningless to add quantitieshaving different measures. Addition of physical quantities is only meaningful whenboth the dimensions and the units are identical. There is no useful meaning in addinga length to a force; equally, nor is there in adding acres directly to hectares. It followsthat an equality is under the same restrictions. This principle, though simple, is thefoundation of the development of dimensional analysis. It is the first stage in thelogic of this subject: it is the primary statement as being an acceptable affirmationfrom 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


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) A0 has the dimensions of [area � length�2];(ii) A0 has the units of acres per square yard: this is written as; a yd�2;(iii) A0 has the numerical value of 1/4840.

To illustrate, the square of side 500 yd has an area of;

1

4840� 500 � 500 D 51:7 a

b) where a is measured in hectares (ha) and ` is measured in metres (m) then:

(i) A0 has dimensions of area length�2;(ii) A0 has units of ha m�2

(iii) A0 has the numerical value of 1/10 000.

Now, the square of side 200 m has an area of;

1

10 000� 200 � 200 D 4 ha

c) when a is measured in square metres, denoted here by m2 and ` is measured inm, then:

(i) A0 has no dimensions;(ii) A0 has no units;(iii) 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 unitsfor area and for length – in this example the use of metres – the factor A0 has nodimensions and no units: it can be excluded from the calculating process. Then onlyone and not two units are needed for this problem of the measure of the concept ofarea. The factorA0 is called a units-conversion factor [2]. This feature, that removalof a unit results in the removal of a units-conversion factor, is a general one indimensional analysis.

Just as the amount 2 � 54 is the value of the units-conversion factor to changeinches into centimetres, so the quantity A0 is called a units-conversion factor andrepresents 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 hasa value of unity and is dimensionless, its existence can go unrecognised; but inprinciple it is present.

From these three examples it is seen that, by writing Equation 1.1, then thisinclusion of the units-conversion factor, A0, makes this equation consistent for thedimensions and valid for all sets of units. In the next chapter it will be shown thatsuch consistency and validity is obtained for all the basic physical equations.

This discussion introduces also the idea of the combination of units and dimen-sions. For case (a) (i) above, the dimensions of area being denoted by the symbol Hand those of length by L, then the dimensions of A0 are H=L2. In a similar contextJeffreys pointed out that the dimensions of ` and of A0 could be specified so thatthose of a would follow: the choice is arbitrary [3]. With case (c)(i) the dimensionsof area are L2 and so those of A0 are L2=L2 D 1: that is it is dimensionless. Thisintroduces the concept of cancellation of dimensions.

Another elementary example of this cancellation is that in which velocity multi-plied by time equals length. Denoting the dimension of time by the symbol T, thenin 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 asM, for length as L, and so on [4].

The full set of dimensions in the Systeme International d’Unites together withthe symbols of dimensions is now listed in Table 1.1.

Table 1.1

Quantity Unit DimensionSymbol

Length metre; m LArea metre2; m2 HTime second; s TForce Newton; N FMass kilogram; kg MElectric current ampere; A ATemperature Kelvin; K �Luminous intensity candela; cd CQuantity mole; mol nPlane angle radian; rad ˛Solid angle steradian; sr S


Table 1.2

Quantity Dimensions Dimension symbols

i) Velocity Length/time L T�1

ii) Acceleration Velocity/time L T�2

iii) Velocity gradient Velocity/length T�1

iv) Density Mass/volume ML�3

v) Force Mass � acceleration MLT�2

vi) Pressure, stress Force/area ML�1T�2

vii) Viscosity Stress/velocity gradient ML�1T�1

viii) Work Force � length ML2T�2

ix) Power Work/time ML2T�3

x) Electric potential Power/current ML2A�1T�3

Table 1.3

(1) ıx � x

(2) dy

dxD Ltıx!0

ıy

ıx� y

x

(3) dny

dxn D Ltıx!0ıy

.ıx/n � y

xn

(4)R

ydx D Pyıx � yx

This listing is justified later. The dimensions of physical quantities can then bedeveloped 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 shouldbe noted here that items (i) to (iv), (vi) and (viii) to (x) come from the definitionsof these physical properties. Items (v) and (vii) come from a physical law, they arerespectively Newton’s law of motion and that of the Newton–Navier statement ofviscous shear [5].

Dimensions in the calculus are indicated by the examples set out in Table 1.3where again the symbol � means ‘is dimensionally equal to’.

1.5 Synthesis of Experimental Data

The power of dimensional analysis in ordering and synthesising experimental datais 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 oncominguniform 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- andwhen 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 bya length, `, that could be taken, for example, as the diameter, d . As a result of theflow a drag force, D, is exerted upon the disc.

Over limited ranges – which can only be quantified later in this discussion – ifthe stream velocity alone was varied in this experiment then it would be found thatD / V 2. This result would plot as sketched by the full line in Figure 1.2. Repeatingthe 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. Oneis not immediately justified in concluding that both these results combine to giveD / `2V 2 for this implies a complete family of straight lines on Figure 1.2. To getthe 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 dottedin Figure 1.2, then for five lines, the number of data points required is 52 D 25. Morethan five might be judged necessary for each line [6]. Repeating the experiment byindividually varying the density, � and then the viscosity, �, gives in turn, D / �

and D ¤ f .�/. Extending the previous illustration, in total the number of datapoints required is 54 D 625.

Figure 1.2 Sketch of theexperimental values of thedrag as a function of thesquare of the velocity


It is worth pausing here to counter an erroneous conclusion on the physics ofthis phenomenon. To say that D ¤ f .�/ is not to imply that the viscosity has noinfluence upon the drag. A well known analytical result shows that in any fictitiousflow that has no viscous shear stress, the drag and indeed also any force includinga lift and a thrust, must be zero [6–8]. What has happened in the present example isthat the numerical value of the viscosity is of no influence, which is something quitedifferent. Viscosity causes the flow to separate from the sharp edge of the disc andthis is independent, as is the rest of the flow pattern in the region of the plate, of thenumerical value of the viscosity: then so also is the drag.

A similar result is found, for examples, in the case of flow past buildings, bridgesand other so-called bluff bodies such as some road vehicles [9].

The foregoing experiments will not ensure that all physical quantities that mightinfluence the value ofD have been taken into account. A full consideration requiresan understanding of the physics of the phenomenon. In the present one, recourseto the equations of motion shows the presence of forces from pressure and viscousstresses that are balanced by inertia terms which contain � and V . Forces involve thesize of the system, `, whilst the viscous stresses derive from �. This more carefulreasoning gives preliminary assurance that no important parameter has been omit-ted. Later discussion shows how this conclusion has to be amplified in one importantrespect.

The final result of this particular experiment is that, over some limited range,D / �V 2`2, or:

D D K1�V2`2 : (1.2)

Checking the dimensions of the quantities on both sides of Equation 1.2, throughreference to Table 1.2, gives the dimensions ofD as:

D � MLT�2 :

Also,

�V 2`2 � ML�3 � �LT�1�2

L2 D MLT�2

A comparison with Equation 1.2 shows that, as both terms above have identicaldimensions, then for this equation to be meaningful by having an equality of di-mensions, the coefficient K1 must be dimensionless; it is a pure number. This iswritten 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, `, alsoset in a uniform stream. Following through the arguments and the test as previouslyfor the disc, and again over limited ranges, the result obtained from a series of ex-periments would be:

D / �V `

or,

D D K2�V ` : (1.3)

Reference again to the prior tabulation of dimensions gives

�V ` � ML�1T�1LT�1L D MLT�2 :

Comparison with the dimensions of D from above shows that K2 also is a non-dimensional 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 rangesthe result would be found to be:

D D K3�1=2V 3=2�1=2`3=2 : (1.4)

A check as before, left as an exercise, would show that again K3 is a dimensionlessconstant.

Experiment d)

The final test would be one on the flow through a length of smooth pipe of varyinglength, `, and diameter whilst holding the ratio of length to diameter constant. Againover limited ranges the result obtained would be,

D D K4�V `

with, as for experiment (b),K4 being a dimensionless constant.


1.6 Comparison of Results

The results of all four experiments are collected in Table 1.4.

Table 1.4

(1) (2) (3) (4)D=�V ` D=�V 2`2 CD

a) Disc D D K1�V 2`2 K1

��V `

�

�K1 2K1

b) Sphere D D K2�V ` K2 K2

��V `

�

��1

2K2=Re

c) Plate D D K3�1=2V 3=2�1=2`3=2 K3

��V `

�

�1=2K3

��V `

�

��1=2

2K3=R1=2e

d) Pipe D D K4�V ` K4 K4

��V `

�

��1

2K4=Re

Column 1 of Table 1.4 lists the foregoing results. If it was wished to comparethe drags of these four objects of different shape under like conditions this couldhardly be done by inspection of this column except for items (b) and (d). This latteris only possible because the forms of the two equations are identical and so therelative values of the non-dimensional constants,K2 andK4 give the required directcomparison. However, this comparison is now a much simpler one in not requiringseparate 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. Threemost useful results are now revealed.

First, inspection shows that comparison of resistances is now possible becausethis 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 log-arithmic scales are used to give the four straight lines at the appropriate slope foreach. The general comparison of drag is clearly seen in this greatly condensed pre-sentation of all the proposed four sets.

Secondly, following the previous suggestion of sets of five data points this nowshows a remarkable condensation of the 2500 data points down to only those neededto determine four curves on a single graph: possibly twenty points.

And thirdly, the limits, determined experimentally, over which the data is validare now seen by the end marks on each curve and so are now quantified in a verygeneral form as values of only the combination .�V `/=�, with corresponding lim-iting values of D=.�V `/ and not as separate limits to each of the four variables inthis product group.

1.7 Re-ordered Functions

It follows from the prior discussion of Equations 1.2 and 1.3 that the groupsD=.�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 multi-plication and division of variables forming products which are then non-dimensionalare, 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 theshapes.

The practice of multiplying and dividing non-dimensional groups of variableswhilst retaining a functional relationship will be used here. Teaching experienceshows 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 val-ues of y can be plotted against those of x even if the function is multi-valued: forevery value of x the value of y is known and vice versa. So for every value of xany combination of y and x can be calculated so that this combination can be plot-ted against any other one.; for example, y=x D f2.x

2y/. The changed suffix on fmerely indicates a fresh form of the function; a different shape graph. This is illus-trated in Figures 1.4(a) and (b). The point ‘A’ in Figure 1.4(a) transforms into thepoint ‘A’ in Figure 1.4(b). In the same way, other forms of presentation of the com-parison in column 2 are possible. For example, and bearing in mind Equation 1.5,

Figure 1.3 Experimental results for the four different shapes


Figure 1.4 Illustration ofa transformation of variables

˘1 D ˘2=˘3 D f Œ˘3� and this gives the set of relations in column 3. Now com-parison 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

CD � D.�1

2�V 2`2

�

whilst the non-dimensional number .�V `/=� is called the Reynolds number andgiven the symbolRe. Making these changes gives the final column 4 and the graphsin these terms are shown in Figure 1.5.

This final synthesis of all the data of at least 625 data points representing justone object, into just one curve, requiring a mere five data points, is only one of theremarkable powers of dimensional analysis. It will be noted that the reproductionin Figure 1.5 of the lines in Figure 1.3 are indicated by the markers on extendedcurves that cover much greater ranges than shown in the latter figure and which areno longer straight lines over the whole length of each. That such general curves canbe demonstrated to exist for each shape of these objects will be shown later as yetanother power of dimensional analysis.

2 In the older European literature, which is still used for data, kD is defined without the factor of1=2 so that kD D CD=2.

1.8 Preliminary General Analysis 13

Figure 1.5 Experimental results for drag coefficient against Reynolds number for the four differentshapes

1.8 Preliminary General Analysis

Equation 1.5 is a particular example of a general result that can be obtained by useof what is known as the pi-theorem. The principles of this theorem have been givenelsewhere as a generalised proof [11]. It will be illustrated here initially by justa 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 elasticspring 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 singlejerk to set the solid oscillating vertically with a frequency of !.

The physical laws involved are:

a) Hooke’s law of linear elastic springs.b) Newton’s law of motion.

For (a) the force exerted by a linear spring can be represented by the spring elastic-ity, e, which measures the force per unit deflexion. For (b) the acceleration can berepresented by the frequency, !, and the amplitude of the oscillation, a. The forcefrom the spring is accounted for under (a). Preliminary inspection recognises thatthere is also a weight force acting on the mass. However this force merely causesa fixed deflexion of the spring; obtaining vibration requires an oscillating force andthis comes from the spring alone. There is no aerodynamic force because a perfect


Figure 1.6 Sketch of thearrangement for a springmounted mass

vacuum is specified. Finally, the mass of the spring is taken as negligible comparedwith that of the suspended solid.

Setting out the dimensions gives the following lines (a) and (b):

(a) — Variables ! m a e

(b) — Dimensions 1T M L M

T2

It has just been specified that:

! D f1Œm; a; e� (1.6)

so that:

!2 D f2Œm; a; e� (1.7)

the subscripts on the function indicators emphasising the difference of these twofunctions. Then, noting line (b) above, Equation 1.7 is rewritten as:

!2

ee D f2Œm; a; e� : (1.8)

Setting out the dimensions now gives:

(c) — Variables !2

em a e

(d) — Dimensions 1M M L M

T2

By dividing !2 by e, a quantity has been obtained for which the dimension in T hasbeen cancelled out.

It is specified that Equation 1.8 must be such that the dimensions of both sidesof this equation are identical. Considering first the dimension of T then, observingline (d) in conjunction with Equation 1.8 shows that now only the variable e containsthe dimension T. This means that this equation must be of the form of:

!2

ee D ef3Œm; a� (1.9)

1.8 Preliminary General Analysis 15

for the dimension in T to balance; there is no other permissible form. The variablee then cancels out leaving Equation 1.9 as:

!2

eD f3Œm; a� : (1.10)

Going back to lines (a) and (b) above, it is seen that by cancelling and thus eliminat-ing the dimension in T, the variable, e, on its own, is now eliminated. It is combinedas effectively the single variable of .!2=e/.

Now considering dimensional equality in M and noting line (d) above, Equa-tion 1.10 is rearranged to the form:

!2m

e

1

mD f4Œm; a� : (1.11)

Again setting out the dimensions gives:

(e) — Variables !2me

1m

a

(f) — Dimensions 1 1M L

where the unit dimension means that the quantity is non-dimensional. Now Equa-tion 1.11 has to take the form of:

!2m

e

1

mD 1

mf5Œa� (1.12)

for the dimension in M to balance. Thus the variablem cancels so that:

!2m

eD f5Œa� : (1.13)

Now there is the situation that it is not possible to balance the dimension in L fromthe variable a. It is simply concluded that a is not a relevant variable in this problemso that finally:

!2m

eD constant (1.14)

a result that is most useful. A detailed formal analysis would confirm that the vari-able a does not arise because the spring is taken as being linearly elastic. For a non-linear 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 layoutof Compact Solution 1.1.

This can be condensed further, for convenient working, into the form of CompactSolution 1.2 [6]:

This demonstration is an illustration of the pi-theorem and its application. Thistheorem with various ramifications is considered in a general form in Chapter 3.


Compact Solution 1.1

(a) — Variables ! m a e

(b) — Dimensions 1T M L M

T2

(c) — Variables !2

em a e

(d) — Dimensions 1M M L M

T2

(e) — Variables !2me

1m

a

(f) — Dimensions 1 1 L

M


Variable ! m a e

Dimensions 1T M L M

T2

Variables !2

e–

Dimensions 1M –

Variables !2me

– –

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 incompress-ible 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 MLT2

ML3

LT 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 illus-trate acceptable rearrange-ment of variables

It is seen that now both F and � have been divided and multiplied by �. Setting outthe variables and dimensions shows why this has been done as follows:

F�

� V ` ��

L4

T2ML3

LT

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 transformedinto 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 illustrateswhy Equation 1.16 can now be transformed into:

F

�� D f

�

�; V; `;�

�

: (1.17)

Figure 1.8 Sketch to il-lustrate rearrangement ofvariables



F � V ` �

MLT2

ML3

LT

MLT

F�

– �

�

L4

T2 – L2

T

F�V 2 – – �

�V

L2 – – L

F�V 2`2 – – – �

�V `

1 – – – 1

Following the previous discussion and noting the dimensions of the variables givenabove, 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 CompactSolution 1.3.

This layout becomes the operational one for this solution of the pi-theorem. Itgives the final result of:

F

�V 2`2D f

��

�V `

which is rewritten as:

F

�V 2`2D f

��V `

�

: (1.18)

This is the general form illustrated for the drag in Figure 1.5. Thus dimensionalanalysis can both prove the existence of those curves and show the general non-dimensional nature of them.

This example illustrates two important matters. It shows how two or more groupsof variables can be rearranged to cancel a single dimension whilst retaining a func-tional relationship. It is not acceptable merely to operate on variables to canceldimensions. 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 importantmatter is returned to in Chapter 3.

1.10 Benefits of Dimensional Analysis

The principal benefits of dimensional analysis to experiment now start to be re-vealed. In the first example of elastic vibration an associated experiment would notrequire separate measurement of the effects of the variables m, e and a. It is onlynecessary to set up an experiment in whichm and e are fixed at a single value each,then to read the corresponding single value of ! so that the value of the constant inEquation 1.14 is determined. The value of ! is then known for all values of m ande 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 seenfrom Equation 1.18 that each curve can be obtained by varying in the experimentonly one of �, V ,�, and ` and measuring the corresponding values ofD. Clearly, thevariable 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 wouldrequire the following:

a) three variables require three readings for each experimental point;b) say five points as a minimum are required for determination of each curve;c) graphically, this would require a family of curves on a single graph;d) therefore the number of readings would be:

3:52 D 75 readings

Again, to illustrate the fluid drag experiment we have:

a) there are five variables so each point requiring five readings;b) graphically, this would require a family of a family of a family of curves;c) 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; thatis, of three fixed variables and two varied ones.

In general the number of readings for n variables becomes:

nr .n�1/

where r is the number of points needed to determine a single curve.


So as well as this economy of experiment, there is an equally valuable synthesisof the resulting experimental data.

Exercises

1.1 Using the system of dimensions of Table 1.1, derive those for the following:

a) elastic strain energy;b) Young’s modulus;c) Poisson’s elastic strain ratio;d) work;e) heat;f) thermal internal energy;g) coefficient of specific heat;h) thermal conductivity;i) electrical voltage;j) electrical field strength;k) electrical charge;l) electrical permittivity;m) electrical conductivity;n) diffusion coefficient;o) ionic concentration;p) ionic mobility;q) work function;r) Avogadro constant;s) Planck constant;t) Faraday constant;u) Boltzman constant;v) luminance;w) illumination.

1.2 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 quantitiesin that Table have been listed in a logical order of derivation. Set out a similarorder for this change of dimension.

1.3 Using the force, length and time or FLT system of dimensions derive thosefor the items of Exercise 1.1.

1.4 Show that the Reynolds number is non-dimensional.1.5 Check the dimensions of Equation 1.4.

1.10 Benefits of Dimensional Analysis 21

1.6 Show that the following equations which govern the linear motion of a par-ticle satisfy the equality of dimensions:

v D uC f t ;

s D ut C .1=2/f t2 ;

v2 D u2 C 2f s :

Then rearrange each equation so that all the terms are non-dimensional andshow that in each of these three equations three variables are reduced to twonon-dimensional groups.

1.7 Show that the Bernoulli equation, that is:

p

�C ghC .1=2/q2 D constant :

satisfies the equality of dimensions.1.8 From the three pi groups of Section 1.6 derive one that does not contain V

and check that it is non-dimensional.1.9 Consider the physics of the example of Section 1.8 and show how that phe-

nomenon might be extended to the case of the oscillation of a large elasticstructure.

1.10 Consider the physics of the example of Section 1.8 and show how that phe-nomenon might be extended to the case of the time of swing of a pendulum.

1.11 The power, P given by a ship propeller of a diameter d rotating at n radiansper second in water of a density of � is quoted as being given by:

P D constant�n3d 5

�:

What are the dimensions of the constant?1.12 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

nd 3D a numerical constant :

Check the dimensions of this equation.1.13 The force between the two plates of a condenser can be expressed by:

F

"�2D f

�A

d 2

:

where the force is F , the potential difference is �, the permittivity is ", theplate area is A and the distance between the plates is d . Check the dimen-sions of this equation.

1.14 An electrical circuit having a resistance, R, a capacitance, C , and an induc-tance, L, can be made to oscillate at a frequency of !. This can be repre-


sented by:

!L

RD f

�CR2

L

:

Check the dimensions of the two terms.1.15 The equation of state of an ideal gas is given by:

p

�D RT

M0;

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. Determinethe dimensions of R.

1.16 The equation of state of a real gas is given approximately by the equation:

�

p C a�2

M 20

�M0

�� b

D RT :

Determine the dimensions of a and of b and show that this equation then hasa balance of dimensions.

1.17 The “fine-structure constant”, a, which indicates the strength of the electro-magnetic force, is given by:

a D e2

2"0chp;

where,

e is the elementary charge;"0 is the permittivity of free space;c is the velocity of light;hp is Planck’s constant.

Show that a is dimensionless.1.18 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

�n3d 5D f

�V

nd

:

An experiment in a wind tunnel enables a graph of results to be plotted asa single line in the form of this equation. However this is not convenientfor design studies that have to determine a design diameter. How could theresults 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 applicationsof 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 equa-

tions, 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 2Concepts, Dimensions and Units

The results of this investigation have both a practical and aphilosophical aspect.Osborne Reynolds, 1883

Notation

a Area, accelerationA;B QuantitiesAm Unit atomic massA0 Units-conversion-factor for areac Velocity of light, particle velocitye Electron chargeE , Ex , Ey Electrical field strengthEr, Ek, El Energy; radiation, kinetic, luminancef Frequency, accelerationF ForceFa Faraday constantg0 Inertia constant; motionG0 Gravitational constanthp Planck constantH Magnetic field strengthI Electrical currentJ Mechanical equivalent of heatk, k1, k2 MultiplierskB Boltzmann constantkf, ks Scaling factorsK Constant` LengthL Luminancem Mass, electron massme Electron massM0 Molecular ‘weight’N Sound levelNa Avogadro numberp Pressure


26 2 Concepts, Dimensions and Units

q Electrical chargeQ Heat, quantityr RadiusR Gas constantRy Rydberg numbers Distancet TimeT Temperatureu Velocityv, V Velocityw � � C i x; y Unit measuresz � x C iy

˛ Angle"0 Permittivity; vacuum� Field direction� Luminous flux�0 Permeability; vacuum� Density� Potential function Flux function! Angle

Units-conversion factors:c Velocity of lightg0 MassG0 Gravityhp Radiation; Planck constantJ HeatkB Temperature; Boltzmann constantP0 Luminous fluxR Particle-quantityˇ0 Plane angle�0 Solid angle"0 Electrical charge�0 Magnetic field

Dimensions notation:A Electric currentC Luminous intensityE FieldF ForceL LengthM Mass

2.1 Summary of Basic Logic 27

n QuantityH AreaT Time˛ Plane angle� Temperature� Solid angle

2.1 Summary of Basic Logic

In the previous chapter stress was laid upon the necessity in dimensional analysisthat reasoning be quite logical. Thus as a first starting point, such discussion shouldcommence 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 hereis to develop the logic of dimensional analysis from very first principles so that itsbasis 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 betrivial: 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 toerror in more advanced applications; examples appear later in this text: here thefoundations are laid firmly.

Dimensional analysis is founded upon basic principles of science. Not doing thishas in the past resulted in difficulties with and dissensions over this subject. AsJeffreys pointed out during an extended controversy [2], confusion even has comefrom semantic misunderstandings. Other problems arise from the use of definitionsthat are lacking in rigour; one such is the use of mass both in a momentum contextand 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 ofspecific heat should be mass or volume based.

Here the fine details and practices of measurement in science are not consid-ered. But the fundamental principles of measurement have to be discussed becauseonly after that can matters, which are basic in dimensional analysis, be dealt withadequately.

A complete order of the logic of dimensional analysis is now set out. Each itemis discussed in turn in this and the succeeding chapter, each step in the argumentbeing referred back to this list. The order of the logic follows the successive stagessummarised in the following list.

(i) The full definition of a scientific concept is in two distinct and consecutiveparts: first, the nature of a concept is defined; then the definition of themeasure of that concept follows from the first.

(ii) The idea of a ‘primary’ physical concept is defined.(iii) The unit reference measure is defined.(iv) The measure of a ‘primary’ concept is defined as addition of unit measures.


(v) The measure by addition of unit measures, together with a specified originof zero measure, results in a linear scale of measure.

(vi) The existence of linear scales together with common origins results in theconstancy of relative magnitude.

(vii) There is a principle of the meaningful addition of numerical values, thisrequiring both dimensional and unit equality.

(viii) Dimensions are assignable to products of variables so that dimensions inthese products can be multiplied and cancelled.

(ix) The multiplication and cancellation of dimensions is required for Item (vii)so that variables can be combined only as products.

(x) The constancy of relative magnitude is retained for products of concepts.(xi) Dimensional and unit equality is required by Item (vii) for the addition of

products of variables.(xii) There are limitations in dimensional analysis to the permissible functional

operations on equations.(xiii) The complete equation is a consequence of item (xi).(xiv) Derived concepts are in the form of products of concepts with units-

conversion factors and are defined through either an arbitrary defining re-lation or an observed physical law.

(xv) The arbitrary choice of a dimension with its unit measure for a derivedconcept requires the introduction of a units-conversion factor to retain di-mensional and unit equality.

(xvi) The foregoing of an arbitrary measure with its unit measure results inthe removal from a dimensional equation of the corresponding units-conversion factor together with the corresponding dimension.

(xvii) The appearance of units-conversion factors, in a functional statement be-tween variables that govern a phenomenon, is directly related to the inclu-sion of the corresponding defining relations in the analytical model.

(xviii) Within an equation, variables can be grouped in products, the limit to thisgrouping occurring when the products become non-dimensional.

These steps in the logic have been set out before [4]. This particular ordering andtotality 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 care-ful distinction, necessary in science, between the nature of a physical concept andthe definition of its measure is not made. One good example, described later, comesfrom the various attempts by philosophers to define the concept of time where thenature of time is not separated from the measurement of it. The nature of a concepthas to be defined before its measurement can be considered.

There is a difference in approach between the ‘philosopher’ on the one handand 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 rigorousdefinition of the nature of the concept on which to build a rigorous definition of howto measure values of it. The importance of this is made clear in the quotation fromKelvin given in Chapter 1, Section 1.2. Fourier thought of a physical quantity firstbeing defined as being a concept and then as having a numerical measure [7]. Thisapproach is used here.

2.3 The Definition of Primary Physical Concepts

In science, concern is with observation of material things forming a system, as thatterm is defined in a generalised thermodynamic sense [8,9], and in the way that theybehave during a process. Quantification of the state of a system is by quantificationof its properties and these properties derive from concepts. In taking the first stepof defining a concept it is recognised philosophically that all words are defined interms of other words so that there is no absolute of verbal definition [10].

Starting from the foundation of absolute basic ideas, consider first the conceptof 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. Thisdifference is defined here as the concept of extension. Before a definition of itsnumerical measure can be formed it is necessary to define the straight line. This isleft as the Exercise 2.1.

The definition of the numerical measure must then be a consequence of these twodefinitions; it is measured by counting a set of standard lengths along the straightline joining the two end positions; here this is called the length. Basically, this mustbe along a straight line to achieve a unique value. Analytically, using the infinites-imal calculus, the distance along a curve can be calculated as distinct from beingmeasured.

This example illustrates why it first is necessary to define a concept and thento follow this by a separate definition of its measure. This logical approach is thereverse of that proposed by Taylor [11] who wrote that ‘it is important that theconcept – as determined from the rules for its measurement –’. The precision ofmeasurement is controlled by the size of the smallest standard length used in themeasurement; in the language of physics, by the degree of division of the standardmetre. This specifies a limiting tolerance on the accuracy.

The definition of extension is described as a ‘primary’ one because it is quiteindependent of the definition of any other measurable quantity. Extension is a pri-mary concept. These definitions must contain the requirements of the secondarydefinition of length measurement. That is why, in this definition of extension, onehas 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 twopositions. There can be three positions, A, B , and C successively along a straightline as sketched in Figure 2.1. Because the measure of length is specified here by


simple addition, it follows that this measure is by a linear scale as illustrated inFigure 2.1. Then the length AC is the sum of those of AB and BC. If it was by a log-arithmic scale this would not be so. The measurement of length is a difference andthis is analogous to the First Law of Thermodynamics in which internal energy ismeasured 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 theobserver having made the measurement is not concerned, remembering tolerancesand mistakes, that another would record otherwise.

In contrast, the definition of angle is a derived one in that it requires the priordefinitions 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 ofmeasure 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 seemsthat some philosophers use motion and time as synonyms; to a physicist motionis the relation between space and time measured, for example, as a velocity. Ploti-nus [5] criticised Aristotle’s definition as being circular in that it relied on the def-inition of motion which in turn relied on the definition of time. This difficulty isstill discussed in modern times [6]. An approach analogous to that just given forextension and length is now presented.

We can appreciate an accumulation of experience. Then we can appreciate a dif-ference in a total of experience. This difference is now defined as time [13]. It isnot then a definition linked with motion and is another ‘primary’ concept because itstands 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 intime by observing ‘b’ which contains some or all of ‘a’ so that some events either

Figure 2.1 The linear scaleof extension with an arbitraryorigin

2.4 The Definition of Time 31

side of the ‘point’ are common to both. Then ‘A’ and ‘B’ can agree on this ‘point’ intime. 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 pointin time gives a time difference. Again there is an analogy here with the statement ofthe Zeroth Law of Thermodynamics.

The present definition overcomes further philosophical difficulties concerningthe 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 extensionis.

Thus the present definition is analytically precise unlike Bergson’s philosophicalcomment that ‘present is that which is acting and, for which Russell pointed out, ledto a circular definition [15].

As before, the numerical measure of time then follows; in physics it is measuredby counting events such as the oscillation of the current across a crystal. Here thereappears an uncertainty. We have to assume that these events are identical measuresof 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, thereis some evidence for this constancy in that highly accurate measurements showingthat the rotation of the earth is slowing are supported by observations of total solar

Figure 2.2 Illustration of the definition of time


eclipses going back to 700 BC [16]. A further support for this constancy of the unitof 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 experiencedfrom the gravitational attraction upon an object held in one’s hand. One can senseno such attraction when the object is removed from the hand.

It follows from this definition that, unlike those for extension and time, force isan 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 grav-itational attraction upon a reference object in a reference location under referenceconditions. Then using a balance, another force can be measured by the summingof a number of unit balancing forces. Thus the measure of force defined in this wayprovides a linear scale.

As before, this definition of force is independent of other concepts and so isa primary concept.

2.6 The Definition of Quantity

The concept of particle-quantity is defined as the number of elementary entities ina system, such as the number of molecules. The unit of measurement of quantity canbe a single entity or a specified number of them. This definition makes the origin ofthe 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 allothers and so particle-quantity is a primary concept. However, this concept standsalone 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 in-dependent of any other definition of a concept. The first three can also be compre-hended by the human senses. The measures of the first two are defined in terms ofdifferences whilst the latter two each have an absolute zero. These four concepts arehere called ‘primary’ concepts. A unit measure for each of these concepts also canbe 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 inlinear 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 wefollow Bacon who wrote “And it is a grand error to assert that sense is the measureof 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 defi-nition of each concept is independent of all others. Here again we follow Baconwho wrote “For information begins with the senses. But our whole work ends inPractice –” [19]. The definition by the idea of independence of measurement doesnot seem to accord with the choice by Esnault-Pelterie of mass, rather than force asused 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 linearscale of measurement with a scale zero at zero amount. Then if x is the size of theunit measure, for example 1 mm, then a length, `, can be written ` D kx where k isan integer number. A power of ` has a numerical value of à D kaxa. The ratio ofthese powers of two lengths, `1 and `2, is:

à1

à2

D�k1x

k2x

�a

D�k1

k2

�a

;

which is independent of x and so, as Bridgman pointed out [1], the ratio of two mea-sures 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 theCelsius or in the Réamur scales of temperature but not between either and the Kelvinscale.

From the definition of the measure of a primary concept, a multiple of the small-est distinguishable amount is chosen as a convenient unit amount. In principle, allother amounts are then taken as the arithmetical count of the content of smallestamounts.

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. Bywriting:

A D B


four different equalities are commonly implied as follows:

a) The measurement scale of A is identical to that of B. The unit measure must beidentical as must be the origin of the scale.

b) In units that are common to both, the numerical value of A equals that of B.c) The dimensions of A are identical to those of B. Here that is written as:

A � B

d) The vectorial direction of A is identical to that of B. Usually, a special symbol-ism is used to indicate vectorial equality. This can be

A D B or EA D EB :The symbols for dimensions as introduced by Maxwell [20] have been described inChapter 1. Also, the full set of dimensions in the Système International d’Unités (SIunits) 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 primaryconcepts just discussed. All the concepts with their units, excluding those of areaand of force, are those of the SI system.

It is incorrect to identify the use of the dimension equality symbolism with a nu-merical equality. It is as different from the latter as is the directional equality of twovectors.

2.10 Units-conversion factors

As discussed in Chapter 1, to add two quantities having different unit measuresrequires the introduction of a units-conversion factor. Problems have arisen in theliterature over the necessity and choice of introduction of these factors in dimen-sional analysis. Their use requires care and so will now be considered in some de-tail. Some of them, like J in the First Law of Thermodynamics and g0 in Newton’slaw have fallen into disuse largely as a result of the adoption of metric measures. Inthose countries that still use the old Imperial system of units the use of these twounits-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; thatis of 103. One arises from the measurement of volume in litres, the other from theuse 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 coefficientin kg � cal per sq.ft. per ıF per h. The ’X’ Co. Ltd. used its own ’X’ litre based on the volumeof a chemical it produces, whilst a lecturer once quoted a specific impulse for ion thrusters forspace vehicles in W per � lb. Also conversion of units is sometimes taken very seriously: onan archway leading into a British government department a traffic notice once read ’Maximumwidth 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 1.2through 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. Fromthe above example of Equation 2.1, those of area being H and of Length, L, thenthose of the units-conversion factorA0 are HL�2. Also it was shown how dimensionsin products of concepts can be cancelled.

The ability both to cancel and to multiply dimensions is a requirement at a laterstage of dimensional analysis. It follows that concepts are required to be combinedonly as products; no other form of combination allows dimensional cancellation.With linear scales of measure, such products retain the idea of constancy of relativemagnitude. For example, with the units of pressure being FL�2, a change in the sizeof the unit quantity for force means that the ratio of two pressures stays constant justas 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 lengthby y, and using the scaling factors of kf and ks respectively, the two pressures canbe written as:

p1 D kf1x

.ks1y/2I p2 D kf2x

.ks2y/2:

Then the ratio of these two pressures is given by:

p1

p2D kf1

kf2

�ks2

ks1

�2

:

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 isuniversally valid.

This part of the logic is the reverse of Bridgman’s when he specified retentionof absolute relative magnitude and then deduced the requirement of products ofconcepts [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 becauseone group measured force in Imperial pounds whilst another used Newtons [21]


2.12 Dimensional Equality in Functional Relations

To retain meaningful expressions, the additive terms in an equation containing prod-ucts 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 t2 : (2.2)

Then here each term has the following dimensions:

s � LIut � LT�1 � T D LI 12f t

2 � 1 � LT�2 � 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 twoare 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 manipula-tion of complete equations during the generation of solutions is limited to certain op-erational rules. This has been discussed by both Bridgman [22] and by Taylor [11].Addition and multiplication are permissible but other functional operations, such astaking logarithms and forming binomial expansions, whilst generally acceptable nu-merically, dimensionally are only acceptable if all arguments are non-dimensional.The latter restriction is seen from comparison of the dimensions of the terms ofa series. For example:

sinx D x � x3

3ŠC x5

5Š� x7

7ŠC : : :

From the above discussion, and in particular from the right hand side of this equa-tion, it is only dimensionally acceptable when x is dimensionless.

The measures of some concepts are not in the form of products of quantities. Onesuch is that of a regular scale of optical luminance which is related to surface areaby [23]:

lnL D KaC constant : (2.3)

A logarithm of a dimensional quantity is readily seen not to be dimensionally ac-ceptable from the relation of:

ln s D .s � 1/� .s � 1/

2

2

C � � � (2.4)

This shows that this equation is only dimensionally acceptable when s � 1.

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 mea-

sured in a logarithmic scale. This obeys the relation for sound pressure intensity, p,of:

log10 p D n

10C constant ; (2.6)

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 humansight and hearing [23, 24].

Another case where care has to be taken is in the use of functions of complexvariables. From the complex number of:

z D x C iy

we have

x2 C y2 D r2

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

dzD �Ex C iEy (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:

lndw

dzD lnE C i. � �/ : (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 dimensionalanalysis.

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 hasto be of those quantities that are measured in both the same dimension and referred


to a common unit measure. Esnault-Pelterie required them to be of the ‘same phys-ical nature’ which seems much too restrictive [18]. Where this type of addition is asa multiplication then the multiplier must be a dimensionless number.

It has now been specified, for the purposes of dimensional analysis, that dimen-sional equality is required for the addition both of quantities and of products ofquantities. Such equations containing these were called ‘complete equations’ byBridgman [1].

The size of the unit measure of any dimension appearing in a complete equationcan 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, fora linear scale, the equation is unchanged in algebraic form. In semantic terms, if anequation is to provide a universal statement of a real event then this feature of thecomplete 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 lim-its of precision with which the equation models a phenomenon [26]. He introducedthe equality of dimensions as a separate idea. Bridgman, in contrast, clearly linkedthe notation of a complete equation with that of equality of dimensions and givesit the meaning, used here, that its algebraic form remains unchanged by changes inthe size of the unit measure. But Bridgman and other authors do not go as far asspecifying a complete equation as a requirement. For example, Bridgman and Se-dov [27] said that physical regularities (laws) are ‘– generally independent of theparticular system of units of measurement selected from among a set of such sys-tems – That this should be so is plausible — it is so exceedingly improbable as tobe practically impossible – if it did depend on one particular system of measure-ment.’

Bridgman’s order of logic is different from that presented here in that he startedwith the acceptance of the complete equation, then derived the pi-theorem – whichis to follow here – and finally deduced equality of dimensions [1].

The present logic starts from the basic premiss of the requirement of meaningfuladdition and leads on to the result of the complete equation. This procedure auto-matically arrives at Bridgman’s assumed result with no exceptions.

Bridgman’s example in support of his discussion [1] has been repeated by othersbut seems not to be a powerful one. He added the linear-motion relations, v D f t tos D 1

2f t2 to get vC s D f t C 1

2f t2, stating correctly that it is numerically correct.

There is however no equality of either dimensions or units. Also, the latter equationcould be written as v D v.s; f; t/ which is analytically wrong there being a surplusof independent variables. Also the two equations are uncoupled in that with f and tspecified the first equation can be solved for v and then the second independently fors. This latter is an important point first dealt with elsewhere [13] and to be returnedto later.

It will be noticed that the present discussion does not specify the idea that it isessential that an equation must be a complete one for it to represent validly a phys-ical 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 analysisand 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 previouslydefined primary concepts and each is in the form of products of quantities. Somedefinitions introduce the appropriate units-conversion factor. In this table the defin-ing 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 fromtwo primary concepts which are those of length and of time; column six gives thesymbols to be used for the dimensions as being respectively L and T; column sevengives the unit measures in the Système International d’Unités (SI units). Velocitycan require a units-conversion factor when, for example, velocity is measured inknots, that is nautical miles per hour, and length and time are measured respectivelyin metres and seconds.

The definition of angle, as the difference in orientation between two straight linesobserved in the plane containing both, has been left as part of Exercise 2.2. Whenone line is rotated from a position of coincidence with the other until coincidenceis again achieved then that standard angle can be called any one of 360ı, 400 grador 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 UnitsConversion

Dimen-sion

SIunit

Primary Derived Factors Symbols

1 V � s=t 2 L, T m, s2 ! � .1=ˇ0/.s=r/ 1 1 ˛ rad3 Er D hpf 1 14 Q D .1=J /F s 2 15 F D �

G0m2�

=r2 2 16 q D r.F "0/1=2 2 17 a � V=t 1 18 I � q=t 1 1 A A9 m D .F g0/ =a 1 1 1 M kg

10 Ek D 12 mV 2 W Ek � Er 2

11 T D 12 mc2=kB 2 1 � K

12 F D .1=P0/ .dE1=dt/ W E1 � Er 2 1 C cd13 H D F=�0sI 2 1 114 M0 D R�T=p W Q � m=M0 2 2 1 n mol


a straight line to that of a curve we have to invoke the idea of the sum of an infinitenumber 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 thatof the corresponding radius, r . To allow any of the above three acceptable units wehave to write this definition as:

! � 1

ˇ0

s

r; (2.9)

where ˇ0 is the units-conversion factor for angle. This is shown as the second def-inition in Table 2.1. An example given later further illustrates the necessity for thisunits-conversion factor.

Similarly the definition of solid angle is derived from those of length and area asa=.�0r

2/ with �0 as the units-conversion factor: conventionally this latter is taken ashaving a value of unity as it is the practice to take a single unit of length to measureboth a and r .

The third derived concept in Table 2.1 is defined by a physical law. It is Planck’slaw which relates energy of radiation to the frequency and so involves the primaryconcept of time. The constant, hp, can be regarded as a units-conversion factor be-tween those of time and those of energy.

The fourth concept in Table 2.1 relates heat to work and is again an expressionof the physical First Law of Thermodynamics together with mechanics [12]. Againa units-conversion factor is introduced which is J and this is founded upon twoprimary concepts. These are the primary definitions of length and force. Again it isa statement of a physical law.

The fifth definition is the statement of Newton’s law of gravitational attraction. Itis an experimental law which gives a definition of mass. Now the universal gravita-tional 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 isagain based upon a physical law and founded upon the primary concepts of forceand 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 accelera-tion. Otherwise, no units-conversion factor is involved.

The eighth definition, like the seventh, is an arbitrary definition of electrical cur-rent not involving a units-conversion factor. It is based upon one primary and onederived concept.

The definition number nine shows that mass, m, is a derived concept. AcceptingNewton’s law, that acceleration, a, of a certain system is proportional to the causingforce, F , the constant of proportionality is a universal constant for that system. Thatconstant is called the mass of that system. To exclude relativity effects the constantis specified as corresponding to the limit of the relative velocity of the observer


tending to zero. Thus it is written as:

m � g0

aF ; (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 putg0 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 thisrespect. It is still used.

The tenth definition is of kinetic energy with the supplementary equating of thisenergy with the Planck energy of definition number three. The first part is an arbi-trary definition, the second comes from a physical law.

Definition number eleven gives the relation from the kinetic theory of gases that:

T D 1

3

mc�2

kB(2.11)

and so is a physical law defining temperature. It contains the units-conversion factorof kB, the Boltzmann constant.

The twelfth definition is of luminous flux in terms of luminous energy wherenow the mechanical equivalent of light forms the units-conversion factor. It is sup-plemented by again identifying the energy with the Planck energy, which latter isa physical law.

Definition number thirteen is of magnetic field strength in terms of electricalcurrent. The units conversion factor is now the magnetic permeability.

The final definition, number fourteen, is of quantity. The units-conversion factoris the universal gas constant, R.

The molar quantity is introduced from the relations from gases. Those to be ac-counted for in dimensional analysis are:

p

�D

�R

M0

�

T ; (2.12)

m D M0Am ; (2.13)

Am D 1=Na ; (2.14)

kB D R=Na ; (2.15)

.1=2/mc�2 D .3=2/kBT : (2.16)

Using again the symbol � to mean ‘is dimensionally equal to’ and by assigningm � M then the dimensions of kB are, from Equation 2.16:

kB � ML2T�2��1 (2.17)


Also from Equation 2.12

R=M0 � L2T�2��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 de-

noted by ‘n’, then assigning

Na � n�1 (2.20)

as required by the physical concept of Avogadro’s number and by the definition ofamount 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 perunit reference particle. Further, the molecular mass has, from Equation 2.13, thedimensions of

M0 � Mn�1 (2.22)

so that it is not purely a numerical factor; it has dimensions. Then from Equa-tion 2.15

R � ML2T�2��1n�1 : (2.23)

An illustration of the need for the unit of amount of substance to be included in theSI 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 unitsconversion 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 showsthat R is the one for particle-quantity and kB is that for temperature. This mightbe expected from inspection of Equation 2.17 and 2.23; only the latter contains thedimension of n.

The defining relations, listed in this table, are seen to be of two kinds. One, suchas the first one listed, is a definition. The other, such as the third one listed, is theexpression of a physical law.


Of the fourteen definitions given in Table 2.1, seven are listed in the SI sys-tem covering eight dimensions. In that system angle is classed as a supplementaryconcept as once was quantity. The present discussion makes clear that both theseconcepts are indistinguishable in status from the other five.

In summary the types of concepts are now listed:

a) A primary concept where the definition of the concept is followed by the defi-nition of its measure. These are those of extension, time and force.

b) A definition of a primary concept relying on some physical knowledge for thespecification of the particle, for examples, of atom, electron and molecule. Thisis a definition of the measure and the example is that of molar quantity.

c) A definition of a derived concept of which there are two categories, viz.:

(i) A definition based upon primary concepts; examples are, velocity, accel-eration and angle.

(ii) 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, vir-tually proposed the definitions of all concepts to be inseparably linked with thedefinition 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 consideringthe meanings of time and of extension that the present line of argument for primaryquantities 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 expressionof a physical law. This approach would appear to be consistent with Bridgman’sview that “– operations which give meaning to our physical concepts should prop-erly be physical operations, actually carried out –” [29] as long as the operationsleading to the present definitions of primary concepts are recognised as not, inthe first instance, requiring numerical measure. It is relevant that Bridgman goeson to say that “It must not be understood that we are maintaining that as a neces-sity of thought we must always demand that physical concepts be defined in termsof physical operations –” [29]. This is consistent with the present distinction justmentioned, between defining relations formed from definitions and from physicallaws.

Whether a definition or a physical law, all these relations are single proportion-alities 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. Thatis, they can be used to obtain solutions of processes that govern systems. That iswhy the list excludes the Second Law of Thermodynamics. If the use of governingequations leads to more than one solution, as can happen for example in the com-pressible flow of a gas [12], then the Second Law of Thermodynamics can be usedto determine which of the multiple solutions could occur in reality.

The logical steps numbers (xiv) and (xv) have now been explained.


2.16 Dimensions of Units-conversion Factors

From definition number seven in Table 2.1 we can write the dimensions of accel-eration as a � LT�2. Then from definition number nine we get g0 � MLT�2F�1.If we assign zero dimensions to g0, that is g0 � 1, then one of two consequencesfollow. Either we can put F � MLT�2 or m � FT2L�1. By making the units-conversion factor non dimensional we remove one dimension from those needed todescribe the dimensions of the concept. Logical step number (xvi) has now beencovered.

This can be carried further. Several writers have in effect suggested that the num-ber of primary dimensions is one more than the number of primary units-conversionfactors, so that the number of dimensions can be reduced to one. Buckingham sug-gested this in a little-known response to the Rayleigh–Riabouchinsky discussion tobe described later [30]. Later writers such as Wilson [31, 32] have repeated the ideathough without reference to Buckingham.

The total number of dimensions, and hence by the above demonstration, the totalnumber of units-conversion factors is, to some extent, at choice. For example, it haslong been common practice to assign zero dimensions to g0 and J . The commonestform 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 correspond-ing 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 units-conversion factors can be expressed in terms of members of this set. For example,there is the physical relation that,

�0"0 D 1=c2 : (2.27)

Now �0, which is a units-conversion factor listed at item thirteen in Table 2.1, hasbeen assigned the value of 4 �10�7 H m�1. The entry number six in that Tableshows "0 as being the units-conversion factor for charge. Thus c becomes an al-ternative to "0; it is a units-conversion factor where physically c is a limiting valuefor a perfect vacuum. In this sense it is analogous to R. It is related to Maxwell’sderivation in 1860, from his studies of electro-magnetism, that the value of c is inde-pendent of the velocity of the observer. Einstein then made the general assumptionthat c is a universal constant, independent of the choice of axes [33, 34]. This willshow later the comforting result that all units-conversion factors are of universalvalue.

2.16 Dimensions of Units-conversion Factors 45

It follows that one of the entries in Table 2.1 could have been replaced by theEinstein one of,

E D mc2 : (2.28)

Suppose that a problem in dynamics involved force, mass, length and time. Theunits would be those respectively of F , M , L and T . By putting g0 � 1 then fromitem number nine in Table 2.1, the dimensions of force now become,

F � MLT�2 : (2.29)

By making the units-conversion factor, g0, dimensionless the dimension F is notrequired. This idea has been extended by the use of ‘atomic units’ as being conve-nient 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 ob-tained by putting; m D 1, hp D 2 and c D 1. From this practice springs thepractice in astronomy of measuring extension in units of light-years simply to avoidthe continual inconvenience of recording large numerical values.

This process can be continued. If in item number five in Table 2.1, we putG0 � 1then from that relation there is:

F � M2L�2

Comparing this with Equation 2.29 gives:

(i) M D L3T�2

Putting hp � 1 in item number three gives:

(ii) E � T�1

Also:

(iii) E � ML2T�2

Using item (i) gives that:

(iv) E � L5T�4

Comparing items (ii) and (iv) gives:

L � T 3=5

Then from (i):

(vi) M � T�1=5


From item number eleven in Table 2.1 and putting kB � 1 gives:

� ML2T�2

And using (v) and (vi) gives:

(vii) � T�1

Again in item number 14 0f Table 2.1, by putting:

R � 1

Then:

M0 � �L�2T2

So that using items (v) and (vii) gives:

M0 � T�1=5

From Equation 2.22,

Mn�1 � T�1=5

So that:

(viii) n � T�1=5T1=5 D 1 :

From item number 12 in Table 2.1, there is:

P0 �E

t� 1

C

so that putting P0 � 1;

(ix) C �ML2

T3 D T�16=5

Finally, putting ˇ0 D 1;

(x) ! � 1

Thus all seven dimensions as listed in Equation 2.25 can, in principle be replacedby just the single dimension of time. Equally, from this use of linear algebra, sevencould be replaced by any one of the eight dimensions.

However, carrying the reduction of dimensions as far as this is invariably notacceptable for real analytical modelling of real events and processes; this will nowbe explained.

2.17 The Inclusion of Units-conversion Factors 47

If the previous evidence, just mentioned, for the constancy of the measure of thetime and of the velocity of light, c is accepted, then it follows that there is a con-stancy 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 ofevidence.

The idea that the so-called universal constants of science are really just units-conversion 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 ofthem. Further, he makes the philosophical point that “... we can only work withinstantaneous 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 conceptsis 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 generalfunctional statement describing a phenomenon; if indeed any. This has been stressedin the comment by Prandtl that forms the heading to Chapter 6. Sedov also put thematter of units-conversion factors as a prime problem [37]. He said that these factorsmust 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 inpart has been illustrated by the above examples. Philosophically, in any real phe-nomenon, to have absolute precision of understanding, all possible physical phe-nomena must be accounted for. This would require the inclusion in an analysis of allthose relations listed in Table 2.1 which contain the units-conversion factors listedin 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 consid-erations which require an approximate modelling of a real event in order to make ittractable, but rather consider the most general position as follows:

a) Real events on a continuum scale must be influenced to some degree by allphysical phenomena.

b) Therefore all concepts arising from the analytical expression of all these phe-nomena will appear in the list of variables in a general functional statementgoverning the complete event.

c) Therefore all the units-conversion factors associated with all the concepts willhave to be included along with the physical variables.

d) The number of dimensions is of arbitrary choice; the number of units-conver-sion factors is one less.


e) The minimum number of dimensions that can be used is one; the one to be usedis of arbitrary choice.

Before going further, there is an intriguing philosophical corollary to the above. Sup-pose that another completely independent ‘fundamental’ relation – like for exampleNewton’s law of gravitation, F D �

G0m2�=r2 – is discovered, with its correspond-

ing primary units-conversion factor – in this exampleG0. Then either:

(i) we would have no dimension left for measurement; all that would be left toscience would be counting;or

(ii) a completely new concept would have been discovered with its own unit ofmeasurement dimension and not related to any existing physical concepts ex-cept through its units-conversion factor.

If item (i) was true we would not be able to measure and so could not observeanything; nothing would exist. Have we then reached the position where all ‘funda-mental relations’ are now known?

The practical study of a phenomenon requires, as already stated, approximatemodelling and so not all physical phenomena are to be accounted for. Then only thephysical 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 withtheir corresponding units-conversion factors and dimensions, are to be included ina list of the variables and universal constants – that is units-conversion factors –required to describe the phenomenon. This gives a quite rigorous means of assess-ing which units-conversion factors are to be included in the list of variables: unlikeprevious claims, the matter is neither one that is arbitrary nor a pragmatic choicenor is the choice an art rather than a science. It is strictly controlled by the approxi-mations introduced in the modelling of the true physical event. However care has tobe 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-conversionFactors

In the literature, and over a long period, a surprise has been hinted that a non-dimensional group can be formed from hp, e, "0 and c. This group is e2="0hpc

and is called the fine-structure constant. This has even attracted philosophical atten-tion. The fact that this non-dimensional group can be formed from units- conversionfactors 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 fac-tors 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 group-

ings. This derivation is conveniently obtained by the cancellation technique usedin the present operation of the pi-theorem though it is not to be confused with theapplication of that theorem because no functional relation is set out. The tabulationis as follows in Table 2.2.

Thus the groups are as follows:

"0�0c2;

R

NakB; NaAm;

Fa c

Na e;

g20"0c

4

Ge2R2y

;hp"0c

e2; with

m2eG"0

e2:

Taking the square root of the reciprocal of the product of the fifth and seventh groupsand multiplying by the cube of the sixth group gives the group of:

Ry"20h

3pc

g0mee4:

This with the next four of these non-dimensional groups are known relations fromanalysis. Of numbers six and seven groups, the former is the reciprocal of the ‘finestructure 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 isa measure of the strength of the electromagnetic force and the latter is a ratio of thegravitational and Coulomb forces between electrons.

Thus, as Bridgman would have said, we see nothing esoteric about the fine-structure constant being non-dimensional.

2.19 Summarising Comments

The discussion in this chapter has set out the foundation principles of dimensionalanalysis. The whole logic starts from the self-evident premiss of meaningful ad-dition. The basic requirements for further development in the coming chapters are


Tab

le2.

2

g0

hp

G0

"0

kB

R�

0c

eN

aA

mF

am

eR

y

FT2

ML

FLT

FL M2

A2T

2

FL2

FL �FL �

nF A

2L T

AT

1 nn

AT

2

nLM

1 L

g2 0

G2 0

� �

R Na

� �

NaA

mF

aN

a

m2 eG

0

FT4

L4

� �

FL �

� �

1A

T2

LFL

2

� �

"0

e2

� �

RN

ak

B

�0e

2� �

� �

Fa

Nae

� �

1F

L2

� �

1F

T2

� �

� �

T L

g2 0"

0

G0e

2h

p"

0

e2

� �

� �

� �

"0�

0� �

� �

Fac

Nae

m2 eG

0"

0

e2

T4

L6

T L� �

� �

� �

T2

L2

� �

� �

11

g2 0"

0c

4

G0e

2h

p"

0c

e2

� �

� �

� �

"0�

0c

2� �

� �

� �

1 L2

1� �

� �

� �

1� �

� �

� �

g2 0"

0c

4

G0e

2R

2 y

� �

� �

� �

� �

� �

� �

� �

1�

��

��

��


now available. In particular, the place of units-conversion factors has been rigor-ously laid down so that there is a clear method for deciding on their inclusion ina general functional statement that governs a phenomenon.

Exercises

2.1 Define a straight line. As a hint, find out how Whitworth first made a flat sur-face [39] and so define a plane surface. Then consider the intersection of twosuch planes [40]. Then define a circular arc. Consider how these definitionsare needed for a definition of angle. Then go on to demonstrate that a straightline forms the shortest distance between two points. Satisfy yourself on therigorous logic of your argument. Out of interest, compare your definition ofa plane with the logic used in forming the Zeroth Law of Thermodynam-ics [12].

2.2 Define the concept of angle; use similar logical steps of Exercise 2.1. Bronow-ski rather suggested [41] that the ancients made use of the Pythagoras theoremfor a right-angle triangle, in constructing a right angle for their building con-struction. But such is the precision of the ancient temples and the medievalcathedrals that the then scales of length might possibly be not sufficiently pre-cise. Suggest means of constructing, in those times, a very large builders setsquare 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 inuniform temperature change its precision?

2.3 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 pressuredifference of�p, against the gravitational constant g. Which of the following

Figure 2.3 The sketch ofa pump system


groups are non-dimensional?

PQg`2

;PQ

.g`/1=2;

PQg1=2`2:5

�pV

�`;

�p

�V 2;

�p�`6

V 2

�

�V `;

��`5

V

2.4 Using the dimensions listed in column six of Table 2.1, determine the dimen-sions of the units-conversion factors in that Table.

2.5 Repeat the analysis of Exercise 2.4 but replacing the dimension of M with thatof F.

2.6 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.

2.7 The induced electro-motive force (in volts) applied by a solenoid, ", is givenby:

" D ��0An2

`

di

dt;

where:

�0 is the units-conversion factor (Table 2.1);A is the cross-sectional area of the solenoid;n is the number of coil turns;` is the solenoid length;i is the current;t is the time.

show that this is a complete equation.2.8 The photo-electric equation is:

hpf �W D �12

�mv2

m ;

where:

hp is the Planck constant;f is the frequency;W is the work-function energy;m is the electron mass;vm is the emitting velocity.

Show that this is a complete equation.2.9 “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, Decem-

ber 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 equa-

tions, 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 Philosoph-

ical Magazine and Journal of Science (Philos. Mag.), Vol. 33, 7th Ser., No. 216, pp. 26–33,January 1942.


32. W. Wilson. Note on dimensions, The London, Edinburgh and Dublin Philosophical Maga-zine and Journal of Science (Philos. Mag.), Vol. 33, 7th Ser., No. 226, pp. 842–844, Novem-ber 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 AlbertEinstein, 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 meth-

ods, 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 3The Pi-Theorem

In relation to engineering, the qualities required ina mathematical process are utility, generality, and simplicityand of these the greatest is simplicity.Sir Charles Inglis

Notation

a AmplitudeA Surface areaai r Powers of dimensionsC ConcentrationCp Coefficient of specific heatd Pipe diameterD Drag forceD1. . . Di Dimensionse0, e1 Elasticity coefficientsE Electric field strength, elastic modulusf AccelerationF Forceg Gravitational accelerationh Heat transfer coefficientH Magnetic field strengthj Current densityK1, K2 Coefficientsk Thermal conductivityL Pipe length` Scale sizem MassPnn Diffusion raten Number of dimensionsN Number of variablesp Pressure, number of groupsPQ Quantity flow rate, heat rate

Q1. . . Qi VariablesR Resistances Surface area element; distance


56 3 The Pi-Theorem

S Shape variablet TimeT Temperatureu, v, w VelocitiesU Energy per unit volumev Local velocityW Loadx; y VariablesX Body force

Fa Faraday constantg0 Inertia constanthP Planck constantkB Boltzmann constantme Electron massNa Avogadro numberR Gas constantRy Rydberg numberG0 Gravitational constant

ˇ Angleˇ0 Units-conversion factor� Angleı Deflexion"0 Vacuum permittivityP� Heat rate intensity� Conductivity�, �0 Viscosity, magnetic permeability˘ Non-dimensional product� Density� Surface tension� Shear stress Electrical potential! Frequency

G Number of non-dimensional groupsk Number of cancellations for a non-dimensional groupK Number of cancellation levelsm Number of variables in a non-dimensional groupn Number of dimensionsN Number of variables

A Current dimensionL Length dimensionM Mass dimension

3.2 The Basic Outcome 57

T 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 introducedin outline in Chapter 1 through a solution to two simple problems. It generatesthe ability to derive, using a functional transformation, an equation containing onlyproducts of variables and sometimes, a single non-dimensional product. The fullproof of this theorem, in a general and rigorous form [1], provides the final logicalstep 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 itscorrect 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 thevery 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 prop-erty 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 principleof 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 2.1and, or, formed from complete equations of properties such as [2]:

a) The Newtonian equation for viscous shear in a fluid, that is:

� D �@u

@yI

1 Strictly speaking, a units-conversion factor is a universal constant and so should not properly becalled a variable; but to do so here is to follow a general practice by many writers: still it goesagainst the grain not to accord with The Concise Oxford Dictionary.

58 3 The Pi-Theorem

b) The relation for molecular diffusion, that is:

Pnn D �c D@

@x

�cn

cI�

c) The relation for thermal conductivity, that is:

� D �k@T

@xI

d) The relation for electrical current, that is:

j D ��@

@x

with other similar relations. It is noted that these physical relations are eacha 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 ad-equately describes the phenomenon to the desired precision in the sense of the dis-cussion of Chapter 2. The physics must be fully understood so that the physicalvariables together with the units-conversion factors are listed precisely and com-pletely.2

But otherwise the form of the function in Equation 3.1 is not limited: it couldhave a singularity; be a multi-valued one; even have a discontinuity. This generalityis important to the wide application of the pi-theorem.

The pi-theorem of dimensional analysis provides a transformation of Equa-tion 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-dimensionalproduct 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 thatthe pi-theorem transforms the former into the latter whilst still retaining a func-tional relation between the variables in the latter. Also Equation 3.2 retains theformat of being a ‘complete’ equation because all the variables are formed intonon-dimensional products. So important are these transformation features, yet be-ing 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-theoremwill now be given as a generalised proof.

For N variables, there is Equation 3.1 which is to be a ‘complete’ equation. Thedimensions 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 thediscussion of Chapter 2, tabulation is set out as Table 3.1.

Table 3.1

Variable Dimensions

Q1 Da111 Da12

2 : : : Da1rr : : : Da1n

n

Q2 Da211 Da22

2 : : : Da2rr : : : Da2n

n

��

Qi Dai11 Dai2

2 : : : Dairr : : : Dain

n

��

QN DaN 11 DaN 2

2 : : : DaNrr : : : DaNn

n

Here the air are non-dimensional numbers. Some of the variables, Qi , may notrequire all of Dn to describe their dimensions. That is, for one or more i , some orall of air 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,

Qai11 =Q

a11i �Da11ai1

1 =Dai1a111 D 1 ; (3.3)

or generally,

Qai1j =Q

aj 1i � D

aj 1ai11 =D

ai1aj 11 D 1 : (3.4)

Equation 3.4 is effectively a transformation relation. The result of this transfor-mation is tabulated in Table 3.2.

By writing Equation 3.1 as:�Q

ai11 =Q

a11i

�Q

a11i D f

˚�Q

ai12 =Q

a21i

�Q

a21i ; : : : ; Qi ; : : : ;

�Q

ai1N =Q

aN 1i

�Q

aN 1i

�

(3.5)

or

�Q

ai11 =Q

a11i

�Q

a11i D f

˚�Q

ai12 =Q

a21i

�; : : : ; Qi ; : : : ;

�Q

ai1N =Q

aN 1i

��: (3.6)

60 3 The Pi-Theorem

Table 3.2

Variable Dimensions

Qai11 =Q

a11i

�Da12

2 : : : Da1nn

�ai1.�

Dai22 : : : Dain

n

�a11

��

Qi Dai11 Dai2

2 : : : Dainn

��

Qai1N =Q

aN 1i

�DaN 2

2 : : : DaNnn

�ai1.�

Dai22 : : : Dain

n

�aN 1

This retains the previously prescribed operational limits so that a ‘complete’ equa-tion is retained.

Inspection of Table 3.2 shows that Qi is the only variable in Equation 3.6 thatcontains 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:

�Q

ai11 =Qa11

i

�Qa11

i D Qa11i f

˚�Q

ai12 =Qa21

i

�; : : : ;

�Qai1

N =QaN 1i

��: (3.7)

Thus the term in Qi cancels out as does D1 leaving only:

f˚�

Qai11 =Qa11

i

�; : : : ;

�Q

ai1N =Q

aN 1i

�� D 0 : (3.8)

This transformation process may be continued for successive cancellations ofD2, D3, . . . , Dn until there are no more dimensions left for cancellation and all thegroups of variables are in the form of products and are non-dimensional. There isnothing mandatory about the completion of this transformation process; prior stagesgive equally valid transformed equations. Also the order of the cancellation of thedimensions, Di , is of arbitrary choice. This is because Di can represent any one ofthe physical variables.

If each of the air are not zero then, with exceptions to be described, each timea cancellation of a variable is made then:

a) one variable is added by multiplication to each group of variables;b) one dimension is removed.

From item (b) there will be a total of n cancellations and after all possible cancella-tions each group will contain (1 C n) variables and there will be (N � n) groups.

If one of air is zero, say ajr , then the dimensions of the group with Qj will notcontain 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:

ai1=air 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 willcancel also the dimension Dr . The number of groups then becomes .N � n/ � 1 DN � n � 1 because there will be one fewer set of cancellations. These results willnow be illustrated by examples.

This is the generalised proof of the pi-theorem and covers the final logical stepnumber (xviii) as listed in Chapter 2.

It is important to emphasise again that the pi-theorem is a mathematical trans-formation process. This ensures that, in the transformation from Equation 3.1 toEquation 3.2, a functional relation is retained. Also, because only products are de-rived, then from the discussion of Chapter 2, a complete equation is retained. As isto be described in Section 3.5, procedures by Rayleigh, Bridgman and Birkoff wereall clear about this. These matters are emphasised because there are many proce-dures that have been published in which a set of variables is merely reformed intoa set of non-dimensional groups with no reference to the outcome being a functionalrelation 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 inChapter 1 and is illustrated again in Figure 3.1. Returning to this example, thatanalysis led to the result of:

!2m

e

1

mD f .m; a/ :

To balance the mass dimension in this equation it must then take the form of:

!2m

e

1

mD 1

mf .a/

Figure 3.1 The oscillation ofa mass on a spring

62 3 The Pi-Theorem

so that:

!2m

eD f .a/ : (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 nonecan be observed from the physics of this phenomenon. Secondly, the quantity a isa false independent variable. Thus to obtain a balance of dimensions in Equation 3.9the quantity a has to be removed. In any example where this sort of thing happensit is always important to investigate the first possibility. However, inspection of thelisting of the dimensions of the four variables in Equation 1.6 shows immediately inthis case that the variable a is the only one with a length dimension so that it mighthave 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 ex-tension:

F D e0ı C e1ı2 :

The transformation of this equation to a function of non-dimensional groups fol-lows the procedure of the preceding proof. It can be tabulated as before as describedin Chapter 1.

Then the pi-theorem solution is of Compact Solution 3.1.Thus the pi-theorem gives the result that:

!2m

e0D f

e1a

e0

:

Now the amplitude of the oscillation is correctly present in the functional rela-tion.

The following notation is now used

G number of non-dimensional groups;k number of cancellations in each group;K number of levels of cancellation;m number of variables in each group;n number of dimensions;N number of variables.

In the above Compact Solution 3.1 are included the numbers of levels of cancel-lation as denoted by K , and the number of cancellations for each group as denotedby k.



K ! m a e0 e1

1T M L M

T2M

LT2

me0

�

�

e1e0

1 T2�

�

1L

�

�

!2me0

�

�

2 � 1 �

�

�

�

�

�

�

e1a

e0

3 � � � 1

k 2 2

Conclusions from this example are as follows:

(a) The number of variables, N , less the number of levels of cancellation equalsthe number of non-dimensional groups, G; or, G D N � K .

(b) The number of cancellations for each group is less than the number of dimen-sions, n; or, k < n.

(c) The number of variables in each group, m, is given by m D k C 1.(d) 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 noz-zle, into the atmosphere and at the position where the jet is breaking up. This isillustrated in Figure 3.2. The governing equations will be the following:

a) the relation for the viscous force;b) the relation for the surface tension force;c) the Newton law.

The variables will be the surface tension, � , the viscosity, �, the density, �, the jetvelocity at the point of break-up, V , and a size, `. The pressure does not appear inthe equations because the flow is in an atmosphere of uniform pressure and so thereare 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 ofa jet flow


K F � V ` � �

MLT2

ML3

LT L M

LTMT2

1 F�

�

�

�

�¢�

L4

T2�

�

L2

TL3

T2

2 F�v2

�

�

�

�

�

�V�

�V 2

L2� � L L

3 F�V 2`2 � � �

�

�V `�

�V 2`

1 � � � 1 1

k 3 3 3

The transformation of this equation to a function of non-dimensional groups fol-lows the procedure of Compact Solution 3.2.

From this solution Equation 3.10 is transformed to:

f

F

�V 2`2;

�

�V `;

�

�V 2`

D 0 : (3.11)

For this example:

(i) G D N � K;(ii) m D k C 1;(iii) K D n;(iv) k D K .

3.4.4 Electromagnetic Field Energy

The next example is that given by Buckingham [4]. It has presented special difficul-ties. It is of the energy per unit volume, U , of an electromagnetic field. Buckingham



K U E H " �

MLT2

MLAT3

AL

A2T4

ML3ML

A2T2

1 U�

E�

"�

�

�

�

A2

L2AT

T2

L2�

�

2 E"12

�12

�

�

�

�

AL

�

�

�

�

3 U�H 2

E"12

H�12

�

�

�

�

�

�

1 1 � � �

k 2 3

proposed that:

f .U; E; H; "; �/ D 0 : (3.12)

The solution procedure is shown in Compact Solution 3.3.The transformed equation becomes:

U

�H 2D f

"E

H��

"

�

�1=2#

(3.13)

Though the dimensions used to described the variables of Equation 3.12 are fourin number, that is, M, L, T, and A, in this case the dimensions can be taken aseffectively being those of the three cancelling quantities, that is:

ML=A2T2; T2=L2; A=L :

These can be simplified to:

M=L3; T=L; A=L : (3.14)

For this example:

(i) G D N � K;(ii) m D k C 1;(iii) K < n;(iv) k � K .

Now the condition number (iii) indicates the result that the number of effectivedimensions of Equation 3.14 is less than the number of those originally listed whichwere M, L, T and A.

66 3 The Pi-Theorem

Again, the number of non-dimensional groups obtained is independent of theorder 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-flowheat exchanger. This is illustrated in Figure 3.3, showing the cold fluid in the centraltube 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 eachstream is:

PQ D PmhCph.Thi � Th0/ � PmhCph�Th :

Similarly, for the cold stream and for the same heat rate to the cool flow, theenergy equation is:

PQ D PmcCpc.Tc0 � Tci/ � PmcCpc�Tc :

The heat rate can be expressed by:

PQ D hA�T ;

where �T is the temperature difference across the area A.These equations show that the mass flow rate, Pm, and the coefficient of specific

heat, Cp are combined as PmCp. Also the variables A and h are combined as theproduct hA.

The Compact Solution 3.4 is set out as follows.This shows that:

�Tc

�ThD f

PmhCph

PmcCpc;

hA

PmcCpc;

�T

�Th

(3.15)

Inspection of Compact Solution 3.4 shows that the effective dimensions are twoin number and not the original four. These two are seen to be:

ML2

T3; ;

Figure 3.3 The contra-flowheat exchanger

3.5 Prior Proofs of the Pi-theorem 67


K PmhCph PmcCpc �Th �Tc hA �T

ML2

T3�ML2

T3�

� � ML2

T3�

�

PmhCph

PmcCpc

hAPmcCpc

1 1 � 1

�

�

�Th�Tc

�

�

�T�Tc

2 � 1 � 1

k 1 1 1 1

which reduce to:

ML2

T3 ; ;

Now for this example:

(i) G D N � K;(ii) m D k C 1;(iii) K < n;(iv) 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 existingones of which four basic variants are described.

The earliest version relies on the initial functional relation, such as that of Equa-tion 3.1, being in the form of a single power product of the variables. This provisionwas used by Rayleigh [5] in 1885 and still appears in several texts. Experimentson all sorts of phenomena show that this simple form of the function does not atall represent the experimentally determined one. The flow examples illustrated inFigure 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/at2 :

68 3 The Pi-Theorem

This is normalised to the form:

s

utD 1 C 1

2

at

u: (3.16)

The dimensions of these variables are:

Variables s u t a

Dimensions L L=T T L=T2

Then Rayleigh’s assumption would be to propose that

suatbac D K ; (3.17)

where K is a non-dimensional constant. Requiring the left hand side of Equa-tion 3.17 to be also non-dimensional leads to:

for the dimension in L:

1 C a C c D 0

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 ofusing determinants to obtain a solution gives the denominator as:

ˇˇˇˇ

1 0�1 1

ˇˇˇˇ D 1 :

The first numerator is then:ˇˇˇˇ.�1 � c/ 0

2c 1

ˇˇˇˇ D .�1 � c/

and the second is:ˇˇˇˇ

1 .�1 � c/

�1 2c

ˇˇˇˇ D c � 1 :


Substitution into Equation 3.17 then gives:

s

utD K

at

u

�c

: (3.18)

This, in comparison with Equation 3.16, is the wrong answer. Without even theaccompanying clarification that both s=ut and at=u are each dimensionless, com-monly it has been assumed that all equations derived by this method of the form ofEquation 3.18 lead to:

s

utD f

at

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 theWeierstrass approximation theorem, and so again this approach is approximate andrequires the existence of a function that is continuous and so is not general. Inpresenting his version, Buckingham additionally separated all those variables hav-ing a common dimension and, dividing by one of them, expressed them as non-dimensional ratios before applying the pi-theorem to the remaining variables. Nei-ther 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 notedthat Rayleigh was careful to retain a necessary functional relationship. It is not clearhow Buckingham’s mixed procedure retains this requirement.

A second approach is that discussed, for example, by Birkhoff [8, § 64] which isto express the original function as an infinite Maclaurin series. But not all examplesare 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 begenerally demonstrated and without convergence a series is meaningless.

Some phenomena are known analytically not to be represented by a Maclaurinseries. One example is the simple one of a vehicle coasting to rest. By writing theresistance as being represented by:

R D K1V C K2V 2 C K3V 3 C � � �and then putting:

mdV

dtD �R D �K1V � K2V 2 � K3V

3 C � � �

70 3 The Pi-Theorem

gives on integration that:

t

mD � 1

K1ln V C K2

K21

V C 1

2K1C"

K3

K1��

K2

K1

�2#

V 2 C � � � C constant :

Thus the leading term is a logarithmic one and so cannot be represented bya Maclaurin series in V . Equally, the use of the Weierstrass approximation theo-rem 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 rep-resentation of the drag force involved a series containing logarithmic terms in theReynolds 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 bodyin a flow [8]. Putting:

D D f .�; V; `; �/ :

Following the example in Chapter 1, this can then be rewritten as:

f

D

�V 2`2;

�

�V `; �; V; `

D 0 : (3.19)

A change in the size of the unit quantity can then be made in turn so as to givea numerical value of unity to each of �, V and `. Taylor [10] describes this as a useof units that are intrinsic to the problem rather than a use of extraneous units. It isthen stated that Equation 3.19 can be replaced by:

f

D

�V 2`2;

�

�V `; 1; 1; 1

D 0 ; (3.20)

so that:

f

D

�V 2`2;

�

�V `

D 0 : (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 bea single-valued function of the other ones. Neither of these very severe limitationsto applications appear in the present generalised proof.

A further difficulty with this third approach arises from the transformation be-tween Equations 3.19 and 3.20. An equation that represents a real physical event isunderstood to be a description of the relation between variables that retains truth asthese variables change in numerical value, these values being related to fixed valuesof each and every unit quantity: it is a matter of semantics. It is also the requirement


for a ‘complete’ equation as set out in Chapter 2. A fixed linear transformation ap-plied to such an equation retains this meaningful representation by the transformedequation. But the transformation between Equations 3.19 and 3.20 is one that mustchange continuously as the last three variables of Equation 3.19 are each contin-uously changed: this has the effect of continuously changing the unit quantity ofthese variables. Then the representation is no longer meaningful as just describedand also a constancy of relative magnitude is not retained.

Finally, a fourth approach, such as by Bridgman [12], relies on the function ofEquation 3.1 being differentiated in turn with respect to each and every one of thevariables. But the differentiability of a function is not generally demonstrable. Fur-ther, some of the ‘variables’ of Equation 3.1 can be units-conversion factors havingfixed values and so are not differentiable. Esnault-Pelterie [13] advanced furtherdetailed adverse criticisms of Bridgman’s proof by use of an example ascribed toVillat. What is quite surprising is that Bridgman commenced his discussion [12] bytwo simple examples, each of which resulted in a single non-dimensional group.He basically introduced the present method of the removal of variables by applyingthe condition of the balance of dimensions in his starting functional relation. Andyet, instead of following this through to produce a general proof, he developed thelimited one just described.

Staicu proposed a general dimensional analysis which claimed to derive more in-formation than the standard application of the pi-theorem [14]. It has been adverselycriticised 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 byIpsen in 1960 [16]. He used the procedure of sequential cancellation of the dimen-sions and was clear on the retention of a functional relation but he did not make thebasic statement that he had a new proof of the pi-theorem just describing the Buck-ingham demonstration as something different. He neither generalised his proof norset out the logical steps required as is done here in Chapter 2. His statement thata variable, being the sole one having one particular dimension, cannot appear inthe functional relation is not as clear as the present one that it must cancel out. Allthis may explain why Ipsen’s early valuable work has generally been overlooked bylater writers. For example, both Roberson and Crowe [17] and Barr [18] also usedthe cancellation of dimensions technique but seemingly only as a means of obtain-ing the non-dimensional groups directly from a list of the variables, for they say thatit is used without resort to the pi-theorem. Taylor similarly uses the method alsoonly as a means of obtaining non-dimensional groups noting that the technique al-ways results in the correct number of groups of the correct composition [10] but notobserving that it does so because it is an intrinsic part of the present demonstrationof the pi-theorem.

So again, these writers were only assembling variables into non-dimensionalgroups without reference to the retention of a functional relation.

Taylor noted that the current technique avoids the difficulty of some other meth-ods 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 arrivingat the correct number of non-dimensional groups, and so have led to means for de-termining these groups such as that described by Van Driest [19]. But, as Taylor haspointed out [10, p. 29], such discussions do not always lead to a determination thatis straightforward. Even Buckingham, for his example of Section 3.4.4, in order toachieve a solution chose unusual combinations of dimensions to form his effectiveones but without explanation. Because of the intrinsic link between the present proofand the operation of the pi-theorem, the latter always gives the correct number ofnon-dimensional groups each of a correct form meanwhile automatically accountingalso for a variation in the number of variables from group to group.

From the pioneering idea by Ipsen, the present generalised and rigorous proofovercomes 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 flowof 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

@xC v

@u

@yC w

@u

@zD � 1

�

@p

@xC X C �

�r2u ;

where:

r2 � @2

@x2C @2

@y2C @2

@z2

Figure 3.4 The steady in-compressible flow in a pipe

3.6 The Careful Choice of Variables 73


K �p � � PQ L d g

MLT2

ML3

MLT

L3

T L L LT2

1 �p

�

�

�

�

�

L2

T2�

�

L2

T

2 �p

� PQ2�

�

�

� PQ

�

�

gPQ2

1L4

�

�

1L

�

�

1L5

3 �pd 4

� PQ2�

�

�d

� PQ

�

�

Ld

�

�

gd5

PQ2

1 � 1 � 1 � 1

k 3 3 1 2

with similar equations for the other two direction components [20]. These equationsshow 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 vari-ables are � and g. Also the pressure force operates against the surface friction sothat a further variable is �. The pressure and friction forces vary with the flow rate,PQ and with the sizes of d and L.

Then,

�p D f .�; �; PQ; L; d; g/ : (3.22)

The solution is formed in Compact Solution 3.5.From this solution there is:

�pd 4

� PQ2D f

�d

� PQ;L

d;

gd 5

PQ2

: (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 thehorizontal flow though the pressure does vary vertically according to the hydrostaticrelation. Therefore the variable g can be excluded so that Equation 3.23 is reducedto:3

�pd 4

� PQ2D f

�d

� PQ;L

d

: (3.24)

3 The simple combination of p with g� as is done in elementary hydraulics has to be used withcare in a turbulent flow [20, 21].

74 3 The Pi-Theorem

Secondly, at positions well downstream of the pipe entry the flow pattern of ve-locity is unchanging along the pipe so that the wall friction is constant along thelength. Then from momentum considerations [2], a pressure difference is propor-tional to the associated local length [20]. This variable can then be replaced by thepressure gradient so that:

�p

LD @p

@L

and so Equation 3.24 is replaced by:

@p

@L� d 5

� PQ2D f

�d

� PQ

: (3.25)

This relation is correct for both laminar flow and turbulent flow. Thirdly, whenthe flow is purely laminar, then downstream there are no accelerations of the flowelement, the flow being controlled by just viscosity. This means that the density doesnot enter into the governing equations and so Equation 3.25 reduces to:

@p

@L� d 5

� PQ2� � PQ

�dD @p

@L

d 5

� PQ D constant : (3.26)

This equation is seen to be valid only as a specially restricted case of Equa-tion 3.23. It illustrates the care to be taken in a choice of variables. Also, the resulthas been obtained for a specified circular cross section. For a different shape, a vari-able, 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 Sec-tion 2.15, can now be demonstrated. It is shown in Taylor’s example of the loadingof 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 bytwo angles of the triangle, ˇ and � , and whose size is measured by the length ofone 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 thepi-theorem is as in Compact Solution 3.6.

This gives that:

ı

`D f

ˇ

�;

W

`2E;

A

`2

: (3.27)

4 There is a long-standing habit amongst scientific authors of using the word ‘geometry’ whenthey mean ‘shape’; surely both Fowler and Gowers would not have approved as ‘geometry’ is thescience of ‘shape’ as is defined in the Concise Oxford Dictionary.

3.7 The Necessity for a Units-conversion Factor for Angle 75


K ı W ˇ � ` A E

L MLT2

˛ ˛ L L2 MLT2

1 WE

�

�

L2 –

2 ı`

WEı2

�

�

A`2

�

�

1 1 �

�

1 �

�

3 ˇ

�

�

�

�

�

�

�

1 � � �

k 1 2 1 1

This is the result that Taylor obtained from the pi-theorem and which, as hepointed out, is incorrect. For if ˇ and � are each changed by the same factor thentheir ratio is unchanged; but clearly this change in the angles would change the shapeof the frame and hence the deflexion. By introducing the units-conversion factor, ˇ0,the extra group ˇ0ˇ necessary to resolve Taylor’s difficulty is obtained; for now, thesolution comes from Compact Solution 3.7.

This gives the result that:

ı

`D f

ˇ0ˇ;ˇ

�;

W

`2E;

A

`2

(3.28)

A change in the angles is now reflected by a change in ˇ0ˇ and hence in thedeflexion.

It is now seen that the dimension and units of angle, with its units-conversion fac-tor, is required in dimensional analysis so that it is not to be regarded as subservientin 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 ofa 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 non-dimensional groups.

The results from inspection of the operation of the pi-theorem together with theexamples are now listed in Table 3.3.

From the foregoing general demonstration of the pi-theorem and the illustrativeexamples just given, general conclusions are:

(i) G D N � K;(ii) m D k C 1;(iii) K � n;(iv) k � K.

In Table 3.3, the last three examples are cases where the effective number ofdimensions 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 aregiven as Exercises 3.4–3.7 and 3.12 and 3.17. The above four conditions form somegeneral rules for the formation of non-dimensional groups to satisfy the pi-theorem.


K ı W ˇ � ` A E ˇ0

L MLT2

˛ ˛ L L2 MLT2

1˛

1 WE

�

�

L2 –

2 ı`

WEı2

�

�

A`2

�

�

1 1 – 1 –

3 ˇ0ˇ ˇ0� – – –1 1 – – –

k 1 2 1 1 1

Table 3.3

Example G k K m n N

Linear spring 1 2 2 3 2 3Non-linear spring 2 2 3 3 3 5Pipe flow 4 1, 2, 3 3 2, 3, 4 3 7Vertical jet 3 3 3 4 3 6Field energy 2 2, 3 3 3, 4 4 5Structure 5 1, 2 3 2, 3 4 8Heat exchanger 4 1 2 2 4 6



The final stage in the full logic of dimensional analysis has been considered in thischapter. It is completed by a proof of the pi-theorem that is both general and rigor-ous. Operation of it thus gives non-dimensional groups that are automatically of thecorrect number, each being of the correct size and correct composition.

The various examples given bring out the problems that have exercised writersin the past particularly when the effective dimensions are different from those origi-nally set out: this has been resolved here by straightforward procedures in operatingthe present demonstration of the pi-theorem.

An illustration of the care to be taken in the choice of units-conversion factorscomes from the demonstration that has been given of the case for inclusion of thedimension 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 ourpoint 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=L3I Y D T=LI Z D A=L

Then re-do that problem of Equation 3.12 but now using these dimensionsof X , Y , and Z to show that the answer is still given by Equation 3.13.

3.2 An aerofoil; is in forced pitching oscillation within the flow of a low-speedair-stream. The mean value of the oscillating force on the aerofoil, NF , isa function of the oscillation frequency, !, the viscosity, �, the density, �, thestream velocity, V , and the scale size, `. Show that:

NFpV 2`2

D f

!`

V;

pV `

�

:

3.3 The emitted energy of a black body per unit area, per unit time, E , is a func-tion of the temperature, T , the velocity of light, c, the Planck constant, hP,and the Boltzmann constant, kB. Show that:

Eh3pc

2

k4BT 4

D constant ;

which is Stefan’s law.3.4 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, � , thethickness of the double-layer, d , and the dielectric constant of the liquid, ".

78 3 The Pi-Theorem

Figure 3.6 The end-heatedrod in the atmosphere

Show that:

& � "

� � dD constant :

What are the effective dimensions in this derivation?3.5 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 heatrate of PQ is applied. The end temperature is Te. Inspection of both the gov-erning equation for heat transfer along the rod and that to the atmosphere,shows that only temperature differences occur. Thus .Te � Ta/ is a functionof .T0 � Ta/, the thermal conductivity of the rod, k, the heat transfer coeffi-cient to the air, h, the rod diameter, d and the rod length, L. Show that:

Te � Ta

T0 � TaD f

k

hd;

L

d

:

What are the effective dimensions?3.6 The chemical constant, k, has the dimensions of M=

�LT25=2

�. It is a func-

tion of the Boltzmann constant, kB, the Planck constant, hP, and the molec-ular mass, m. Show that:

kh3P

m3=2k5=2B

D constant :

What are the effective dimensions?3.7 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, thedielectric coefficient of the intervening space, ", the area of each plate, A,and the spacing between the plates, z. Show that:

fe"

q2AD f

z2

A

:

What are the effective dimensions?There are edge effects upon the plates so that q is not uniform there. If theplate spacing is sufficiently small these edge effects become negligible. For


this case simplify the above result. Hence show that then the force is inde-pendent of the plate spacing.

3.8 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, theuniversal gravitational constant, G0 and the speed of light, c. Show that:

shPG0

kBc3D constant :

3.9 The radius of the event horizon of an astronomical black hole, r , is a func-tion of the mass, m, the velocity of light, c, and the universal gravitationalconstant, G0. Show that:

rc2

G0mD constant :

3.10 An object of mass, m, is suspended vertically on a spring and is oscillatingvertically at a frequency, !, with an amplitude of a. Show that the meankinetic energy of the object, E, is given by:

E

m!2a2D constant :

3.11 The armature of a DC motor has a diameter, d , a length, `, and carries a cur-rent 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

iB!d 2D f

`

d

:

Ignoring end effects of the magnetic field at the ends of the armature, reducethe above result to one non-dimensional group.

3.12 A fine tube is inserted vertically into a still liquid so that the meniscus inthe tube rises up by the height, �h. With this height being a function ofthe weight per unit volume of the liquid, w, the surface tension, � , and the

Figure 3.7 The parallel platecondenser

80 3 The Pi-Theorem

diameter of the tube, d , using the MLT system of units show that:

�h

dD f

wd 2

�

:

What are the effective dimensions? Repeat the analysis using the FLT di-mensional system where F is the unit of force. Compare the two methods ofsolution.

3.13 Fans and windmills produce high sound levels because the fundamentaloperating mechanism is to produce a fluctuating pressure field. The soundpower, 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

�!3d 5D f

!d

a

:

The non-dimensional group on the right-hand side of this equation is a mea-sure of the Mach number at the fan tips. If this is limited to a specified sub-sonic value then show that the acoustic power is proportional to the squareof the diameter.

3.14 An electron of mass, m, and charge, e, passes across a magnetic field ofmagnetic flux density, B , at a velocity, v, so that its path has a radius ofcurvature, r . Show that:

B e r

m vD constant :

3.15 The effect of an atomic explosion initially is from the rapid expansion of anextremely strong spherical shock wave. The radius of curvature of this shockwave, r , is a function of the explosion energy, E , the initial air density, �0,and the time, t . Show that:

Et2

�0r5D constant :

3.16 The stability of a co-axial spray generated by an inner co-axial jet insidea co-axial electrode, is controlled by the ratio of the electric relaxation timeto the hydrodynamic capillarity time. with the notation of:

r jet radiustc capillarity timete 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

tc/ "

�

�

�r3

1=2

:

3.17 A liquid electrolyte is contained in a conductivity cell. This is rectangular inshape with electrodes of area A on opposite facing sides spaced apart a dis-tance `. An electric field is applied across the electrodes. Above a criticalvalue of this field, Ec, the liquid is set in motion under electro-hydrodynamicforces. With the notation of:

A Electrode areaEc Critical fieldkBT Thermal energy` Electrode spacing" Dielectric coefficient�0 Zero-charge conductivity

and then:

Ec D f .A; kBT; `; "; �0/

show that Ec is not a function of �0. In this application of the pi-theoremwhat are the effective dimensions?

3.18 A pipette is used to issue the liquid slowly, a drop at a time, from the sharpnozzle outlet of diameter, d . The drop size is then controlled by the surfacetension of the liquid, � . Show that the weight of each drop, W is given by:

W

d�D constant :

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 equa-

tions, 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 4The Development of Dimensional Analysis

There is a yearly battle going on for students’ minds; historymight help to convince them that the use of the Reynolds numberas an independent variable is an application of a basic truth,and not just a useful convention for a handy diagram.N. Rott

Notation

b Plate breadthd Pipe diameterK Resistance coefficient` Plate lengthn Resistance exponentQ Volume flow rateRe Reynolds numberRf Resistance forceV 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 improvingcurrent practice. For example, the study of the steady decline in British engineeringindustry from the Great Exhibition of 1851 onwards [1] still has not always beenacted upon in many instances up to this day.

As Rott has written, “ – the peculiar history of the ideas on hydrodynamic sim-ilarity in the epoch around 1900. The ideas were initiated by the great men aroundthe turn of the century, but were originally met with indifference, for an unbelievablelong period.” [2]. This history, then, illustrates that remarkable tardiness in adoptingthe use of dimensional analysis which continues in some branches of science andengineering.


84 4 The Development of Dimensional Analysis

4.2 The Onset of Similitude

The idea of similar systems long preceded the development of dimensional anal-ysis. Pomeranz goes so far as to propose relevant contributions by Ptolomy andGalileo [3], whilst Szucs adds the name of Kepler [4]. However most of the concernof these early workers was with the dimensions of measurement [5].

Long before this, Euclid in his studies of geometry had dealt in detail with theidea of similarity in the shape of geometrical figures, variation being in size only sothat relative representation could be by the value of a single ratio of lengths. Yet, asseen in Exercise 4.1, Pythagoras could have benefited from an elementary knowl-edge of dimensional analysis in deriving his famous theorem, which Bronowski hasdescribed in the words of; “To this day, the theorem of Pythagoras remains the mostimportant single theorem in the whole of mathematics.” [6].

Newton took all these ideas much further in his famous Principia [7]. He con-sidered similarities in the trajectories of bodies in motion under his laws of motion,going on to derive the similarity between accelerations, momenta and forces. Hethus introduced the ideas of similarity between kinematic and force systems. Thispart of Newton’s work was not acted upon and so rather lapsed from use until takenup by Thomson in the late 19th century who then applied it to problems in hy-drodynamics and structures [8]. In the meantime earlier in that century, the idea ofsimilarity of biological systems was put to extensive use by D’Arcy Thompson inhis 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 intoNewton’s law of motion. But again, as Macagno says; “It is remarkable that Euler’sideas 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 dimensionalanalysis was created by Fourier [11]. He made the basic point that an equation thatrepresents a real physical event must have the same dimensions for all its additiveterms. Correspondingly, quantities on both sides of an equation had to have thesame dimensions. He then went on to point out that these forms of equation remainunchanged when a change is made in the size of the unit measure. He used thisprinciple as a means of checking the validity of an equation. But he did not extendthis finding further to derive results.

An early signpost to dimensional analysis was set up by Stokes. In 1823 Navierhad set out the governing equations for a viscous flow and Stokes produced a fur-ther derivation in 1845 [12]. From this, in 1856 Stokes applied these equations tothe problem of viscous friction from the air upon the swing of a pendulum. Heshowed that there was a single independent non-dimensional group controlling thisphenomenon; 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, todenote dimensions of a quantity. That is that of M for the dimension of mass, L forlength and T for time [14].

It was a further half a century after Fourier’s lead before the use of dimensionalanalysis really got started. The great breakthrough was made by Lord Rayleigh whoin 1871 used the ideas of similitude in a study of the colour of the sky [15]. He lateradvocated dimensional analysis in 1885 [16] and though he usually referred to it asa 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-theoremwas made by Vaschy in 1892 [19]. He followed this by a more general statementin 1896 [20]. Yet this work was seemingly ignored as described by Macagno [5]who pointed out that as a consequence Riabouchinsky rediscovered this theoremindependently in 1911 [21]. His assumption of a series solution would seem to havebeen influenced by Rayleigh’s work. Even then Buckingham, who produced hisproof of the pi-theorem in 1914 [22], did not acknowledge Riabouchinsky’s priorityuntil 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 experi-ments on the flow through pipes using a visualisation method to determine a crite-rion that distinguished whether the flow was a steady laminar one or an unsteadyturbulent one.1 This work was and remains of considerable practical importance be-cause the friction loss in the flow greatly increases in going from a laminar flow toa turbulent one. [25].

In either of these types of flow we have the relation of:

� D f Œ�;Q;�; d � : (4.1)

The pi-theorem solution is set out in Compact Solution 4.1.This gives the result that

�d 4

�Q2D f

��Q

�d

�

D f .Re/ : (4.2)

This shows that, when the friction jumps in value in going from laminar flow toa turbulent one, then the criterion dividing these two regimes is given by the cor-responding value of the Reynolds number. This was Reynolds most significant dis-

1 Because Reynolds’ pipe-flow experiment was such a seminal contribution to the early develop-ment of dimensional analysis, it is important that this flow is described correctly. This is done inAppendix 4.1



� � Q � dM

LT2ML3

L3

TMLT

L

��

�

�

�

�

L2

T2�

�

L2

T

��Q2

�

�

�

�

�

�Q

1L4

�

�

�

�

1L

�d 4

�Q2�

�

�

�

�d

�Q

�

�

1 � � 1 �

covery; it was a finding that could only be revealed by dimensional analysis boththen and to this day.

There is another important parameter which is the presence of disturbances in theform of residual turbulence in the incoming flow entering a pipe. Reynolds recog-nised this as well but found that with a high degree of such disturbance the Reynoldsnumber fell to a minimum value of about 2000. This priority of dimensional analysiswas acknowledged by Foppl in 1910 [26].

Prior to this, Sommerfeld had, in 1908, given the name of Reynolds number tothis non-dimensional group not having taken account of Stokes original observationof some three decades earlier as mentioned above [27].

Dimensional analysis was first used extensively in aerodynamics [28]. FollowingReynolds’ work, Rayleigh made a clear application of dimensional analysis to aero-dynamics when he used it in 1904 to correlate values, measured by Zahm, of thedrag of a flat plate at zero incidence [29]. Before this the forces of both lift and dragon aerofoils were given in the literature as forces per unit wing area. This practicewent back to at least Wenham who carried out the very first model tests in his windtunnel in 1871 [30].

The first attempt to evaluate a Reynolds number scale effect between model testsand full-scale was made by Wilbur and Orville Wright who tethered one of theirfull-scale man-carrying gliders in a steady wind and compared the measured forceson it with those that they had obtained in their wind tunnel [31]. This provided themwith a form of correction factor which they applied in their calculations with theirconsiderable success [32].

Rayleigh brought his influence to bear in two ways. First, he was regarded in hisday as a pre-eminent scientist, certainly in Great Britain. Secondly, he was the firstchairman of the Advisory Committee for Aeronautics. This enabled him to bringthe application of dimensional analysis to aerodynamics to the attention of the staffof the Aeronautical Department of the National Physical Laboratory at Teddington.Lanchester was aware of the application of dimensional analysis at least by 1909 forin the Committee’s Report and Memoranda No. 1 it is made clear that at 12th May


1909 Prandtl was not properly using this analysis for the correlation of the drag ofairships [33]. He was using Froude’s law and it was pointed out in a footnote byLanchester that this law was not applicable.

Yet in his lengthy paper of 1910 Prandtl gave an equation for the measured fric-tional resistance on a plane surface of [34]:

Rf D Kb`nV nC1 : (4.3)

Figure 4.1 Reproduction of Fig. 1 of ACA R&M No. 15. (See [38])


Here Rf is the frictional resistance, K is a coefficient, b is the breadth and ` is thelength of the surface, V is the velocity and n is an exponent which for smooth sur-faces was quoted as n D 0:80 � 0:85. What is now intriguing is that this expression



can be readily rewritten as:

kf � Rf

�b`V 2/ Rn�1

e (4.4)

which is the standard non-dimensional form.



Doubts still existed about the validity of dimensional analysis to aerodynamics.The evidence indicates that these were finally resolved in 1911 by Bairstow andBooth in R&M No. 38 [35]; by Rayleigh in R&M No. 39 [36]; and in R&M No. 40by Melville Jones [37].

The source of these doubts can be illustrated by the data for the drag of flat platesnormal to the flow which had been correlated on the basis of size alone. This resultis shown in Figure 4.1 by the original figure from R&M No.15 [38]. Then this wascompared with the less successful correlation in terms of the Reynolds number asshown in Figure 4.2 [35]. Rayleigh obviously felt the need for an apologia whichwas issued in R&M No. 39 [36]. This was followed by Melville-Jones’ successfulcorrelation for the drag of smooth wires given in R&M No. 40, Pt.1 [37]. His orig-inal figure is shown as Figure 4.3. It seems quite likely that this latter figure wasthe means of finally convincing those working on aerodynamics of the validity ofdimensional analysis to their studies [28].

From then on dimensional analysis was gradually developed. The advances indeveloping the final general proof of the pi-theorem are described in Chapter 3. Thefirst attempt to set out a logic of the full analysis was by Bridgman in his classic andpioneering book of 1922 [39]. This was taken further to give a full logic sixty yearslater [40].

4.5 The Place of Dimensional Analysis

The place of dimensional analysis in science has been put by Rott who described thissubject; “ – as a (or as ’the’) discipline of science.” [2]. Pomerantz, in his introduc-tion 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 yearsfrom Fourier for it to be widely adopted and then only in aerodynamics. Yet, asexplained in Chapter 3, further developments in the theory continued for over halfa century. Yet still and so often, the analysis is not used by the physicist, the physical-chemist 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 flowwas 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 misinterpretationof the onset of flow turbulence.

The present writer repeated this experiment in his department. The flow wasfrom a tank into an analytically profiled entry cone set flush into a side wall of thetank [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 show-ing tank vorticity. (b) Sketchshowing laminar flow.(c) Sketch showing fluctu-ating laminar flow. (d) Sketchshowing turbulent flow

of localised flow separation which could introduce upstream disturbances. The glasstube was specially drawn to have a bore of precision diameter and straightness andthe metal inlet was machined to match this glass bore quite accurately. The outletfrom the nozzle for the stream of dye was located in the position of high accelerationof the main flow at entry to the main-flow nozzle so that the dye flow was laminarthere.

As found by Reynolds, the water in the tank had to stand for many hours atleast to allow all the vorticity from the filling of the tank to die out. If this was notdone, a false flow pattern was obtained, the dye line having a waviness in it whichtravelled with flow whilst retaining its distinct outline. Typically, the picture was assketched in Figure 4.4(a). Clearly the vorticity in the tank remained for this longtime. Additionally, the opening of the dye inlet tube had to be operated with care toavoid a significant input disturbance to the flow.

Initially, as the flow was speeded up from a laminar one, the dye line sometimesdeveloped from a straight line form, as shown in Figure 4.4(b) to one with a regularwaviness, each wave travelling downstream with the flow; typically as sketched inFigure 4.4(c). But the dye filament remained quite distinct. This was an indication ofa fluctuating laminar flow. This is the diagram that appears in many texts but it wasnot the turbulent pattern. That it was not a rotational flow was further confirmed bya bang on the side of the tank, when the filament was straight, resulting in a singlewave form of the dye travelling along the flow as in Figure 4.4(c). Such an impulsewould, by potential flow theory, have set up a potential flow disturbance and not anyvorticity ( [25], Section 119).

As the velocity was increased, eventually the distinct dye line was quite sud-denly diffused into a cloud of dye across the pipe flow with no discernable anddistinct pattern as sketched in Figure 4.4(d). This was described by Piercy as “it


presented a frosted appearance of the outlet jet” and was the true picture of turbu-lence. It seemed to originate quite suddenly in the pipe inlet region. Further, whenthe velocity was decreased, this dispersion of dye remained to be swept downstreamby the flow. The vorticity did not dissipate within the flow but by Kelvin’s theoremwas transported with it [25].

It was not found possible to move the transition front up and downstream byvariation of the flow velocity suggesting that the onset of turbulence was controlledby the initial development of the pipe flow at entry coming from the wall boundarylayers. 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 theproduction of turbulence in straight pipes with well rounded mouths. The initialvortices whose existence has been postulated owe their origin to disturbances whichoccur before the fluid enters the pipe.” [42].

Exercises

4.1 Using dimensional analysis, prove the famous result of Pythagoras concerninga right-angle triangle. (Hint; from the point of the right angle, drop a perpen-dicular on to the horizontal hypotenuse.)

4.2 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 thatthe buoyancy force is proportional to the weight of water displaced. Considerthen 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, 6thAugust 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 aerody-

namique, L’Aerophile, pp. 407–408, 1911.22. E Buckingham. On physically similar systems; illustrations of the use of dimensional equa-

tions, 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 experimentson the resistance of square plates normal to a current of air, ACA Report 1910-1911, R & MNO. 39, 26th March 1911.

37. B Melville-Jones. The resistance of wires and ropes in a current of air, ACA Report 1910-1911, R & M No. 40, Pt. 1, p. 40, March 1911.

38. Lord Rayleigh. Note as to the application of dynamical similarity, Report of the AdvisoryCommittee 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 5The Choice of Dimensions

– the principle known as Occam’s Razor: essentia non suntmultiplicanda praeter necessitatem (hypotheses are not to bemultiplied without necessity).R.V. Jones

Notation

a Area, amplitudeA Amplitudec Velocity of lightc Mean molecular velocityCV Coefficient of specific heat; mass basedCH Coefficient of specific heat; volume basedd Tube diameterD DragE Energy; electromotive forceF Forceg Gravitational accelerationg0 Gravitational constantG Number of non-dimensional groups; modulus of rigidityH Magnetic field strengthi Electric currentI Luminous flux intensitykB Boltzmann constant` Length scale, tube length`m Mean-free path lengthL Self inductancem Mass, electron mass; mass per unit lengthPm Mass-flow rateM Dipole momentM0 Molecular ‘weight’n Number of dimensionsN Number of variables; number densityp Radiation pressureP Power outputP0 Mechanical equivalent of light


96 5 The Choice of Dimensions

Q Heat ratePQ Quantity flow rater DistanceR Gas constantRe Reynolds numbert Timev Velocityz Distance

�p Pressure drop@p=@x Pressure gradientˇ0 Units-conversion factor for angle� Angle� Thermal conductivity! Frequency� Frequency� Viscosity�0 Magnetic permeability�L Liquid density�S 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 byno means mandatory. For example, in problems in statics the mass dimension mightmore conveniently be replaced by the dimension of force. In gravitational problemsthe replacement may well be by the weight force.

These changes sometimes require care because replacements such as those cancontrol the inclusion of units-conversion factors. Such changes are not alwaysstraightforward 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 arederived 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 ofbeing unduly influenced by a prior knowledge of the solution existing from formalanalysis of the governing equations. Arising from this observation, there have beenseveral 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 sepa-rately that of force in one phenomenon. An example dealt with the fall under gravityof a very small solid sphere through a liquid. The forces acting are the drag,D, theweight, W , and a buoyancy force, FB. The event is illustrated in Figure 5.1. Thisrequires the variables of:

v; `; �S; �L; �; g :

A straight attack on this problem without any further reference to the governingequations, and using the dimensions of M, L, and T results in the Compact Solu-tion 5.1 of the pi-theorem.

This gives:

�L�`

�D f

��2

g`;�S

�L

�

(5.2)

In this equation the first non-dimensional group is a Reynolds number and the sec-ond is a Froude number.

In an endeavour to reduce this result to the known analytical solution it has beenproposed that the dimensions should have that of force, F , added. Certainly thisreduces the non-dimensional groups to two in number but the falsity of this approachis seen by the discussion of Chapter 3 where in Newton’s law of motion, when thedimensions of force and of mass are separate then there is a need for the introductionof a units-conversion factor, g0. So this extra variable occurs and then the solution

Figure 5.1 Illustration of theslow fall of a small sphere ina fluid



v ` �S �L � g

LT

L ML3

ML3

MLT

LT2

�S�L

�

�

�

�L

1 �

�

L2

T

�

�

�

�

�

�L�

g

�2

�

�

�

�

L 1L

�

�

�

�

�

�

�

�Lv`

g`

�2

�

�

�

�

�

�

1 1


� ` �S �L � g g0

LT

L ML3

ML3

FTL2

FM

MLFT2

�g0 gg0 �

�

MLT

LT2

�

�

�S�L

�

�

�g0�L

�

�

1 �

�

L2

T�

�

�

�

�

�

�g0�L�

gg0�2

�

�

�

�

�

�

L 1L

�

�

�

�

�

�

�

�

�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�`

�g0D f

��2

g0g`;�S

�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


� ` g�S g�L �

LT

L FL3

FL3

FTL2

�S�L

�

�

�

�Lg

1 � LT

�

�

�

�

��

�Lg

� � L2

�

�

�

�

�

�

��

�Lg`2

� � � 1

density, �L, is excluded from the expression for the drag, so that both densities arerelated to weight forces in the combination (�g/. Then Equation 5.3 reduces to;

� D f .`; �Sg; �Lg;�/ :

It is now proper to use the dimensions of F, L, T without the need of the units-conversion factor, g0, giving the Compact Solution 5.3.

This gives:

��

�Lg`2D f

��S

�L

�

(5.5)

By recognising from the initial assessment of the physics that no momentum effectsare acting but only weight forces, the three groups of Equation 5.2. are now reducedto two in number. Also the use of just the dimensions of F, L, and T gives a neatersolution over that of Equation 5.4.

A full inspection of the equations governing this phenomenon shows the wayforward. There are two sets of equations which are found to be uncoupled. One setis the hydrodynamic one for incompressible flow given in Chapter 1 but now withthe density excluded for the reasons just given. In this set there is for the drag on thesphere:

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 aretwo cases. For a sphere of comparatively high density falling at a steady and low


velocity through a gas the drag force is equal to the weight of the sphere, W , as thebuoyancy force from the air is negligible. Because W / g�S`

3, now the densitiesare required in the independent variables. Then substitution into Equation 5.7 gives:

g�S`2

��D constant : (5.8)

When the sphere is falling through a liquid, then the weight is balanced by thesignificant buoyancy force from the liquid given by g�L`

3, plus the drag force. Sub-stituting 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 thefurther important feature of these equations that they are uncoupled, a most usefulcharacter that will recur later.

If the sphere is accelerating after its initial release then a further term arises. Thiscomes from the inertia effect of the acceleration of the fluid around the sphere whichis 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 ap-plication of momentum, then the physics of such events would suggest that the di-mensions 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 uniformtemperature. Then one end is raised by a fixed temperature difference of � so thatthe temperature along the bar rises with the time, t , resulting in the heat applied tothe end face being a function of time. Putting this input Q as the heat per unit area

Figure 5.2 Illustration of theend heating of a bar

5.4 Mass and Volume Dimensions 101


Q � t k CV

HL2

� T H�LT

L2

�T2

tk �

�

CV

k2

H�L

�

�

�L4

H2

Q

tk

�

�

�

�

t2CV

�L

�

�

�

�

L2

�

Q

tk�

�

�

�

�

�

�

t2CV �

1L

�

�

�

�

�

�

L2

Q2CV

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:

Q2CV

k2�D constant : (5.11)

This result presents a serious omission. For it does not contain the time, t . It wouldbe most surprising if this were to be correct as it would be quite expected that thetotal heat would steadily increase as time passes. So now Equation 5.10 has to beinspected for any fault.

It is noted that, though it does not contain a dimension in M , the coefficient ofspecific heat is defined through the First Law of Thermodynamics which is a mass-based expression. Here the mass dimension has been excluded in favour of that ofheat. Thus there are physical grounds for defining a coefficient of specific heat basedupon 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

tkCH�2D constant : (5.13)



Q � t k CH

HL2

� T H�LT

H�L3

tk �

H�L

�

�

Q

tk

�

�

�

�

CHtk

�L

�

�

�

�

1L2

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 problemsfor which formal solution exists. If that had not been so here, resolution of thediscrepancy between Equation 5.11 and Equation 5.13 would have had to rely on anadequate understanding of the subtle physics.

Expanding on this explanation, it is noted that dimensionally, in the M, L, T sys-tem 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 re-quired. By a similar reasoning CH � M=LT2� or CH � H=L3� . Now if CH ischosen then either M or H dimensions are required. The conclusion is that in usingthe heat dimension in place of the mass one then the coefficient of specific heat mustbe based upon a volume and not on a mass. For this example that necessary under-standing has been described. Yet it is clear that this physical understanding can bequite subtle and has relied here not only upon a suitable choice of variables but ofalso the appropriate dimensions.

It is noted that this approach of using CH instead of CV was used by Rayleighwhich enabled him to obtain a single group for the problem described later in Chap-ter 6.

5.5 Temperature and Quantity Dimensions

A similar problem arises in the example of the thermal conductivity of a gas asdescribed again in the discussion of the kinetic theory of gases in Chapter 8. In that

5.5 Temperature and Quantity Dimensions 103


� N `m c m R

ML�T3

1L3

L LT

M L2

T2�

�m

�

�

LT3�

�

�

�mR

�

�

�

�

1LT

�

�

�

�

�mRc

�

�

�

�

�

�

1L2

�

�

�

�

�

�

�`2m

mRc

N`3m �

�

�

�

�

�

�

�

1 1 � � � �

discussion, the expression for the thermal conductivity will be derived as:

�

NkBc`mD constant : (5.14)

That derivation relies on the use of kB as the units-conversion factor because thevariables have the dimension of temperature. Also mass is involved to describe themotion of the elementary particles. If instead from inspection of item 14 in Table 2.1it is considered that the gas constant, R, is to be used as the units-conversion factorfor temperature and no account is taken of the mole as a unit,M0 being dimension-less by Equation 2.13, then the solution would go as in Compact Solution 5.6.

This gives that:

�`2m

mRcD f

�N`3

m

�:

As in the discussion of Chapter 8, putting � / N , then:

�

mRcN`mD constant :

Now:

Rm D kBM0


so that:

�

M0kBcN`mD constant :

Comparing this equation with Equation 5.14 shows that it is incorrect. This illus-trates again a case where care has to be taken in the choice both of the variables andhence 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.The variables were chosen carefully and of them, PQ, �,� and�p had a dimension inM. Thus when as in an approximation for laminar flow the density, � was excludedthere were still three variables left with a dimension in M so that there was noproblem in cancelling this dimension in application of the pi-theorem.

It is thus acceptable to use the mass flow rate, Pm as the dependent variable. Then:

Pm D f .d; `;�p;�; �/ : (5.15)

The pi-theorem goes as in Compact Solution 5.7.This gives that:

Pm�d

D f

�`

d;�d 2�p

�2

�

or rewritten as:

Pm��d 3�p

D f

�`

d;�d 2�p

�2

�

(5.16)

As in Section 3.6 this solution is the correct one for the full pipe flow and applies toboth laminar and turbulent flow. Again, as in Section 3.6, far downstream from theentry and for laminar flow in a horizontal pipe the density, �, is excluded from the

Figure 5.3 Illustration of theflow into a circular tube

5.6 Mass and Quantity Dimensions 105


Pm d ` �p � �

MT

L L MLT2

MLT

ML3

Pm�p

�

�

�

�p

�

�p

LT �

�

T T2

L2

Pm�

�

�

�

�

��p

�2

L �

�

�

�

1L2

Pm�d

�

�

`d

�

�

�

�

�d2�p

�2

1 � 1 � � 1

list of variables. It now becomes appropriate to consider a flow rate as a volume rateand not a mass rate. Thus the variables are:

PQ D f .�; d; @p=@x/ :

Also it is now appropriate to use the dimension of force rather than to use the massdimension. The solution then goes as in Compact Solution 5.8.

This results in the single non-dimensional group of:

PQ�d 4.@p=@x/

D constant (5.17)

as obtained in Section 3.6.


PQ � d @p=@x

L3

TFTL2

L FL3

�

�

@p=@x

�

�

�

1LT

PQ�

@p=@x� �

L4 – –

PQ�

d 4.@p=@x/� � �

1 – – –


This is another example where care is required in choosing the variables to accordwith the physics of the phenomenon and hence a suitable choice of dimensionsfollows. The association in the above tabulation of the force dimension with onlythe terms respectively in the viscosity and the pressure is consistent with the fluidmechanics 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 therequirement of a corresponding units-conversion factor, ˇ0 was demonstrated. Thisneed 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 desirableresult is obtained by using these four dimensions in Equation 5.18 thus resulting injust two non-dimensional groups. The error here is that Equation 5.18 in effect putsthe units-conversion factor for angle at unity and so the angle is measured in radiansand then is of zero dimensions. This reduces the number of dimensions to threeand so results in three non-dimensional groups. The way forward is to add the units-conversion factor to Equation 5.18 so that the solution is as in Compact Solution 5.9.

This gives:

Tˇ0

Gd 3D f

�`

d; �ˇ0

�

: (5.19)


T G d ` � ˇ0

ML2

T2M

LT2˛

L L ˛ 1˛

Gˇ0

�ˇ0 �

�

MLT2

1 �

�

Tˇ0G

�

�

�

�

L3� �

Tˇ0Gd 3

�

�

�

�

`d

�

�

1 � � 1 �

5.8 Electrical Dimensions 107

The effective dimensions are just three in number: that is; M=T 2, L and ˛ thusresulting in the three non-dimensional groups. Three groups are still obtained withˇ0 D 1. Equation 5.19 is the correct solution and progress can only be made byintroducing two approximations. First, to be compatible with the definition ofG thedeflexion is assumed to be purely elastic so that the angular deflexion is proportionalto the torque. Secondly, if there are no significant end effects, which is not alwaysthe case, then there will be a distance uniformity so that the angular deflexion willbe proportional to the length. These two approximations then reduce Equation 5.19to:

T `

G�d 4D constant : (5.20)

Thus Equation 5.20 becomes a special case of Equation 5.19 being then an approx-imate solution.

5.8 Electrical Dimensions

Taking as an example the force,P , acting between two parallel plates of a condensereach of area, a, spaced apart a distance, z, and under a potential difference, '. Thisis 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 Solu-tion 5.10, which gives that:

P

"'2D f

h a

z2

i(5.21)

In this phenomenon there is no mechanical motion and no current flow. Thus itseems more appropriate to replace the dimensions of M and A by those respectivelyof force, F and of charge, Q. Using the dimensions of F, Q and L, gives CompactSolution 5.11, which again gives Equation 5.21.

Figure 5.4 Sketch of the field lines between the two plates of a condenser



P " ' a z

MLT2

A2T4

ML3ML2

AT3L2 L

"'2�

MLT2

�

�

�

�

"'2

P

�

�

� 1 �

�

�

�

�

az2

�

�

� � 1 �


P " ' a z

F Q2

Fl2FLQ

L2 L

"'2

F�

�

P"'2

�

�

�

�

1 � �

�

�

�

�

az2

�

�

� � 1 �

This is the correct solution. There are edge effects because there the field linesare not straight and perpendicular to the condenser plates. This is illustrated in Fig-ure 5.4. When a � z2 such effects can be negligible so that from the uniformityacross the plates, then P / a. This reduces Equation 5.22 to:

Pz2

"'2aD constant : (5.22)

This again is a relation that embodies an approximation. The latter use of the di-mensions for the variables of this problem is again consistent with the physicalsignificance 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 vectorialidentities to the numerical ones though he appeared not to have made use of thisidea in deriving results [2]. Bridgman quoted Williams paper and his work has beenquoted in support of this proposal [3]. However, when Bridgman applied this idea heshowed that the desired result was obtained without its use but instead by a properassessment of the physics of the problem together with a proper use of a units-conversion factor.1 A further example by Bridgman is unfortunately incorrect andso cannot be adduced in support of this idea.2

Brooke Benjamin discussed the application of vectorial identities by describingin detail the case of the drag resistance of a sphere in a flow at a very low Reynoldsnumber [4]. This was described in Chapter 1 where the result derived analyticallyby Stokes was obtained as:

D / �V `

writing this as:

D D k�V ` : (5.23)

Brooke Benjamin observed that this equation can be taken as a vectorial identitybetween D and V , both being aligned in the direction of the free stream so thatEquation 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 theReynolds number tends to zero. A more detailed analytical solution he quoted asbeing:3

D D k�V `

�

1 C 3

8Re C 9

40R2

e lnRe CO�R2

e

�

; (5.24)

where the Reynolds number is given by:

Re � �V `

�(5.25)

so that the solution given by dimensional analysis would be:

D

�V `D f .Re/ :

1 See pp. 59–65 of [3].2 See pp. 65–67 of [3].3 This is a good example where the assumption of a simple power series cannot be made in orderto develop the Rayleigh–Buckingham version of the pi-theorem.


Retaining V as a vector in Equation 5.25, then Equation 5.24 cannot be a vectorialidentity.

It has to be concluded that the result of using dimensional analysis gives an an-swer that can refer only to numerical equality. This is consistent with the full logicpresented here in Chapter 3 which is developed from the starting point of just mean-ingful numerical addition.

There have been the strongest of objections raised against assigning vectorialcharacteristics to length dimensions in dimensional analysis. An early advocate ofthis idea referred to the extended lengths as vector lengths [5].

Gessler has strongly queried this as, for just one example of many, some authorshave allocated various directional length dimensions to the essentially scalar quan-tity of viscosity [6]. Further, in the literature there are examples of the assignment bya single author of different directions to viscosity depending upon the phenomenonbeing considered, this despite the accompanying statement that “. . . there should bea one-to-one correspondence between physical quantities and dimensional formu-lae.” Further, even different assignments for viscosity have been given by differentauthors for the same phenomenon. Some quite ignore the physics of viscosity inthat, 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 doesnot “. . . 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 paperrecorded a critical review of twenty five examples of supposed use that had appearedin three different texts. It was stated that every single example had been successfullyresolved without the use of extension of the length dimension. It is relevant to notethat 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) The derivation given had either an incorrect or a poor formulation of thephysics in drawing up the initial functional statement.

b) The result was only valid for certain approximations which were not inherentin the original functional statement.

c) The result was only valid for a special case which was not specified in thefunctional statement

d) An inadequate expression of the physics had preceded the formulation of theoriginal functional statement.

e) There was an uncoupling of the equations governing the phenomenon whichwas not reflected in the original functional statement.

f) Variables were missed from the original functional statement.g) Results were obtained that incorporated expressions of either definitions or of

basic physical laws neither of which, by their natures, are provable by dimen-sional analysis.

h) 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 lessonhas not been quoted by later writers.

In using this idea of extended length dimensions, different authors adopt differentdimensional representations of the same variables. For example, in considering theproblem of the energy of vibration of a stretched wire for the approximation ofa small amplitude, it is noted that the physics of this phenomenon is of the totalmass, m, of the wire oscillating over an amplitude of a. The energy will be kineticand so of the form of a mass times the square of a velocity represented by thefrequency. Thus the length of the string will not be a relevant variable. Then thefollowing Compact Solution 5.12 gives the known answer:


! E a m

1T

ML2

T2L M

Em

�

�

L2

T2�

�

�

�

Em!2

�

�

� L2�

�

�

Em!2a2

�

�

�

�

� 1 � �

Thus:

E

m!2a2D constant :

For this problem, the following dimensional scheme introducing orthogonal lengthdimensions, Lx and Lz , has been adopted in the literature:

E LaxLb

zMT�2 ;

! T�1 ;

m M ;

a LcxLd

z :

4 See pp. 105–108, particularly p. 107, of [10].


A single power product would take the form of:

LaxLb

zM

T2� 1

TA� MB

hLc

xLdz

iC

� 1 : (5.26)

Meeting the required dimensions of the individual variables as listed in CompactSolution 5.12, gives that:

In E, LaxLb

z D L2 so that a C b D 2 ; (5.27)

In a, LcxLd

z D L so that c C d D 1 : (5.28)

Then for:

MI 1 C B D 0 so that B D �1 ; (5.29)

TI �2 � A D 0 so that A D �2 ; (5.30)

Lx I aC cC D 0 ; (5.31)

Lz I b C dC D 0 : (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!�2m�1a�2 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 illustratefurther, two solutions by two different authors to the problem of natural thermalconvection, which both come under the cases (b) and (d) above, used the followingscheme of dimensions:

(i) For the kinematic viscosity, �,

Dimensions Lx Ly Lz T H �

Author A � 1 1 0 �1 0 0Author B � 1 1 �1 �1 0 0

(ii) For the thermal conductivity, k,

Author A k 0 0 �1 �1 1 �1Author B k �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 setspurported 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 theabsence of a general supporting argument, the case for the extension of the lengthdimension has rested on the validity of examples. – there is no need for the extensionof 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 thismaxim; “ – if everything in some science can be interpreted without assuming thisor 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 determi-nation of both the variables and their appropriate dimensions. The discussion showsthat when dimensional analysis is used to determine an unknown solution, con-siderable care and understanding must be exerted: the initial steps of dimensionalanalysis are not always simple ones.

Exercises

5.1 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 inthe field, the field strength is proportional to the moment and in the field thefield 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.

5.2 Light applies a radiation pressure, p, that depends upon the reflectance, r , ofthe receiving surface. With radiation pressure dependent upon the luminousflux intensity of the source, I; the distance of the source, `, the velocity oflight, c, and the mechanical equivalent of light, P0, show that:

pc`2

P0ID f .r/ :

Use the dimensions of force, luminous flux intensity, velocity and length.5.3 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, !. Showthat:

PL!

E2D f

�Li!

E

�

:

Use dimensions of power, current and time.


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. Ed-

ucation, 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 6Supplementation of Derivations

The task of discovering the physical magnitudes connected withthe phenomena which are decisive and of eliminating thosemagnitudes which are of subordinate importance, is the verycore of the problem.L. Prandtl

Notation

a0 Speed of soundc Mean molecular speedCp Specific heat, constant pressureCV Specific heat, constant volumee Specific internal thermal energyE Young’s elastic modulusf ‘functional’F Forceg Gravity accelerationGr Grashof numberh Heightk Thermal conductivitykB Boltzmann constant` Length scale`m Mean free molecular path lengthm Mass of molecule, of planet, of projectilems Mass of the sunMa Mach numberM0 Molecular massN Number of molecules/unit volumeNU Nusselt number


116 6 Supplementation of Derivations

p PressurePr Prandtl numberq Resultant velocityQ HeatPQ Heat rate

R Gas constant, rangeRe Reynolds numbers Distancet TimeT TemperatureTw Wall temperatureT0 Reference temperatureu, v, w Velocity componentsU Reference velocityv Specific volumeV Resultant velocityw Widthx, y, z Coordinates

ˇ Bulk modulus� � Cp=CV

ı 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 knowledgeof the physics of the phenomenon being studied. Also inspection of the governingequations can sometimes enable a simplification in setting out the variables or incombining 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 isfixed at the other end. The following notation is used:

6.2 The Bending of a Beam 117

E Modulus of elasticity;I Second moment of cross-sectional area;` Length of beam;w Cross-sectional dimension of beam;W Concentrated end load;ı End deflexion.

Then, for deflexion within the elastic limit, there is the relation of:

ı D f .W; `; E; w/ : (6.1)

The pi-theorem solution is set out in Compact Solution 6.1.This gives the relation that:

ı

`D f

�W

E`2;

w

`

�

: (6.2)

This is the general solution. But consideration of the physics enables some approx-imations to be introduced. They are:

a) When the thickness of the cross-section in the plane of bending is not verysmall compared with the width then there is negligible transverse bendingof the cross-section. If this transverse bending is significant then a furthervariable, the Poisson ratio, is added to Equation 6.2 forming its own non-dimensional variable.

Figure 6.1 Sketch illustrationof the bending of a cantileverbeam


ı W ` E W

L MLT2

L MLT2

L

WE

�

�

L2�

ı`

WE`2

�

�

�

�

W`

1 1 � � 1


b) When in addition to approximation (a), the deflexion is small then plane cross-sections remain plane under load.

c) When the deflexion is small, and within the elastic limit for the beam, thenı / W .

With these approximations, item (c) above requires that Equation 6.2 must take theform of:

ı

`D W

E`2f�w

`

�

or,

ıE`

WD f

hw

`

i: (6.3)

The second moment of area, I / w4. Thus Equation 6.3 becomes:

ıE`

WD f

�w4

`4

�

D f

�I

`4

�

:

Subject to the above approximations (a) and (b), E and I appear in the governingequations in the combination EI . This last equation must then take the form of:

ıE`

W/ `4

I

so that:

ıEI

W `3D constant : (6.4)

This is a well recognised result but it is now made clear, and will be returned tolater, that it is a special case of Equation 6.2, being seen here to be subject to severallimiting approximations.

6.3 Planetary Motion

Applying dimensional analysis to the motion of the planets the following notationis used.

F Gravitational force of the sun;` Size of the orbit;m Mass of the planet;ms Mass of the sun;t Time of planetary orbit.

6.3 Planetary Motion 119


t F m `

T MLT2

M L

Fm

�

�

LT2

�

�

�

�

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

m`D constant : (6.5)

There is extra information available in the form of Newton’s law of gravitationwhich 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

`3D constant :

This is Kepler’s famous law which he derived from extensive observation. The abovederivation again makes use of extra information though not in this case requiring anyapproximations.

It is fascinating that had Kepler known of dimensional analysis when he de-rived the above result in 1619 and after lengthy calculation, he could have deducedNewton’s law of gravitation before the latter finally confirmed the accuracy of it in1685 [1].


6.4 Extrapolated Solutions

Examples have been given where a non-dimensional group has been excluded be-cause a variable in that group has been extrapolated to a value of either zero orinfinity so that the group then has a fixed value. This asymptotic approach has to bedone with care as is now illustrated.

The example of natural convection, to be discussed later, involves finally anasymptotic solution as Re tends to zero as Churchill has also discussed [2]. Theidea of an asymptotic solution was used by Prandtl in 1904 for his concept of thethin boundary layer [3].

Care has to be taken because misunderstanding of the physics can arise. Theasymptotic approach can also fail. For example, if one tries to obtain an incom-pressible flow from a compressible flow by letting the Mach number tend to zeroone simply loses the solution because, for example p=p0 ! 1. A further exam-ple is of the laminar flow through a pipe as considered in Sections 3.6, 5.6, and byPankhurst [4].

With the notation of:

L pipe length�p pressure dropPQ flow rate

d pipe radius� viscosity

then:

�p D f�L; PQ; �; L

:

Compact Solution 6.3 gives the solution as:


�p L PQ d �

MLT2

L L3

TL M

LT

�p

�

�

�

1T

�

�

�p

� PQ

�

�

�

�

1L3

�

�

�

�

d 3�p

� PQ

Ld

�

�

�

�

�

�

1 1 � � �

6.5 Uncoupled Equations 121

Thus,

d 3�p

� PQD f

�L

d

�

(6.6)

which is correct because it takes account of the entry region. It would now be incor-rect to make the entry effect negligible by letting L ) 1 because in Equation 6.6this gives the incorrect asymptotic result that:

�pd 3

� PQ D constant : (6.7)

This suggests that the pressure drop is independent of the pipe length which clearlycontravenes the physics. As explained previously, that physics shows that away fromthe entrance region the balance between the pressure and viscous forces requiresthat, �p / �L. Then Equation 6.6 gives the previous correct result of:

�pd 4

�L PQ D constant : (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 fourflows described there. For these flows, the asymptotic solution is not obtained inEquation 1.18 by putting Re D 0. Another writer adopted the assumption that ˘ DŒO�1. This assumption can be seriously misleading as some magnitudes of the pi-groups 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 numericalvalues of the non-dimensional groups.

6.5 Uncoupled Equations

Another example where supplementary information comes from the governingequations is of the motion of a projectile under gravity in a vacuum as illustratedin Figure 6.2.

Figure 6.2 Sketch illustrationof the trajectory of a projectile



R g h m u0 v0 t

L LT2

L M LT

LT

T

�

�

u20

g

v20

g

gt2

� L L L

Rh

�

�

�

�

u20

gh

v20

gh

gt2

h

1 � � 1 1 1

With the notation of:

g acceleration under gravityh height of launch pointm mass of projectileR ranget time of flightu0 horizontal component; launch velocityv0 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:

R

hD f

�u2

0

gh;

v20

gh;

gt2

h

�

and so gives four non-dimensional groups. It is noted that dimensional analysis hasshown that the mass, m, does not appear in this solution.

The governing equations are, for the horizontal component of the motion,

R D u0t

and for the vertical component,

�h D v0t � .1=2/ gt2 :

Inspection of these governing equations shows that the horizontal and vertical mo-tions are uncoupled so that the latter can first be solved alone for t and then theformer 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


t g h v0

T LT2

L LT

t2g �

�

v20

g

L � � L

t2g

h

�

�

�

�

v20

gh

1 � � 1

which gives:

t2g

hD f

�v2

0

gh

�

then:

t2g

h� gh

v20

D f

�gh

v20

�

or:

tg

v0D f

�gh

v20

�

: (6.10)

Then for the horizontal component of the trajectory, R D f .u0; t/ leading to theCompact Solution 6.6.


R u0 t

L LT

T

Ru0

�

�

T �

Ru0t

�

�

�

�

1 � �

This gives:

R=u0t D constant (6.11)


Eliminating t between Equations 6.10 and 6.11 gives the two groups as:

R

u0t� tg

v0D f

�gh

v20

�

or:

Rg

u0v0D f

�gh

v20

�

:

For h D 0, this finally reduces to the single non-dimensional group of:

Rg

u0v0D constant :

A similar case of motion occurs for the stability of an aeroplane. By limiting motionto small perturbations of disturbance, then the solution for the longitudinal motioncan be separated from that for the lateral one. In this case progress has again beenmade by noting a particularity of the governing equations in that the two motionsare uncoupled. Other examples follow.

6.6 Forced Convection of Thermal Energy

An illustration of the need for care in introducing units-conversion factors will nowbe described in the considerable detail which is found necessary to resolve a longstanding problem. It will also show another case of the uncoupling of equationswhich 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 com-pressible stream of an ideal and Newtonian gas, and a solid body. This is illustratedin Figure 6.3. It is a matter of concern, for example, in the design of gas and steamturbines and of aeroplane cooling radiators.

With u, v, and w, the x, y, and z velocity components and q the resultant velocitythen for a continuum and compressible flow and with:

div q D @u

@xC @v

@yC @w

@z(6.12)

with:

q2 D u2 C v2 C w2 (6.13)

6.6 Forced Convection of Thermal Energy 125

Figure 6.3 Illustration of anexample of forced thermalconvection

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

dtD �@p

@xC @

@x.� div q/ C div

�

�@q

@x

�

C div .� grad u/ : (6.15)

The one for the v component is:

�dv

dtD �@p

@yC @

@y.� div q/ C div

�

�@q

@y

�

C div .� grad v/ (6.16)

and for the w component is:

�dw

dtD �@p

@zC @

@z.� div q/ C div

�

�@q

@z

�

C div .� grad w/ ; (6.17)

where p is the pressure, � is the viscosity and � is the second coefficient of viscosity.The energy equation for this compressible flow is:

�de

dtD div k�T � p div q C ˚ ; (6.18)


where e is the specific internal energy, k is the thermal conductivity and ˚ is thedissipation 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

�D RT

M0; (6.20)

where R is the universal gas constant and M0 is the molecular molar mass. Theparticular 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]:

�

�0D f .T /I k

k0D f .T /I CV D f .M0; T / ; (6.22)

where the suffices of ‘0’ on �0 and k0 indicate reference values usually taken asboundary values. For extremes of high pressure these functions can include the pres-sure as an independent variable. Finally, a commonly made assumption, not alwaysvalid, 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 Equa-

tions 6.20 and 6.21, then the three of Equation 6.22 and finally Equation 6.23. Cor-respondingly 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 giv-ing:

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 pi-

theorem solution then is of Compact Solution 6.7.



q U �0 �0 k0 ` CV R M0 T0 Tw p0

LT

LT

ML3

MLT

MLT3�

L L2

T2�ML2

T2�nMn

� � MLT2

RM0 �

L2

T2�

�

�

k0T0 cvT0 RT0M0

�

�

�

�

TwT0

MLT3

L2

T2L2

T2�

�

�

� 1

�0�0

�

�

k0T0�0

�

�

�

�

p0�0

TL2

�

�

L2

T2�

�

�

�

1T

g

U

�

�

U�0�0

�

�

k0T0�0U 2

CV T0U 2

RT0M0U 2

�

�

�

�

p0�0U

1 �

�

1L

�

�

1 1 1 �

�

�

�

1L

�

�

U�0`

�0

�

�

�

�

�

�

�

�

p0`

�0U

� 1 � � � � 1

this shows that:

q

UD f

��0U `

�0;

k0T0

�0U 2;

CV T0

U 2;

RT0

M0U 2;

Tw

T0;

p0`

�0U

�

: (6.26)

The last group in Equation 6.26 comes from the independent variable of p0. Butinspection of Equation 6.20 shows that it is a superfluous variable because the othervariables in that equation have been included in the listing of Equation 6.25. thusthe corresponding last group in Equation 6.26 can be excluded.

Inverting the second independent group of Equation 6.26, dividing the fourth bythe third and also dividing the third by the second produces the following:

q

UD f

��0U `

�0;

�0U 2

k0T0;

R

M0CV

;CV �0

k0;

Tw

T0

�

: (6.27)

Of these independent groups the first is the Reynolds number, Re, and the fourth isthe Prandtl number, Pr. Using the ideal gas relation of:

R

M0D Cp � CV (6.28)

and indicating the ratio of the specific heats by:

� � Cp=CV : (6.29)


Then the third group is:

R

M0CV

D � � 1 (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:

�0U 2

`2

and with the other terms in this equation, all have the dimensions of ML�1T�3. Thenfrom that equation, in non-dimensional form it can be expressed by:

˘˚ � �0U 2

`2� `2

k0T0D �0U 2

k0T0(6.31)

which is the second independent group in Equation 6.27.In summary, Equation 6.27 is now written as:

q

UD f

�

Re; ˘˚ ; �; Pr;Tw

T0

�

: (6.32)

An alternative formulation is obtained by requiring that the compressibility of thegas, @�=@p is related to a process that is independent of the viscosity which latter isrepresented by the variable, �. Such a process is a sound wave as the strength tendsto zero for then the process tends to an isentropic one [8]. In this limit,

1=a20 D @�

@p;

where a0 is the velocity of sound. This is given by [9]:

a20 D �

R

M0T0 : (6.33)

Then by combining the second, third and fourth independent groups in Equa-tion 6.32 there is:

˘˚

�.� � 1/PrD 1

�

�0U 2

k0T0

M0CV

R

k0

CV �0

D U 2

�.R=M0/T0(6.34)

D U 2

a20

D M 2a ;

where Ma is the Mach number.


Replacing then the variable ˘˚ by Ma in Equation 6.32 reforms this equationinto:

q

UD f

�

Re; Ma; �; Pr;Tw

T0

�

(6.35)

which is the commonly quoted form. Yet it is now seen that the representation ofthe 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 governingequations so that the full set of independent variables of Equation 6.25 can be de-duced. It is also important to understand the physics represented by each individualnon-dimensional group.

Equation 6.26 has another point of interest in that it contains non-dimensionalgroups 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 thesegroups in a quite straightforward manner.

There is a thermodynamic requirement that an ideal gas that obeys Equation 6.20needs three thermodynamic properties to fully describe it; this is satisfied here. Theindependent variables in Equation 6.25 are classified as:

Properties; T0, �0, M0

Units-conversion factor R

Boundary conditions U , Tw, `

Derived variables �, k, CV .

In forming this table it has been recognised that �0, k0, and Cv0 are known from thegas properties as listed.

For the case of determination of heat transfer, an alternative dependent group canbe the Nusselt number, Nu, which is:

Nu � PQ= Œk0` .Tw � T0/� : (6.36)

It is noted that kB does not appear among the variables of Equation 6.25 despite itbeing the units-conversion factor for T . It is seen that it does not appear amongstthe above governing equations. These contain R and M0 in the combination R=M0.However,

R

M0D kB

m(6.37)

so that R could be replaced by kB whilst m would then replace M0. This wouldgive an unsatisfactory formulation because, there being one less dimension required,that of quantity, there would be one extra non-dimensional group. This would besuperfluous as will now be shown. Tabulating with these revised variables gives theCompact Solution 6.8.



q U �0 �0 k0 ` CV kB m T0 Tw p0

LT

LT

ML3

MLT

MLT3�

L L2

T2�ML2

T2�n

M �

�

� MLT2

k0T0 CV T0 kBT0 �

�

TwT0

MLT3

L2

T2ML2

T2�

� 1

�0�0

�

�

k0T0�0

kBT0�0

m�0

�

�

p0�0

TL2

�

�

L2

TL3

TLT �

�

1T

q

U

�

�

U�0�0

�

�

k0T0�0U 2

cvT0U 2

k0T0M0U 2

mU�0

�

�

p0�0U

1 �

�

1L

�

�

1 1 L2 L2�

�

1L

�

�

U�0`

�0

�

�

�

�

kBT0�0U `2

mU�0`2

�

�

p0`

�0U

� 1 � � 1 1 � 1


q

UD f

�U�0`

�0;

k0T0

�0U 2;

CV T0

U 2;

kBT0

�0U `2;

mU

�0`2;

p0`

�0U

�

: (6.38)

Dividing the third non-dimensional group of this equation by the second gives:

CV T0

U 2� �0U 2

k0T0D CV �0

k0

which is the Prandtl number.Combining the first, the second and the fourth independent groups of Equa-

tion 6.38 gives:

�0

U�0`� �0U 2

k0T0� kBT0

�0U `2D kB�0

k0�0`3: (6.39)

This group can be assessed by substituting, from the kinetic theory of gases (Sec-tion 8.28), the relation:

�0 / �0c`m

and the one for the conductivity (Section 8.29) which is:

k0 / N kBc`m


then:

kB�0

k0�0`3D 1

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 heat-ing would give a time dependent phenomenon so that the time, t , becomes anotherindependent variable. It would bring in the non-dimensional group of .Ut/=`.

The final independent group containing p0 can be excluded for the reason givenin the case of Equation 6.26.

It is now seen that this different formulation leads to the same result of Equa-tion 6.26.

6.6.2 Incompressible-flow Energy Transfer

The energy transfer in an incompressible flow is relevant, for examples, to the per-formance 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

dtD �@p

@xC ��2u ;

�dv

dtD �@p

@yC ��2v ; (6.41)

�dw

dtD �@p

@zC ��2w :

A further assumption made is that temperature changes are small so that Equa-tions 6.22 are not required. Then, inspection of these four equations for the fourunknown independent variables, that is; p, u, v and w, shows that now they canbe solved on their own. This is a further example of the importance of equationsbecoming uncoupled. In this case the momentum and continuity equations are un-coupled from the energy equation which now is:

�cvdT

dtD k�2T C ˚ : (6.42)

Thus in principle the continuity equation and the momentum equations can besolved for the velocity field and then the energy equation can successively be solvedfor the temperature field.



q U � �0 `

LT

LT

ML3

MLT

L

�

�0

�

�

TL2

�

�

q

U

�

�

�U

�0

�

�

1 �

�

1L

�

�

�

�

�U `

�0

�

�

�

�

� 1 � �

This incompressible assumption means also that a solution does not require theequation of state, Equation 6.20, and so the units-conversion factor of R does notappear amongst the variables. Inspection of the dimensions of the variables in thepi-theorem tabulation following Equation 6.25 shows that only the two variables ofR 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 ap-pears only as a difference. Thus this dependent variable occurs only as the combina-tion .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 butnow the reason is a completely different one. So an alternative solution to the conti-nuity 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 forthe 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 fifthand sixth from the energy equation and the last two are boundary conditions to theenergy equation.



T � T0 CV �0 � U Tw � T0 k0 `

� L2

T2�MLT

ML3

LT

� MLT3�

L

T �T0Tw�T0

CV .Tw�T0/ �

�

k0.Tw�T0/

�

1 L2

T2�

�

MLT3

�

�

�

�0

�

�

k0.Tw�T0/

�0

�

�

TL2

�

�

L2

T2

CV.Tw�T0/

U 2�

�

�U

�0

�

�

�

�

k0.Tw�T0/

�0U 2

1 �

�

1L

�

�

�

�

1

�

�

�U `

�0

�

�

�

�

�

�� 1 � � � �

However, in Equation 6.42 the temperature appears as only a difference. Thusthe variables T0 and Tw are combined into the independent variable of .Tw � T0/ sothat 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

Tw � T0D f

�CV .Tw � T0/

U 2;

�U `

�0;

k0 .Tw � T0/

�0U 2:

�

(6.46)

Dividing the first independent group by the third and also taking the reciprocal ofthe third gives:

T � T0

Tw � T0D f

�CV �0

k0;

�U `

�0;

�0U2

k0 .Tw � T0/

�

: (6.47)

In this equation the first independent group is the Prandtl number, the second is theReynolds number and the third represents the dissipation function.

Now for the incompressible flow, the dissipation function in the energy equationappears as a separate non-dimensional group. As before, commonly in formal anal-ysis it is taken as being negligible. This is equivalent to neglecting this last group inEquation 6.47 so that finally:

T � T0

Tw � T0D f

�CV �0

k0;

�U `

�0

�

: (6.48)



PQ k0 ` Tw � T0 CV � �0 U

ML2

T3MLT3�

L � L2

T2�

ML3

MLT

LT

k0 .Tw � T0/ � CV .Tw � T0/

MLT3

�

�

L2

T2

PQ

k0.Tw�T0/

�

�

�

�

�

k0.Tw�T0/

�0k0.Tw�T0/

L �

�

�

�

T3

L4T2

L2

�

�

�

�

CV .Tw�T0/

U 2�U 3

k0.Tw�T0/

�0U 2

k0.Tw�T0/

�

�

�

�

�

�

1 1L

1 �

�

PQ

k0`.Tw�T0/

�

�

�

�

�

�

`�U 3

k0.Tw�T0/

�

�

1 � � � 1

Alternatively, the dependent variable can be chosen as the rate of heat, PQ. The so-lution is in Compact Solution 6.11.

This gives:

PQk0` .Tw � T0/

D f

�CV .Tw � T0/

U 2;

`�U 3

k0 .Tw � T0/;

�0U 2

k0 .Tw � T0/

�

:

Multiplying the first independent group by the third and dividing the second by thethird gives:

PQk0` .Tw � T0/

D f

�CV �0

k0;

�U `

�0;

�0U 2

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 theapproximations involved, the care needed over the determination of the use of theappropriate units-conversion factors and the use of the uncoupling of the governingequations 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 applicationto the problem of forced convection heat transfer. In the original study [9], Rayleighhad considered the relation for the heat rate, PQ. He had omitted �0 from the list

6.7 The Rayleigh–Riabouchinsky Problem 135


PQ k0 ` Tw � T0 CV � U

ML2

T3MLT3�

L � MLT2�

LT

k0 .Tw � T0/ � CV � .Tw � T0/

MLT3

�

�

MLT2

PQ

k0.Tw�T0/

�

�

�

�

CV �.Tw�T0/

k0.Tw�T0/

L �

�

�

�

TL2

�

�

�

�

CV �U

k0

�

�

�

�

�

�

1L

�

�

PQ

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 theenergy equation, Equation 6.42, it is noted that [7],

D

DtD @

@tC q

@

@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],

PQk0` .Tw � T0/

D f

�CV �U `

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 ofEquation 6.42 shows a justification for his combination of � and CV as a single vari-able. But as described in Section 5.4, this example comes under the same headingof being able to use a dimension of heat, H, rather than that of M. Then the combi-nation of �CV is the equivalent as before of basing CV upon a volume rather thanon a mass. It might be thought that it is equally acceptable to also divide throughthis reduced energy equation by k0 to form a single variable from these three. Butthis would require the approximation at this stage of putting the value of the dis-sipation function at zero. Also, interestingly, doing this would prohibit a solutionbecause in the above tabulation of dimensions an attempted solution would showthe temperature difference as the only variable with a dimension in � . This becomesa case where combination of variables obtained from inspection of the governing


equations cannot be taken as always acceptable for developing a solution obtainedby dimensional analysis.

Riabouchinsky then proposed using the result of the kinetic theory of gases tomeasure 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 furthernon-dimensional group. That is one more non-dimensional group than was obtainedby Rayleigh. It might be expected that this removal of one dimension would corre-spond with removal of the corresponding units-conversion factor, which in this caseis the Boltzmann constant. But the equation:

T / m Nc2=kB (6.51)

is uncoupled from the others so that the energy equation can, in principle, be solvedin 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 nowenlarged to a full explanation.

Riabouchinsky did not give his full analysis but it appears that he used Equa-tion 6.51 in which he set kB � 1. He also seems to have used the energy equa-tion 6.42.

With PQ � ML2T�3, then from Equation 6.51,

k0 � ML2T�3L�2L��1 D MLT�3��1

From Equation 6.51,

� � ML2T�2

and then,

k0 � L�1T�1

From Equation 6.42

�CV � L�1T�1�L�2T��1 D L�3

So Riabouchinsky’s solution is in Compact Solution 6.13.This gives:

PQ`

m Nc2UD f

�k0`2

U; �CV `3

�

:

Dividing the first group by the second and then inverting the second one gives:

PQk0`m Nc2

D f

�U

k0`2; �CV `3

�

(6.52)

6.7 The Rayleigh–Riabouchinsky Problem 137


PQ k0 ` m Nc2 U �CV

ML2

T31

LTL ML2

T2LT

1L3

PQ

m Nc2�

� �

1T

�

�

PQ

m Nc2U

k0U

�

�

�

�

1L

1L2

�

�

�

�

PQ`

m Nc2U

k0`2

U

�

�

�

�

�

�

�CV `3

1 1 � � � 1


PQ �CV

k0` m Nc2 U

ML2

T3TL2

L ML2

T2LT

PQ

m Nc2�

�

1T

�

�

PQ�CV

m Nc2k0

�

�

�

�

U�CV

k0

1L2

�

�

�

�

1L

PQ�CV `2

m Nc2k0

�

�

�

�

�

�

U�CV `

k0

1 � � � 1

which is the result put forward by Riabouchinsky, differing from Rayleigh’s solu-tion of Equation 6.50. But Rayleigh was solving just the energy equation whilstRiabouchinsky was dealing with both the energy and the temperature equations.As with Rayleigh’s solution, the exclusion of the variable �0 has the same conse-quences for both.

Riabouchinsky’s form of the energy equation, Equation 6.42, would be:

�CV

dm Nc2

�

dtD kr2

m Nc2

�: (6.53)

From this equation the grouping of �CV =k can be taken as a single variable. Thenthe solution is given in Compact Solution 6.14.


This gives:

PQ�CV `2

mk0 Nc2D f

�U�CV `

k0

�

: (6.54)

This has just the two groups as were obtained by Rayleigh in Equation 6.50. InRiabouchinsky’s solution there is a difference from the procedure in Rayleigh’sbecause now combining .�CV /=k0 to form a single variable does not change thenumber of dimensions. It also now results in the two groups that Rayleigh obtainedand not the three that Riabouchinsky found. Their two approaches are now seen togive a common result. However, whilst this corrected result of Rayleigh’s applies toNewtonian fluids in general, Riabouchinsky’s is limited, through Equation 6.51, toperfect gases. The full correct solution is that of Equation 6.49.

To complete the discussion of this famous controversy the full solution is nowset out but using Riabouchinsky’s approach. As before, the temperature terms inthe energy equation are as differences so the variable is written as . Nc2

w � Nc20/. The

pi-theorem solution now becomes Compact Solution 6.15:


PQ k0 ` m Nc2

w � Nc20

�CV � �0 U

ML2

T31

LTL ML2

T21M

ML3

MLT

LT

PQ

�

m. Nc2w� Nc2

0/�

CV � �

�

�0�

L5

T3L5

T21

L3�

�

L2

T

PQ

�U 3k0U

m. Nc2w� Nc2

0/�U 2

�

�

�0�U

�

�

L2 1L2

L3�

�

L �

�

PQ

�U 3`2k0`2

U

�

�

m. Nc2w� Nc2

0/�U 2`3

CV �`3�

�

�0�U `

�

�

1 1 � 1 1 � 1 �


PQ�U 3`2

D f

"k0`2

U;

m Nc2

w � Nc20

�

�U 2`3; CV �`3;

�0

�U `

#

:

By dividing the first group by both the second and the third ones, then inverting thefifth 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:

PQm Nc2

w � Nc20

�k0`

D f

"�U `

�0;

CV �0

k0;

�0U2

k0m Nc2

w � Nc20

� ; CV �`3

#

: (6.55)

The dependent group is the one obtained by Rayleigh: the first independent oneis the Reynolds number; the second is the Prandtl number, the third represents thedissipation 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 previouslydiscussed in Sect 6.6.1 as representing the reciprocal of the total quantity of gas andso can be excluded as being zero in value.

Thus adopting Riabouchinsky’s approach in the full solution gives the identicalresult to that of Equation 6.49.

This famous example, which dates back to 1915, is here fully resolved. It isof considerable value in that it brings forward several important matters. First isthe care needed in the initial choice of variables so that all the relevant physicsis accounted for; secondly is the care to be taken in combining variables throughreference to the various governing equations; thirdly is the care to be taken in aninitial inspection of the dimensions of the variables and in particular the care inchecking the effect on the presence of the needed dimensions as resulting from aninitial combination of variables; fourthly is the value in a careful inspection of theform and the meaning of the non-dimensional groups obtained and fifthly is the needof inspection for equations being uncoupled.

In discussing this controversy, Bridgman made the perceptive comment, over thechoice of variables, that [10]:

“We will probably find ourselves able to justify the neglect of all these quantities, but thejustification will involve real argument and a considerable physical experience with thephysical 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. Thisphenomenon is associated with the application to buildings, to heating radiators andto the thermal convection interchange between the earth surface and the atmosphere.

It is again governed by the equations for forced convection. These are Equa-tions 6.12–6.18, and 6.20. The standard formal analysis of these flows introducesseveral approximations to these equations in order to make reasonable progress:these have to be addressed.


The first approximation is to assume a one-dimensional flow so that only theequation of momentum, Equation 6.15 is retained. However it now has a gravity-force term and so becomes:

�du

dtD �@p

@x� �g C @

@x.�div q/ C div

�

�@q

@x

�

C div .�grad u/ : (6.56)

Secondly, the pressure gradient is approximated to by the hydrostatic value so that:

@p

@xD �g�0

Then Equation 6.56 reduces to:

�du

dtD �g

��0

�� 1

�

C @

@x.�div q/ C div

�

�@q

@x

�

C div .�grad u/ : (6.57)

The coefficient of bulk thermal expansion is defined in terms of the specific volumeby:

ˇ D 1

v0

@v

@T:

Then as v D 1=�:

ˇ D �[email protected]=�/

@T

D ��0

�2

@�

@T:

Figure 6.4 Illustration of anexample of natural thermalconvection under buoyancy ofa gas


This is approximated by:

ˇ D � 1

�0

� � �0

T � T0(6.58)

D�

1 � �

�0

�1

T � T0:

With this and the assumption that � / �, then the momentum equation, Equa-tion 6.57 becomes:

�Du

DtD �gˇ� .T � T0/ C �f .u; `/ : (6.59)

Making the assumption of incompressible flow in only the continuity and energyequations, and again with those of negligible temperature effects in Equations 6.22,reduces the energy equation to:

�0Cv0dT

dtD k0�2T C �0f .u; `/ : (6.60)

It further follows from these approximations that Equation 6.20 is not required fora solution so that the variable R=M0 is excluded. Thus a solution comes from thetwo simultaneous Equations. 6.59 and 6.60. Choosing U as the dependent variable,inspection of these two governing equations gives:

U D f Œ�0; gˇ; Tw � T0; �0; `; k0; CV � : (6.61)

The pi-theorem solution is in Compact Solution 6.16.


U �0 gˇ Tw �T0 �0 ` k0 CV

LT

ML3

LT2�

� MLT

L MLT3�

L2

T2�

gˇ .Tw � T0/LT2

��

k0 .Tw � T0/MLT3

CV .Tw � T0/

L2

T2

�

�

�

�

�0�0

k0.Tw�T0/

�0

�

�

�

�

L2

TL2

T2

�

�

�

�

gˇ.Tw�T0/

U 2�

�

�0�0U

k0.Tw�T0/

�0U 2CV .Tw�T0/

U 2

�

�

�

�

1L

�

�

L 1 1

�

�

�

�

gˇ.Tw�T0/`

U 2�

�

�0�0U `

�

�

� � 1 � 1 �


Thus the relation is:

�0

�0U `D f

�gˇ .Tw � T0/ `

U 2;

k0 .Tw � T0/

�0U 2;

CV .Tw � T0/

U 2

�

:

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 leadsto:

�0U `

�0D f

�gˇ�2

0`3 .Tw � T0/

�20

;CV �0

k0;

�0U 2

k0 .Tw � T0/

�

: (6.62)

In this equation the dependent variable is the Reynolds number; the first indepen-dent group is called the Grashof number and the second one is again the Prandtlnumber. The third independent group is the previously described representation ofthe dissipation function.

Again in this case, it is usual practice to regard the dissipation function as a neg-ligible term in the energy equation. Making this assumption then reduces Equa-tion 6.62 to:

�0U `

�0D f

�gˇ�2

0`3 .Tw � T0/

�20

;CV �0

k0

�

: (6.63)

Alternatively, the heat rate can be taken as the dependent variable so that Equa-tion 6.63 becomes:

PQk0` .Tw � T0/

D f ŒGr; Pr� : (6.64)

There is now an interesting comparison between the three thermal convection casesconsidered here. For the compressible flow case the group representing the dissipa-tion function has been shown to be contained within the other independent groupsthat are considered significant. For the other two cases where it is not so contained,the dimensional analysis reproduces the necessary separate group which is usuallyinsignificant.

At very low speeds the acceleration term on the left-hand side of Equation 6.59is negligible. Inspection of that equation with Equation 6.60 gives the variables as:

U; gˇ�0; �0; k0; �0CV ; Tw � T0; ` :

The pi-theorem solution is in Compact Solution 6.17.This gives that:

gˇ�0`2 .Tw � T0/

�0UD f

�k0 .Tw � T0/

�0U 2;

�0CV ` .Tw � T0/

�0U

�

:



U gˇ�0 �0 k0 �0CV Tw � T0 `

LT

ML2T2�

MLT

MLT3�

MLT2�

� L

gˇ�0 .Tw � T0/ k0 .Tw � T0/ �0CV .Tw � T0/ �

ML2T2

MLT3

MLT2

�

�

gˇ�0 .Tw�T0/

�0

�

�

k0.Tw�T0/

�0

�0CV .Tw�T0/

�0

�

�

1LT

�

�

L2

T21T

�

�

�

�

gˇ�0 .Tw�T0/

�0U

�

�

k0.Tw�T0/

�0U 2�0CV .Tw�T0/

�0U

�

�

�

�

1L2

�

�

1 1L

�

�

�

�

gˇ�0 `2.Tw�T0/

�0U

�

�

�0CV `.Tw�T0/

�0U

�

�

�

�

� 1 � 1 � �

Of these three non-dimensional groups, dividing the third by the second, taking thereciprocal of the second and then multiplying the first and the third and dividing thisproduct by the second finally gives:

�0U `

�0� CV �0

k0D f

�gˇ�2

0`3 .Tw � T0/

�20

� CV �0

k0;

�0U 2

k0 .Tw � T0/

�

: (6.65)

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 Prandtlnumber whilst the second independent group is representative of the dissipationfunction.Or this is written as:

Pr � Re D f

�

Pr � Gr;�0u2

k0 .Tw � T0/

�

: (6.66)

If the Dissipation Function is again neglected and putting PQ as the dependent vari-able, then:

PQk0` .Tw � T0/

D f ŒGr � Pr� : (6.67)

This last equation together with that of Equation 6.64 are commonly given in ther-modynamic texts. It is now clear that their derivation depends upon the many ap-proximations to the full governing equations. These approximations have to be in-troduced to get these two valid results.


Certainly the exact solution without these assumptions is the much more complexone of Equation 6.26 which, together with also the Grashof number, contains sixnon-dimensional groups.


The discussion of this chapter emphasises the care needed in supplementing theoriginal functional statement by further knowledge of the physics or by the intro-duction of valid approximations or both. As shown here, the original functionalstatement can enable dimensional analysis to derive a general solution. Then sup-plementation leads to special cases. This latter result is relevant to the previousdiscussion in Section 5.9.

Exercises

6.1 A bi-metallic strip deflects under a change in its temperature. With the nota-tion of:

B Breadth of the strip;E1, E2 Young’s moduli of the two components;H Height of the strip;` Length of the strip;R Radius of curvature of the deflected strip;˛1, ˛2 coefficients of thermal expansion of the

two components;�T Temperature change.

show that:

r

`D f

�h

`;

b

`; ˛1�T; ˛2�T;

E1

E2

�

:

Set down physical conditions so that this result reduces to:

r

`D f

�

˛1�T; ˛2�T;E1

E2

�

:

6.2 A liquid meniscus can rise or be depressed under the action of surface tensionin a vertical tube. The direction depends upon the contact angle. With thefollowing notation:


g Acceleration of gravity;h Displacement of meniscus;R Tube radius;� Contact angle;� Liquid density; Surface tension coefficient.

show that:

h

RD f

�

�gR2; �

�

:

Consider the physical condition that enables this equation to be reduced to theapproximate relation of:

h�gR

D f .�/ :

6.3 The current density from the cathode of a valve oscillator, j , is a function ofthe electron mass, m, the electron charge, e, the cathode-anode distance, `,the applied potential, , and the permittivity, ". Show that:

j 2m`6

e3D f

�"`

e

�

:

Noting that the equation governing the path of an electron is:

md2x

dt2D �e

d

dx:

can you simplify the above result?6.4 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 thetension in the string, � . Show that:

�

m!2`D f

ha

`

i:

For small deflexions of the string the restoring force is proportional to thedeflection. Then, bearing in mind the previous discussion of linear vibration,simplify this relation for the frequency. [Note that this case is unlike that de-scribed in Section 9.5. Whilst the added strain in the string due to the deflexionalso gives a cube relation between the corresponding restoring force and theamplitude, this strain is very small compared with the pre-tension in the stringof musical instruments so that a linear relation is a good approximation: but itis an approximation.]


Then go on to derive an expression for the energy of vibration and show thatthis is not a function of the tension. Finally show that:

E`

�a2D constant :

6.5 Waves on the surface of a liquid can be due to gravity forces and to surfacetension effects. With the notation of:

c Wave speedg Gravitational accelerationh Liquid depth� Wave length� Liquid density Surface tension coefficient

show that:

c2

g�D f

��g�2

;

h

�

�

:

Deduce expressions for very deep liquid when gravity effects dominate andagain for when surface tension effects dominate.

6.6 The temperature, T , of an astronomical black hole is a function of the follow-ing:

c Speed of lightG Gravitational constanthp Planck constantkB Boltzman constantM Mass of black hole

Show that:

T 2Gk2B

hpc5D f

�Mc2

kBT

�

:

In the equation for the hydrostatic equilibrium in stellar structures the vari-ables M and G appear in the combination of the single variable of (MG).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. Inter-

nationalen 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, 19506. 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 7Systematic Experiment

– the partial information given by dimensional analysis may becombined with measurements on only a part of the totality ofphysical systems covered by the analysis, so that together all theinformation needed is obtained with much less trouble andexpense than would otherwise be possible.P.W. Bridgman

Notation

CH Filter coefficientCV Coefficient of specific heatd Fan diameter: pore sizeDC, D� Diffusion coefficientsF Faraday constantg Gravitational accelerationic Electrical convection currentk Trip rod diameterk0 Thermal conductivity` Scale lengthn Rotational speedP Fan powerQ TorquePQ Heat rate

s Electrode spacingt TimeT Thrustv Flow velocityV Flow velocityw Filter thicknessx Distancexk Trip positionxT Position, start of transitionz Valency

˛, ˇ Coefficients, dissociation, recombinationˇ Gas expansion coefficientˇ0 Units-conversion factor for angleı�

k Boundary-layer displacement thickness


150 7 Systematic Experiment

" Dielectric coefficient� Efficiency� Blade angle� Electrical conductivity� Viscosity� Density' Electrical potential

�H Hydraulic head difference�p Pressure difference�T Temperature difference.

7.1 The Benefits of Dimensional Analysis

The elementary introduction to dimensional analysis given in Chapter 1 gave buta glimpse of its power when in partnership with experiment. In this chapter thisgreat benefit will be illustrated in detail as it is a prime case for the usefulness ofthis analysis.

Dimensional analysis can markedly organise well the design and the economyof an experiment; it can greatly clarify and order the output of the data; and it canconsiderably extend the applicable range of investigation. These features, which canhave many valuable consequences for experiment, are now listed to be followed inturn by detailed description.

a) There can be a great reduction in the amount of experimental investigationthrough a reduction in the number of independent variables that are required tobe adjusted.

b) The effect of a variable can most conveniently be determined by the experi-mental variation of another one.

c) An experiment can provide results that apply to the range of a variable that ismuch greater than the range set in the experiment. In the extreme case, thisapplicable range can be achieved by the experimental measurement of onlya single value of one independent variable.

d) The dimensional analysis can show in some cases that a variable has no effectupon the phenomenon and so can be excluded as an experimental variable.

e) The oversight of an independent variable in the planning of an experiment canbe revealed.

f) The experiment can be organised so that data acquisition is set in an orderedand convenient manner.

g) A standard validity test can be applied so that the relationship for the numberof non-dimensional groups is properly satisfied.

h) The presentation of the data is greatly compacted and clarified.

7.2 Reduction of Variables 151

i) The cost of an experiment can be reduced, or even sometimes experimentationcan be made feasible, by enabling tests to be made on reduced-scale modelsof the full-size system. Or, it can ease experimental difficulties by enablingexperiments 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 illustratedby the example of a test of a fan in a duct. Suppose interest is in the power to drivesuch a fan which is pumping air at a fairly low speed along a duct. This is illustratedin Figure 7.1.

It could be reasonable to specify initially that the power, P , is a function of thefan diameter, d , the air density, �, the air velocity along the duct, V , and the fanrotational 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 onlyd would give a single curve on a graph of P plotted against the values of d such asis 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). Furthervariation 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 wouldrequire 54 D 625 experimental points or 1250 readings. This has been described asgoing 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 Solu-tion 7.1.

Figure 7.1 Diagram of an axial-flow fan in a duct


Figure 7.2 Illustration of the complexity of the basic data


P d � V n

ML2

T3L M

L3LT

1T

P�

�

�

L5

T3�

�

P�n3

�

�

Vn

�

�

L5� L �

P�n3d 5

�

�

�

�

Vnd

�

�

1 � � 1 �

This leads to:

P

�n3d 5D f

h v

nd

i: (7.2)

7.3 Further Reduction of Non-dimensional Groups 153

Figure 7.3 Axial-fan powercoefficient

Now the whole experiment gives a single line on a single graph as sketched inFigure 7.3. This results in a reduction of the 1250 readings in the experiment tojust 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 thefan efficiency, �, as the dependent variable. Aeroplane airscrews and many ship pro-pellers have the facility of varying the blade angle, � , so that this becomes a furtherindependent 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:

� D f

�V

nd; ˇ0�

�

; (7.4)


� d � V n � ˇ0

1 L ML3

LT

1T

˛ 1˛

Vn

�

�

L �

�

�

Vnd

�

�

� 1 �

� � ˇ0� �

� � 1


Figure 7.4 Axial-fan efficiency plots

and, as described in Chapter 1 one variable is shown by this application of dimen-sional 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 val-ues is shown in Figure 7.4.

If there is interest in only the peak values of the efficiency, �p, then these aremarked 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

�p D f

�V

nd

�

(7.5)

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 reducedto

�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-dimensionalgroups.

7.4 Alternate Dependent Variables 155


�p d � V n

MLT2

L ML3

LT

1T

�p

�

�

�

L2

T2 �

�p

�n2�

�

Vn

�

�

L2� L �

�p

�n2d 2�

�

�

�

Vnd

�

�

1 � � 1 �

7.4 Alternate Dependent Variables

In the design of an experiment, care has to be taken in carefully distinguishing be-tween dependent and independent variables. It might be assumed that in this exam-ple of the axial-flow fan the power depends also upon the pressure rise across thefan, �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 inan experiment, then the value of �p is fixed. Thus it is not an independent variablethat can be varied in complete independence of all the others but is an alternativedependent one so that this is correctly expressed by:

�p D f .d; �; V; n/ : (7.9)

The solution takes the form of Compact Solution 7.3.This alternative solution is thus:

�p

�n2d 2D f

�V

nd

�

: (7.10)

In the initial assessment of the physics of a phenomenon it can be difficult to deter-mine which of the variables are truly independent and which are possible dependentones. It can be helpful then to visualise what happens in the process. In this case ofthe flow through a fan, for example, once the size of the system as measured by theduct 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 , butalso the pressure rise, �p. This also fixes the Fan thrust, T , and the torque, Q. The


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 designof this experiment; that is; P , �p, T , and Q.

If, however, the test is on a complete fan and duct system pumping to and fromthe atmospheric pressure, then the duct velocity is no longer an independent variablebut becomes yet another possible dependent one. This arises because running the fanat a speed, n, now fixes the value of the duct velocity, V , the value of �p across thewhole 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 ex-periment. First, there is a need to change only one of the four independent variableslisted 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 varyin the experiment. For this experiment this could be the velocity, V , as it could beconvenient to have a simple throttling device at the end of the duct. This could beeasier experimentally than varying the fan speed and certainly more convenient thanaltering the size of the system as measured by values of the diameter, d . So vary-ing V and reading values of P gives a range of values of both the non-dimensionalgroups 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 deter-mined 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 thecorresponding experimental ranges of the non-dimensional groups containing thevariable under consideration. For the example just given, when a fan diameter andthe 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 givesa considerable flexibility of application of the results.

On this last point care has to be taken over the physics of the flow. Experimen-tally, when V=nd D 0 then the power is found to be finite in value. If this equationis 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 interpretedas being for V D 0. For n very large the flow becomes a compressible one so thatEquation 7.1 is an inadequate statement. The same form of argument applies to theother end of the range shown in Figure 7.3.

7.8 Missing Variables 157


� d � V n �

1 L ML3

LT

1T

MLT

�

�

�

�

TL2

1

�V

�

�

�

nV

�

�

1L

�

�

1L

�

�

�

�

�Vd

�

�

�

ndV

�

�

� 1 � 1 �

7.7 Superfluous Variables

Two examples have been given for which dimensional analysis shows that a variabledoes not enter into the problem thus avoiding wasted experimentation.

The first example was given in Chapter 1 where in the case of linear elasticvibration the amplitude of the vibration is not relevant. The second example, thathas just been shown, is that for the expression for the efficiency of a fan where thefluid 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 ef-

ficiency 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 twovalues of the wind-tunnel speed and indicate a failure of Equation 7.3. The conclu-sion is that in this case of small scale flow the variable that has been excluded is theviscosity, �. This brings the density into the list of variables so that the dimensionof mass, introduced by the variable, �, can be dealt with by the pi-theorem as is nowshown in the tabulation below.

Thus the variables now show that:

� D f .d; �; V; n; �; / : (7.11)

Solution of the pi-theorem comes from Compact Solution 7.4.


So that:

� D f

��Vd

�;

nd

V

�

(7.12)

or:

� D f

�V

nd;

�nd 2

�

�

(7.13)

In Equation 7.12 the first independent group represents the Reynolds number asdoes the second non-dimensional group in Equation 7.13. Comparing these equa-tions with Equation 7.4. shows that now the Reynolds number appears as the ex-tra non-dimensional group. Also, because the extra variable, �, is introduced thenalso the density, � is added as an independent variable thus enabling the mass di-mension to be cancelled. The two curves of Figure 7.5 are now seen to representtwo values of the Reynolds number because Equation 7.12 represents a family ofcurves.

Figure 7.5 Efficiency plot of a small-scale airscrew: for Re D .�Vd/=�; lower curve, Re D1:3 � 105, upper curve, Re D 2:5 � 105

7.9 Influence of Variables 159

Figure 7.6 Illustration of theelectrical boundary layer

7.9 Influence of Variables

Another example comes from experiments to measure the electrical boundary layeror diffuse double layer in an electrolyte liquid. This is illustrated in Figure 7.6. Assketched, the positive potential on the electrode tends to attract the negative ions inthe electrolyte and repel the positive ones. There is then a conductivity effect tendingto 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 theion concentration and hence of the potential away from the electrode.

The potential, ', at a position, x, from one electrode, was taken as a function ofthe applied potential between the electrodes, '0, the electrical conductivity at zerocharge density, �0, the dielectric coefficient of the electrolyte, ", the time of appli-cation of the potential, t , the distance between the two electrodes, `, the coefficientsof diffusion of the positive and negative ions, DC and D�, the coefficient of dis-sociation of the ions, ˛, the coefficient of recombination, ˇ, and the product of thevalency with the Faraday number, zF , which latter is an alternative units-conversionfactor.

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 fluidmechanics of gases three regimes are set as the free-molecule flow, the slip regime and the contin-uum regime.



' '0 zF �0 " t ` DC

D�

x ˛ ˇ

ML2

AT3ML2

AT3ATn

A2T3

ML3A2T4

ML3T L L2

TL2

TL A

nAL3

n2

�

�

˛zF

ˇ

z2F 2

�

�

1T

L3

AT2

'

'0

�

�

�

�

�0'20 "'2

0 ˇ

z2F 2'0

� �

1 �

�

�

�

MLT3

MLT2

TLM

�

�

�

�

�

�

"�0

ˇ�0'0z2F 2

� � �

�

�

�

�

�

�

T L2

T2

�

�

�

�

�

�

"�0t

�

�

DC

t D�

t ˛tzF

ˇ�0'0t2

z2F 2

� � � �

� � � 1 � L2 L2 1 L2

�

�

�

�

�

�

�

�

�

�

DC

t

`2D

�

t

`2x`

ˇ�0'0t2

z2F 2`2

�

�

�

�

�

�

�

�

�

� 1 1 1 1


'

'0D f

�"

�0t;

DCt

`2;

D�t

`2;

˛t

zF;

x

`;

ˇ�0'0t2

z2F 2`2

�

: (7.15)

This can be rearranged as:

'

'0D f

��0t

";

DC"

�0`2;

DCD�

;x

`;

˛"

zF �0;� "

zF `

�2 ˇ'0

�0

�

: (7.16)

Figure 7.7 shows results of measurements of the distribution of potential in a liquidelectrolyte under an applied potential difference between two plane electrodes. Theresults show the distribution through the electrical boundary-layers adjoining eachelectrode surface [2, 3]. These layers are sometimes called diffuse double layersafter the Debye discrete double layers of charges, one in the electrode and one inthe electrolyte: this terminology seems a contradiction in terms. In a gas it is calleda plasma sheath. They are shown for two values of the electrode spacing, `, usingstaggered 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 ` � 10�2 m�'0=`2

� � 10�4 V m�2

48.5 2.95 5.5787.5 3.0 9.72

173 3.05 18.655 10.8 0.47

110 10.8 0.94215 10.8 1.84

For a test on a single liquid, the value of DC=D� is constant and the readings atinfinite time reduce Equation 7.16. to,

'

'0D f

�x

`;

DC"

�0`2;

˛"

zF �0;� "

zF `

�2 ˇ'0

�0

�

: (7.17)

The experiments covered the ranges of the parameters as now given in Table 7.1.


The upper set of results in Figure 7.7 shows those obtained in one liquid at oneelectrode spacing and three values of '0. Close agreement between these three setsin this non-dimensional plot is seen so that '='0 D f .x=`/. As Equation 7.17is applicable two conclusions can be drawn. First, ` being held closely fixed, thesecond 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 variationof '0 in the fourth group, then the results show that this group, and hence ˇ, isnumerically 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 theelectrode spacing. Now there is agreement between only two sets and so for thesetwo, 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 forthe origin of the lower set. There is now seen to be a slight difference due to thedifference in the value of `. Thus having excluded an influence of ˛ and ˇ, inspec-tion of Equation 7.17 indicates this to be an effect of the diffusion group. Thus witha ratio in `2 of 13 between the upper and lower sets this ratio of the group containingDC was found to change the sum of the non-dimensional potential drops across theelectrical 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 suggestsan 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 accordwith a prior dimensional analysis, then useful deductions can be made when exper-imental results are studied in the form of non-dimensional groups in conjunctionwith the physics of the phenomenon. This advantage is particularly of help when inthis example the phenomenon is a complex one not being amenable to a full classicalanalytical 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 non-dimensional group, it can happen that it is not possible to obtain a meaningful mea-sure 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 fluidpasses through such a porous medium having very small pores, then the viscousstresses are large so that the flow is a very slow one. This means that the Reynoldsnumbers of these flows are small so that, following the discussion of Chapter 1, thedensity can be excluded from the list of independent variables.

With the following notation of:

7.10 Measurement Limitation 163


�p d � w v

MLT2

L MLT

L LT

�p

�

�

�

1T �

�p

�v

�

�

�

�

1L �

�

�

d�p

�v

�

�

�

�

wd

�

�

1 � � 1 �

CH Filter coefficientd Pore cross-section dimension�H Hydraulic head across filter�p Pressure drop across filterv Mean inlet velocityw 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 flowthrough a porous filter


Thus:

d�p

�vD f

hw

d

i: (7.18)

For the simple arrangement of a filter as shown in Figure 7.8, and when d � w thenover most of the length, w, the average flow conditions will be uniform through thefilter. Then any distortions in the flow pattern from a uniform one at both inlet andoutlet will be of negligible influence. Thus the pressure through the filter will dropuniformly so that �p / w. Then Equation 7.18 reduces to:

d 2�p

�vwD constant : (7.19)

When the flow direction has a vertical component then there is an added gravi-tational force component along the flow so that �p is replaced by the change inhydraulic head given by:

�g�H D �p C �g�z ;

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

�vw� CH ; (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 con-tains a distribution of sizes of the gravel stones together with a distribution of theirarrangement 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 asfollows. 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 useof that equation for evaluating any filter of that chosen material and inlet velocityand 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 offilter material. However, as k has dimensions then these values are of limited usageas they refer only to the flow of water. This is despite the work of such as Yalin whoderived the relation of Equation 7.20 from dimensional analysis [4].

From Equation 7.20 and the definition of Darcy’s k,

CH D � g d 2

� k:

Insertion of values for water in this equation enables Darcy’s k to be converted tovalues of CH.

Equation 7.20 has a physical interpretation. For writing it as:

1

CH� �H

wD �v

�gd 2D �

�vd� v2

gdD Fr

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 ofa single blade angle then Equation 7.4 for the efficiency reduces to:

� D f

�V

nd

�

: (7.21)

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 , needsto be varied in experiment for a full determination of the influence of the other twoto be obtained. And again, adjustment of the velocity, V might much most readilybe 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 forthe correlation of the output data. To continue with this example of a fan flow, if theviscosity, � is added to the list of independent variables in Equation 7.1 then the



P � � V n d

ML2

T3ML3

MLT

LT

1T

L

P�

�

�

�

�

L5

T3�

�

L2

T

P�n3

�

�

�

�nVn

�

�

L5� L2 L �

P�n3d 5

�

�

�

�nd2Vnd

�

�

1 � 1 1 � �

Figure 7.9 Sketch of thevariation of a fan power co-efficient against the Reynoldsnumber

drive power is given by:

P D f .�; �; V; n; d/ : (7.22)

The pi-theorem then leads to Compact Solution 7.7.This gives:

P

�n3d 5D f

�V

nd;

�nd 2

�

�

: (7.23)

In the studies of a fan and an airscrew, the dependent group is known as the powercoefficient, the first independent group is called the advance ratio and the secondindependent group is the Reynolds number.

If an experiment is based upon Equation 7.23 then the desired results should beplotted as shown in Figure 7.9. This would involve the following difficulty. By hold-ing the value of n constant, varying the value of V and measuring the correspondingvalues 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 thevariation of a fan power co-efficient against the advanceratio

Figure 7.11 Illustration ofthe use of a transition rodto trip the laminar boundarylayer

other variables, n, d , � or �. Clearly, the second and the third of these three arethe most inconvenient so the experimenter is faced with providing a variable-speeddrive 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 themost convenient experimentally or an arbitrary one. This matter is now illustratedby three examples.

An experiment was set up to investigate the flow in the boundary layer at incom-pressible speeds over a flat plate. This is sketched in Figure 7.11. Interest was inthe transition from laminar, steady, flow in the boundary layer to a turbulent one attransition between the two regimes; this was influenced by the presence of a circularrod across the flow. Then with the notation of:

xT Distance to the transition start;xk Distance to the rod position;k Transition rod diameter;V Flow velocity;� Fluid density;� Fluid viscosity.



xT xk k V � �

L L L LT

ML3

MLT

�

�

�

�

TL2

�

�

�

�

�V

�

�

�

�

�

1L

�

�

xT�V

�

xk�V

�

k�V

�

�

�

�

�

�

�

1 1 1 � � �

there is:

xT D f .xk; k; V; �; �/ : (7.24)

The pi-theorem gives Compact Solution 7.8.This shows that:

xT�V

�D f

�xk�V

�;

k�V

�

�

: (7.25)

It might be considered, on the basis of the physics of this phenomenon, that the flowwould be better related to the local conditions at the transition rod by introducingthe variable, ı�

k which is the displacement thickness of the boundary layer at the rodposition but in the absence of the rod. Then this thickness is given by:

ı�k D f .xk; V; �; �/ : (7.26)

The solution becomes that of Compact Solution 7.9.This gives the result that:

ı�k �V

�D f

�xk�V

�

�

: (7.27)

Putting this equation into Equation 7.25 gives:

xT�V

�D f

�ı�

k �V

�;

k�V

�

�

: (7.28)



ı�

k xk V � �

L L LT

ML3

MLT

�

�

�

�

TL2

�

�

�

�

�V

�

�

�

�

�

1L

�

�

ı�

k �V

�

xk�V

�

�

�

�

�

�

�

1 1 � � �

This is rearranged to:

xT�V

�D f

�k

ı�k

;k�V

�

�

: (7.29)

Following the prior discussion, with two independent non-dimensional groups thereis the requirement to vary just two of the independent variables in an experiment.Correspondingly, one variable associated with the dependent group is thus mea-sured. In the present case of a total of six variables, three can then be held constantin the experiment.

Designing an experiment on the basis of Equation 7.29, it would be convenientto fix the values of �, � and k. and then to vary the values of V and ı�

k and tomeasure values of xT. Doing this would be expected to vary the values of the threenon-dimensional groups in Equation 7.29 but on doing so these results would plotas only a single curve rather than as a family of curves. The reason would be that thevalue of xk was being held a constant so that a non-dimensional group was beingheld constant; this would be the group of k/xk.

This is the validity criterion that has to be checked before the design of an experi-ment is finalised: it is that a non-dimensional group cannot be formed from amongstthose variables that are to be held a constant.

The second example comes from a study of the electro-hydrodynamic convectionof electric charge in a dielectric liquid which is set in motion by application of anelectric potential applied to an electrode bounding the fluid [5]. This is illustratedby a flow-visualisation picture in Figure 7.12.

With the following notation:



ic s � " � �0 '

A L ML3

A2T4

ML3MLT

A2T3

ML3ML2

AT3

i2c"

�

�

�0"

'2"

ML3

T4�

�

1T

MLT2

i2c

�"

�

�

�

�

�

�

'2"

�

L6

T4�

�

�

�

L2

TL4

T2

i2c "3

��40

�

�

�

�

�"

��0

�

�

'2"3

��20

L6� � L2

� L4

i2c "3

��40s6

�

�

�

�

�

�

�"

��0s2�

�

'2"3

��20s4

1 � � � 1 � 1

ic Convection current.s Electrode spacing.� Liquid density." Dielectric coefficient.� Liquid viscosity.�0 Conductivity at zero charge density.' Electrode potential.

this is expressed as:

is D f .s; �; "; �; �0; '/ : (7.30)

Analysis goes as in Compact Solution 7.10.This results in:

i2c "3

��40s6

D f

��"

��0s2;

'2"3

��20s

4

�

: (7.31)

Figure 7.12 Picture of theelectro-hydrodynamic flowpattern from a sharp point;point at the top of picture,'0 D 8 kV [5]


Table 7.2

� " � '

ML3

A2T4

ML3MLT

ML2

AT3

� '2"

�

�

MLT2

�

�

�

�

�

�

'2"

�

TL2

�

�

�

�

L2

T

�

�

�

�

�

�

'2"�

�2

� � � 1

Writing:

˘1 � i2c "3

��40s6

; ˘2 � �"

��0s2; ˘3 � '2"3

��20s4

then Equation 7.31 can be rewritten as:

˘1=21 D f

"1

˘2;

˘1=21

˘3

#

or,

ic"3=2

�1=2�20s3

D f

"��0s2

�";

ic�1=2s

"3=2'2

#

: (7.32)

Suppose that an experiment is designed on the basis of Equation 7.32. In that exper-iment, 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 de-sign. For a non-dimensional group can be formed from those variables that are heldconstant. This is seen by the following tabulation in Table 7.2. (It is to be noted thatthis tabulation is not here an application of the pi-theorem but merely a convenientmeans of forming a non-dimensional group.)

Thus the variables which were proposed to be held constant can be formed intothe non-dimensional group of:

'2"�

�2:


The proposed design of the experiment has been failed by this validity test. If theexperiment was designed on the basis of Equation 7.31 by holding constant the vari-ables �, 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 theformer 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 sur-

face. This has been studied in Chapter 6. The variables involved there were:

PQ=k0 Heat rate over thermal conductivity;gˇ Combine acceleration due to gravity with the volume

coefficient of gas expansion;Cv Coefficient of specific heat;k0=�0 Thermal conductivity over density;�0=�0 Viscosity over density;` Scale size;�T Temperature difference.

That discussion showed that after making several approximations to the full set ofgoverning equations the result can be expressed by:

PQk0`�T

D f

"gˇ�2

0C 2v `3�T

k20

;gˇ�2

0`2 PQk0�2

0

#

: (7.33)

This indicates that experimental values would be expected to plot as a family ofcurves. The corresponding two variables to be varied in experiment would mostconveniently be the size variable, ` and the temperature difference, �T . Then valuesof PQ, being the dependent variable, would be measured. An experiment could bedesigned by varying the size and the temperature difference so that the quantity,

Table 7.3

� s " �

ML3

L A2T4

ML3MLT

�

�

"� �

�

TL2

A2T3

L4�

�

�

�

"�4

�3�

�

� A2L2�

�

�

�

�

"�4

�3s2�

�

� � A2�


Table 7.4gˇ Cv k0

�0

�0�0

LT2�

L2

T2�L4

T3�L2

T

gˇ

Cv

�

�

k0�0Cv

1L

�

�

LT

�

�

k0�0Cv

�

�

� 1 �

`3�T is held constant thus holding the first independent non-dimensional groupin Equation 7.33 a constant and then enabling a curve between the first and thirdgroups 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 amongstthe 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 functionof the thermodynamic state of the gas. So an experiment would only be valid if atleast one of its thermodynamic properties is varied thus adding to the complexityof the design. The reason why only one curve would be obtained is seen also bynoting that by using the dependent group to eliminate the quantity PQ, from the

Figure 7.13 Sketch of theresults of an experiment onnatural heat convection


second independent group means that that group contains the quantity `3�T whichis 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 tab-ulated in terms of non-dimensional groups. The experimental range of validity ofdata is expressed so simply in terms of the corresponding limiting values of thenon-dimensional groups as distinct from the experimental ranges of individual vari-ables.


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 re-markable reduction of effort and a great benefit in the clarity of understanding ofthe output data.

Also there is a corresponding extreme benefit in the way that expressing this datain the form of non-dimensional groups leads to a great compactness of as well as anincreased clarity in the ordering of the experimental results.

Finally, the validity criterion has been introduced as a necessary check on thechoice of independent variables to be varied for valid experiment.

Exercises

7.1 In an experiment to evaluate Equations 7.12, or 7.13 show how it can be de-signed so that a family of curves is readily determined. The data is to be usedto determine a fan diameter when all the other variables are specified. Ar-range a plot of the experimental results so that the required diameter can bedetermined directly without any trial and error routine.

7.2 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 heightof ", the pressure gradient @p=@x is given by:

d

�V 2

@p

@xD f

��Vd

�;

�V"

�

�

:

A test is to be designed to determine the values of the pressure gradient byvarying the values of d and V . Check this experimental design.

7.14 Concluding Comments 175

Figure 7.14 Illustration ofa cooling fin heated at one end

7.3 Two parallel plates of area, a, and spaced a distance, z, apart, form an electri-cal capacitor. Putting the attractive force between these plates, fe, as a func-tion of the charge on each plate, , and the dielectric coefficient, ", of theintervening medium, show that,

"fe

2aD f

�z2

a

�

:

Ignoring edge effects so that fe / a, show that this force is independent ofthe value of z.

7.4 A hydraulic pump, when running at its maximum efficiency point is rotat-ing at 30 rps and has a flow rate of 0.063 m3 s�1, with a pressure rise of3:8 � 105 kg m�1 s�2 and requires a drive power of 32.5 kW. What would bethese values when running at the same efficiency and at 25 rps? (Exclude theinfluence of viscosity)

7.5 Large water turbines, especially of the Pelton-wheel type, are very efficientso that it is experimentally difficult to measure this efficiency accurately inthe usual way by measuring the output-shaft torque. An alternative way is tomeasure the change in water temperature between inlet and outlet. With thefollowing variables:

Cv Coefficient of specific heatPm Mass-flow rateW Power output�� Temperature change� Efficiency

set out a non-dimensional functional relation for the efficiency. What are theeffective dimensions in this solution?

7.6 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 fromthe fin to atmosphere of h and a thermal conductivity of the material of the fin


Figure 7.15 Sketch ofelectric-field surface-wetting

of k, show that:

PQ.Tf � T /`

D f

�h`

k;

t

`;

A

`2

�

:

What are the effective dimensions? For a fixed shape and steady running con-ditions, then:

PQ

.Tf � T /`D f

�h`

k

�

:

What variables would have to be changed in an experiment to determine thislatter function?

7.7 An electric field when applied to a liquid of very low conductivity can influ-ence the liquid-surface wetting. An experiment to study this is illustrated inFigure 7.15. With the following notation:

d Electrode spacing;g Gravity acceleration;y Liquid rise; Surface tension;" Permittivity;� Density difference, liquid/gas;' Potential difference.

show that:

y

dD f

��gd 2

;

'2"

d

�

:

What variables would be suitably changed in an experiment? How would theabove non-dimensional groups be re-arranged for the design of the experi-ment?

References 177

References

1. J C Gibbings. The dependence of conductivity of a weak electrolyte upon low solute concen-tration 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, MechanicalEngineering 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, liq-

uids, 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 8Analytical Results

I have sought the principles of the resistance of fluids as ifanalysis had not to enter therein, and only after having foundthese 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 beenused to obtain analytical results where formal analysis either is not available oris necessarily semi-empirical. Such results are derived from the information avail-able from the composition of non-dimensional groups. All derived non-dimensionalgroups give relations between the variables contained in those groups but sometimesit is possible to go further. Four examples illustrating these features are now given.

8.2 Example I: Flow Turbulence

Notation for Example I

a Coefficient; Equation 8.4b Jet width; coefficient Equation 8.4B Log-law constantF� � Œf .�/�1=2

K Circulation` Scale lengthPr Prandtl numberq Velocityr Radial ordinatet TimetB Burst timeu Time mean velocityU Reference velocityuc Centre-line velocityus Boundary velocity of viscous sub-layer


180 8 Analytical Results

u� Friction velocityu0 Streamwise turbulence velocityv0 Cross-stream turbulence velocityy Distance from wall

˛ Coefficient; Equation 8.6, 8.12ˇ Integration constant; Equation 8.13ˇ0 Units-conversion factor; angleıs Sub-layer thicknessıC

s Thickness Reynolds number� Phase angle between turbulence components� von Karman constant� Vortex spacing� Viscosityv Kinematic viscosity� Air density� Shear stress�w Wall shear stress

8.3 The Complexity of Flow Turbulence

An outstanding example of an output from dimensional analysis has been in the ap-plication to turbulent flows. This is a major physical phenomenon, which is of con-siderable concern particularly to engineers and meteorologists. Yet despite its verygreat practical importance, this flow characteristic is so complex and detailed in itsnature that practical solutions almost invariably involve a combination of dimen-sional analysis with one, or often more, empirical coefficients. A formal solution ofthe governing equations without such coefficients is rare [1].

Because of its importance, a dimensional analysis for turbulent flows is givennow 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 ofthe 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 6in the discussion of convective heat transfer. So the variables are known. For incom-pressible 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 typicalmean velocities, being three-dimensional in form, of random in occurrence at anyposition and of high frequency. The structure is an intensely packed one. This ex-plains why formal solution of the governing equations presents such a formidablechallenge. But there are some characteristics of the physics which can be deducedfrom experiment on different occurrences of turbulence. Several of these flows arenow listed.

a) Figure 8.1 illustrates the flow in the turbulent shear layer between two uniformstreams of differing velocity. The velocity distribution plotted is the time meanvalue so that the turbulent fluctuations are not shown.In this flow the width of the layer increases with this distance and so theReynolds number of the turbulent portion of the flow, based upon this widthand the constant mean velocity at the mid-point of the shear layer, increasesalong the flow. Yet the normalised velocity distribution is constant along theflow and so there is no Reynolds number effect and so no influence of the nu-merical value of the viscosity upon this profile.

b) Figure 8.2 sketches the flow past a flat plate normal to the oncoming uniformstream.

Figure 8.1 Velocity distri-bution across a shear layerflow: velocity, u, versus thedistance, z

Figure 8.2 Flow past a flatplate set normal to the oncom-ing stream


At a sufficiently high Reynolds number, the turbulent separation is fixed at theedges of the plate and the drag coefficient, as defined in Chapter 1, is a constantindependent of the numerical value of the Reynolds number. So again this flowis independent of the numerical value of the viscosity.

c) Again for the flow of Figure 8.2, there can be a level of turbulence in the on-coming free-stream. Then the measured values of the fluctuating componentalong the stagnation streamline up to the front stagnation point were found toretain a constant value while the mean flow velocity dropped from the free-stream value down to zero [2]. This meant that by Kelvin’s theorem [3, 4], theoverall flow pattern was an irrotational one, the vorticity being conserved.

d) The diagram of Figure 8.3 is of the two-dimensional developed turbulent jetflow.As for case (a), the normalised velocity distribution in the turbulent region ofa developed jet flow, that is downstream of section ’A � A’ as illustrated, isindependent of the value of the increasing Reynolds number along the flowand so of the numerical value of the viscosity.

e) The developed turbulent wake flow downstream of a two-dimensional solidbody is sketched in Figure 8.4.Again as for case (a), from a short distance downstream of the trailing edge ofthe solid body, the varying value of the Reynolds number along the developedwake flow has no effect upon the normalised velocity distribution and so againthe 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 183

f) The temperature recovery factor.The temperature recovery factor is the ratio of the temperature change acrossa boundary layer to the corresponding isentropic change. For a laminar, andhence fully viscous, boundary layer it has the value of P

1=2r which for air is

0.85 whereas for a turbulent layer experiment gives Pr � 1 [5]. This indicatesthat the turbulent boundary layer is largely independent of the value of theviscosity being then an irrotational flow except for the very thin viscous sub-layer that adjoins the surface.

g) The boundary layer in a zero streamwise pressure gradient.The normalised velocity profile of a turbulent boundary layer, outside of theviscous sub-layer, and in a zero streamwise pressure gradient, is universal inshape and so is independent of the value of the Reynolds number. The furtherinfluence of a rough surface upon this profile is merely to shift away from thewall that portion of the profile that is outside the viscous sub-layer. Thus thisprofile, if normalised to the velocity at the edge of the viscous sub-layer, is alsoa universal one again independent of the value of the Reynolds number [6].

h) The turbulent flow in a pipe.The fully developed, normalised velocity distribution across a turbulent flow ina straight pipe of constant diameter and outside of the viscous sub-layer also isindependent of the Reynolds number of this flow.

i) The charging of particles in a turbulent flow.The electrostatic charge induced upon a particle in a turbulent flow has a valuethat is significantly less than the Pauthenier limit. This has been explained asbeing 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 flowthrough an electrostatic precipitator has assumed the absence of rotation of theparticles with a successful result ( [8], p. 133). This again implies an irrotationalflow and hence one independent of the numerical value of the viscosity.

All these examples indicate that turbulence is composed of small vortices containedwithin a flow which in the main is irrotational with viscous effects limited to thesmall 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 ofturbulence 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 thatthe vortices are generated by viscous stresses at the wall and then convect acrossthe flow [1]. Further there is evidence that initially all the vortices are of the samestrength [11]. Finally to complete the physics description, Kelvin’s theorem wouldstate that each vortex is convected without change in strength in the potential flowvelocity 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 beingembedded within a largely irrotational flow.

This detailed consideration of the physics of turbulent flow now enables progressto be made using dimensional analysis. The first step follows from the forego-


ing flow examples which enables the viscosity to be excluded from Equations 8.1and 8.2.

8.5 The Turbulent-Power Law

The nature of turbulence is illustrated in Figure 8.5 showing a typical plot of velocityat 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 tobe closely constant, the mean velocity remaining so at this position. Also the heightof the fluctuation in velocity is also closely constant. The circulation, K, whichmeasures the angular velocity of a vortex and hence the velocity fluctuation, is thusconstant. The intermittency, I , which is the ratio of that part of the signal occupiedby 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 sketchedin 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 areshown for two values of the intermittency, �

Figure 8.6 Variation of inter-mittency of turbulence acrossa boundary-layer flow

8.5 The Turbulent-Power Law 185

Figure 8.7 Variation of vor-tex spacing across a fullyturbulent boundary-layer flow

and,

as y ! 0; I ! 1; u ! 0; � ! 0

A simple representation of this variation of the vortex spacing satisfying theseboundary conditions is given by:

� D ayb (8.4)

with b < 1. This is sketched in Figure 8.7. The validity of this assumption can befurther justified later.

The convection velocity field is formed from the sum of the convection velocitiesof the vortices. The velocity around a vortex falls as the reciprocal of the distancefrom its centre [10]. Thus locally within a flow the velocity is mostly induced fromthose vortices which are fairly adjacent together with their own velocities [3]. Itfollows that in considering the velocity distribution locally away from a surface,the distance from the surface does not influence the local velocity distribution. Thesmall 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; �/ (8.5)

The pi-theorem solution is in Compact Solution 8.1.This gives:

�ıu

KD f

�ıy

�

�

which rearranges to:

ıu�2

ıyKD f

�ıy

�

�



ıu ıy K �

LT

L L2

TL

ıuK

�

�

1L

�

�

�ıuK

ıy

�

�

�

�

�

1 1 � �

As ıy ! 0 then:

�2

K

@u

@yD constant � ˛ (8.6)

Substitution from Equation 8.4, and noting that K is a constant, and integratinggives:

u D ˛K

a2.1 � 2b/:y.1�2b/ (8.7)

This is the power law which is quite a good approximation over all the bound-ary layer except for within the viscous sub-layer. This applicability has been foundalso for the Prandtl solution giving this power law which was based upon the wallshear 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 closelythroughout the greater part–’ [3]. The demonstration here gives validity of the ap-plication across this greater part of the flow.

With the present demonstration of Equation 8.7 it is now possible to reversePrandtl’s analysis so as to derive the corresponding power law for the surface frictionwhich Prandtl had to adopt from experiment.

8.6 Prandtl’s Mixing Length

Reynolds set out time-mean values of the governing equations for turbulence andshowed that a predominant effect of the turbulence in the momentum equations wasfrom terms such as the mean of the cross-products u0v0. Despite this term being onein momentum flux it has come to be referred to as a Reynolds stress. Reynolds’expression for this shear stress is [4]:

� D ��u0v0 (8.8)

8.6 Prandtl’s Mixing Length 187


��

� K � ˇ0

L2

T2L L2

T˛ 1

˛

��K2

�

�

1L2

�

�

��2

�K2�

�

�

�

1 � �

� � ˇ0� �

� � 1 �

In the boundary layer it comes from the component of the vorticity vector that isperpendicular to both the y and the flow directions. Its value depends upon boththat of u0 and of v0 and also the phase angle, � , between these two components. Sofollowing 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:

u0v0�2

K2D f .ˇ0�/ :

Substituting from Equation 8.6 gives:

u0v0�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 assumeda 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 fromexperimental results ( [4], Art. 160). These can now be identified with the assumedvariation for the vortex spacing proposed in Equation 8.4. Both show closely sim-


ilar distributions over most of the outer region of the flow thus giving the furtherjustification for the latter equation.

Goldstein pointed out that using Prandtl’s relation gives acceptable results for thevelocity profile [4]. This arises from differing values of the mixing length leading tovalues of the velocity distribution that are very nearly the same. It is also influencedby 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 mainlyisentropic and so the viscosity is excluded as a variable. The vortex spacing couldbe influenced by the presence of the wall and so by the distance y. Should therenow be momentum effects in the flow then the density becomes a variable but notthe velocity u as it is an alternative dependent variable. As the mean rate of shear ishigh here then the wall value might be significant. Thus for the vortex spacing herewe 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 propor-

tional to the distance from the wall with a universal value of ˛ [4]. Identifying thiswith the vortex spacing confirms Equation 8.12.


� �w � y

ML3

MLT2

L L

�

�

�w�

�

�

L2

T2

�

�

�w�y2

�y

�

�

�

�

1T2

1 �

�

8.7 The Log-law 189

In this region close to the wall the shear stress is a constant. Substituting this andEquation 8.12 into Equation 8.11 gives:

@u

@yD�

�

�

�1=2 1

˛y

1

Œf .ˇ0�/�1=2

or, writing:

F� � Œf .ˇ0�/�1=2 :

Then:

F�u

.�=�/1=2D 1

˛ln y C ˇ (8.13)

which is the log-law for the velocity u, in the near-wall region, with ˛F� now iden-tified with �, the von Karman constant [6]. Further, identifying Prandtl’s mixinglength with � in Equation 8.12 gives a relation that is satisfied by experiment in thenear-wall region for which Equation 8.13 is valid [4].

Detailed experiment shows that � is a constant that is independent of the Reynoldsnumber [6, 12]. Then F� / 1=˛. As experiment related to Equation 8.12 confirmsthat ˛ is not Reynolds number dependent, then neither would be F� .1

Near the wall where the log-law is valid, � ¤ f .y/. Thus:

�

�D �w

�� u2

� : (8.14)

For a viscous sub-layer of thickness, ıs, the boundary velocity, us is given by:

us D ıs

�@u

@y

�

yD0:

As,

�w D �

�@u

@y

�

yD0

then,

ıs D us

u2�

1 The results in [4] that show a Reynolds number dependency are for low values of the Reynoldsnumber which accord with the flow at the end of transition where the turbulence has not fullydeveloped.


giving:

ıCs � ısu�

D us

u�

: (8.15)

In Equation 8.13, putting u D us at y D ıs determines the value of ˇ. Then thisequation with Equation 8.15 becomes:

u

u�

D 1

�ln�yu�

�� 1

�ln ıC

s C ıCs : (8.16)

The constant in this equation is written as:

B � � 1

�ln ıC

s C ıCs : (8.17)

An accepted value of the thickness of the viscous sub layer is ıCs D 11. With � D

0:405 [6, 12], this gives B D 5:08 a value that agrees well with the experimentalvalue that Coles derived after extensive review of the existing data [13].

Equation 8.17 also explains the effect of surface roughness because that increasesthe value of ıs and hence the value of B [12].

Comparison of Equations 8.13 and 8.16 shows that the viscosity enters into thelog-law only from the imposition of the inner boundary conditions for it is notpresent in the former equation. This is consistent with the physics of the flow aspresented here which is one of an isentropic one. It would have been quite inconsis-tent to have entered earlier the viscosity as a variable governing this almost entirely-isentropic turbulent flow.

8.8 Jet Flow

A jet flow is illustrated in Figure 8.3. Beyond the section A-A the flow is said tobe fully developed and from then on all the profiles of u and of u0 are of the sameshape; that is all profiles are identical when non-dimensionalised in term of the jetwidth, b, and of the centre-line values of u and of u0. This is expressed by:

u

ucD f

�y

b

�(8.18)

and by:

u0

uc0

D f�y

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 theoutward convection, there would be:

� / b : (8.20)

Identifying � with Prandtl’s mixing length as before, this is the basic assumptionmade by Tollmien ( [4], Art. 255) which gives good agreement with the experimentalvelocity profiles across the jet. The measured velocity towards the outer edge ofthe jet is lower than values calculated because diffusion of the vorticity becomessignificant [1].

The same assumption of Equation 8.20 was made by Prandtl for the turbulentwake 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 phe-nomenon by developing argument for the largely isentropic nature of turbulenceonce it has been created at surfaces by viscous stresses. Consequently, it is possibleto exclude initially the viscosity as a variable. By postulating that bursts of turbu-lence are in the form of vortices, it is possible to invoke Kelvin’s theorem for theirmovement with the flow and to make use of the expression for the velocity distribu-tion about a vortex. Secondly, it illustrates a case where formal analysis alone with-out the introduction of empirical coefficients, has barely started so that dimensionalanalysis has proved of great value. Thirdly, this example shows how dimensionalanalysis can intertwine with standard analysis making the whole possible.

8.10 Example II: Particle Abrasion in Flows

Notation for Example II

A Surface areaC Number concentration of particlesD Drag forceFe Electrostatic forceFp Pressure forceg Gravitational accelerationL Length size of the flow` Length size of particlesm Mass of particlePm Wear rate


p Pressureq Flow velocityqp Particle velocitys StressU Reference flow velocity

W Weight force" Dielectric coefficient� Fluid viscosity� Fluid density�p Particle density Charge density

Another phenomenon that is of considerable concern to engineers is that of thewear by dust impact within fluid machines [14, 15]. Because of its association withcomplex fluid flows, this problem is amenable to solution by dimensional analy-sis [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 Reynoldsnumber 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 pi-theorem that:

D

��qp � q

�`

D constant : (8.22)

Secondly, the weight force is given by:

W / g�p`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


�p � q � L

ML2T2

ML3

LT

MLT

L

�p

�

�

�

�

�

LT2

�

�

L2

T

�p

�q2�

�

�

�

�

�q

1L

�

�

�

�

L

L�p

�q2�

�

�

�

�

�qL

�

�

1 � � 1 �

Solution of the pi-theorem takes the form of Compact Solution 8.4.This gives that:

L�p

�q2D f

��qL

�

�

: (8.26)

The flow through a fluid machine will usually be highly turbulent so that from theprevious discussion in this chapter viscous effects are not significant so that Equa-tion 8.26 reduces to:

L�p

�q2D constant : (8.27)

Substitution from Equation 8.24 gives that:

FpL

�q2`3D constant : (8.28)

Fourthly, the electrostatic force, Fe, is given by:

Fe D Q2CL

": (8.29)

A comparison of possible numerical values of D, W and Fe each with Fp showsthat for an air flow both D and Fe are small and W is comparable. However, theelectrostatic force would act across the flow and so could be significant, a pointreturned to later. In the case of the weight force, experiment does not distinguishbetween the leading edge deposit of particles on the upper and the lower surfaces atthe leading edge of an aerofoil [18]. This suggests that it can be excluded from thediscussion.



qp � q q �p � ` L �LT

LT

ML3

ML3

L L MLT

�p

�

�

�

�

�

1 �

�

L2

T

qp�q

q

�

�

�

�

�

�q

1 � � L

�

�

�

�

`L

�

�

�

�qL

� � 1 � 1

Then the variables are:

�qp � q

� D f�q; �p; �; `; L; �

�: (8.30)

Application of the pi-theorem gives Compact Solution 8.5.This shows that:

�qp � q

�

qD f

��p

�;

`

L;

�

�qL

�

: (8.31)

This can be rearranged as:�qp � q

�

qD f

��p

�;

�pq`

�;

�qL

�

�

: (8.32)

With the viscous force on a particle being neglected then the second independentgroup, which is the associated Reynolds number, can be excluded. Also, with littleinfluence of the machine Reynolds number upon the overall flow pattern then thethird independent group can also be omitted.

Thus Equation 8.32 reduces to,�qp � q

�

qD f

��p

�

�

: (8.33)

For a flow pattern under these conditions the local velocity, q is given by:

q

UD constant (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, toa good approximation,

qp / U : (8.35)

8.12 The Wear Rate

It follows that for the rate of wear, Pm, there is:

Pm D f�qp; m; `; C; A; s

�

where the stress s, is associated with the material under impact. At the moment it isassumed that the material of the particles is so much harder and stronger than that ofthe surface being abraded so that the numerical value of its material characteristicswill not affect Pm.

The pi-theorem solution is in Compact Solution 8.6.This gives the result that:

Pm`

mqpD f

"s`3

mq2p

; `3C;A

`2

#

: (8.36)

If conditions are supposed uniform across A, then Pm will be proportional to A sothat Equation 8.36 reduces to:

Pm`3

mqpAD f

"s`3

mq2p

; `3C

#

: (8.37)


Pm qp m ` C A s

MT

LT

M L 1L3

L2 MLT2

Pmm

�

�

sm

1T

�

�

1LT2

Pmmqp

�

�

�

�

smq2

p

1L

�

�

�

�

1L3

Pm`mqp

�

�

�

�

�

�

`3C A`2

s`3

mq2p

1 � � � 1 1 1


If each impact is an isolated occurrence and the contribution to Pm is the same foreach impact, then Pm will be proportional to the number of impacts which is in turnproportional to C . Thus Equation 8.37 reduces further to:

PmmqpAC

D f

"s`3

mq2p

#

: (8.38)

Suppose that the particles cause damage by knocking off protrusions upon impactingthe surface. If the relevant breaking stress is halved and the particle exerts the sameforce, then the area of fracture doubles. This implies a doubling of the number ofprotrusions and so doubling Pm. Thus Pm / 1=s and so Equation 8.38 reduces to

Pms`3

m2q3pA C

D constant : (8.39)

Combining Equation 8.35 with Equation 8.39 gives

Pms`3

m2U 3ACD constant : (8.40)

This can be rewritten as

Pms

�2p`3U 3AC

D constant : (8.41)

If Equation 8.33 is taken into account then,

Pms

�2p`3U 3A C

D f

�p

�

�

: (8.42)

It is common practice to define a non-dimensional mass rate by the relation [15],

e � Pm�pUCA`3

: (8.43)

Inserting this equation into Equation 8.42 gives,

es

�pU 2D f

�p

�

�

: (8.44)

Equation 8.42 shows that the wear rate is proportional to the cube of the velocitya result given by Truscott as being the most common value quoted by various authorsfrom the results of experiment and of analysis [19]. Another result from the samesource that the wear is proportional to the particle volume is also shown by thisequation. Further Truscott also quotes the finding by several authors that the wearrate is proportional to the particle concentration for low concentrations and this alsois shown by Equation 8.42.

8.13 Classes of Impact 197

Truscott’s review showed that several sources indicate that,

Pm / ��p � �

�:

Accepting this result then Equation 8.42 becomes,

Pms

��p � �

�`3U 3AC

D constant : (8.45)

8.13 Classes of Impact

It was found by Goodwin et al. [14] that impact occurred in one of two regimes. Inone, for a smaller size of the particles, the erosion coefficient, e, was proportional tothe square of qp. In the other, for the larger particles, e was independent of particlesize, `. 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 offragmentation of the particles whilst Class II accorded with a high amount. A crite-rion for the upper limit to Class I was found to be ` D constant. At the lower limitof Class II qp is proportional to `. The impulse imparted by a particle per unit frontalarea is proportional to

�pqp`3=`2 D �pqp` :

The kinetic energy of the particle per unit volume is 12 �pq2

p . The ratio of thesetwo quantities is then proportional to `=qp. A Class II impact thus corresponds toa higher impulse in comparison with the kinetic energy.

A Class I impact corresponds to Equation 8.44 with Equation 8.35 giving

es

�pq2p

D constant : (8.46)

Whilst this equation gives the above mentioned experimental result that e / q2p [14],

those experiments showed that e / `2. This indicates that the parameter represent-ing the material property of a surface is better chosen as a force, F rather than thestress, s. Thus Equation 8.46 becomes

eF

�pq2p`2

D constant : (8.47)

This gives the representation, shown by experiment, of both the velocity and the sizeeffects for Class I impacts.


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:

t D f

"sp

�pq2p

#

:

Thus introducing this extra parameter, t , Equation 8.46 becomes,

es

�pq2p

D f

"sp

�pq2p

#

: (8.48)

The experimental results of Goodwin et al. show that for Class II impacts e / q2:3p .

This is satisfied by Equation 8.48 if,

es

sp

"sp

�pq2p

#1:15

D constant (8.49)

in which the independence of e with ` is consistent with experiment [14]. BothEquations 8.47 and 8.49 satisfy the experimental result that the erosion rate is littleeffected by concentration [14].

However, the previously mentioned electrostatic force can influence the paths ofthe particles [20] and it has been suggested that this might have caused the 7 % scat-ter 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


t �p ` qp sp

1 ML3

L LT

MLT2

�

�

sp

�p

�

�

L2

T2

�

�

�

�

sp

�pq2p

� � 1

8.17 Example III: Electrostatic Fluid Charging 199

from,

Fe

Fp/ Q2CL

"�q2`3:

This introduces the concentration. Expansion of a space charge cloud [21] wouldlower the local value of C and this would be consistent with an observed slight dropof " 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,

es

sp

"sp

�pq2p.1 � �/2

#1:15

D constant : (8.50)

This then reproduces the experimental determination of the variation of e with qp, `

and � and the independence from C .


Again, dimensional analysis has added to the understanding of a complicated phe-nomenon and enabled the derivation of useful relationships between variables whichhad previously been obtained by several experimenters. It also has given insight intothe nature of the phenomenon and hence of the relevant independent variables.

8.17 Example III: Electrostatic Fluid Charging

Notation for Example III

DC, D� Diffusion coefficients of ionsE Electric field strengthF Faraday constantis Streaming currentk Ion mobility` Reference lengtht Timeu Velocity


U Reference velocityz 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 sur-face then an electrostatic charge exchange can occur at the surface so that a netcharge is left in the flowing fluid [22]. This has led to hazardous incidents goingback 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 thefluid motion are uncoupled from those governing the electrostatics. Thus the formercan first be solved enabling that solution to be introduced into the latter.

Because of the complexity of fluid flows as already discussed, dimensional anal-ysis 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 andearthed reservoir. This liquid contains free charges equally positive and negative sothat the net overall charge, , is zero. The evidence from industrial practice is thatthese ‘ions’ are very small and charged impurity particles [22]. On flowing into thepipe, 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 lattercharge can then form an industrial electrostatic hazard.

Figure 8.8 Diagram of theflow system for the generationof electrostatic streamingcurrent

8.20 The Variables and Groups 201

8.19 The Governing Equations

The governing equations have been set out [20]. They are:

d˙dt

D D˙r2˙ � r � .�˙E/ ;

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 theflow pattern which are,

�; U; ` and � :

Further detailed study has been made of the interaction with the turbulence in theflow [24].

This phenomenon is different to the previous two described in this chapter be-cause now a careful consideration has to be made of the boundary conditions. Sev-eral 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 thephysics and chemistry involved it can be concluded that the variables can be reducedto [20, 23],

is; �; U; `; �; DC; D�; "; � :

Application of the pi-theorem is in Compact Solution 8.8.Thus,

i2s

��U 3`3D f

�"U

�`;

�

�U `;

DCU `

;D�U `

�

:

This can be reformed as,

i2s

��U 3`3D f

�"�

��`2;

�U `

�;

DCU `

;DCD�

�

: (8.51)



is � U " ` � � DC

D�

A ML3

LT

A2T4

ML3L A2T3

ML3MLT

L2

TL2

T

i2s

�"�

�

ML3

T3T �

i2s

�U 3 �

"U�

�

�

U

DC

U

D�

U

M �

�

L �

�

ML2

L L

i2s

��U 3 � � �

�

�U

L3� � � L

i2s

��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 usinghydrocarbon liquids [26, 27]. The tests were performed for the turbulent flow ina tube. For that programme it was intended to study phenomena that might accordwith industrial practice. So the evidence was that the ‘ions’ were charged impurityparticles. It was inferred that then the fourth independent group in Equation 8.51would be a constant. Other calculations suggested that the influence of DC could beexcluded [22]. So these experiments were based on the reduction of Equation 8.51to,

i2s

��U 3`3D f

�"�

��`2;

�U `

�

�

: (8.52)

Writing,

I 01 � i2s

��U 3`3

and with,

Re � �U `

�:

Then a set of these experimental results is illustrated in Figure 8.9. This showsa family of curves giving the relation between the three non-dimensional groupsof Equation 8.52. The value of the streaming current was corrected for an entry

8.21 Experimental Verification 203

Figure 8.9 Experimentalvalues of the electrostaticstreaming current: curves forvarious 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 seenthat the dependent group for the current covers a wide range of nearly three ordersof magnitude. The abscissa is that of the second independent group which is theReynolds number of the flow.

A similar plot of these results is given in Figure 8.10 where now the abscissa isthe first of the independent non-dimensional groups. In both of these figures thereis evidence of linearity on the logarithmic scales and so of the existence of simplepower relations. This enables the final correlation to be shown on the plot of Fig-ure 8.11. There is seen to be a reasonable correlation of the results except for oneset: this is discussed further in the original paper [26]. However, the range of values

Figure 8.10 Non-dimensional plot of Figure 8.9


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 meanvalue of about ˙10 %.


This is an example of a very complex phenomenon for which dimensional analysishas 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 beotherwise unknown.

8.23 Example IV: Kinetic Theory of Gases

Notation for Example IV

a Molecular dimensionai Ion dimensionAm Unit atomic massCˇ Force coefficientc Mean velocityD Diffusion coefficientE Electric fieldFˇ Inter-molecular forcek Radius of gyrationki Ion mobilitykB Boltzmann constant

8.25 Mean-free Path Length in Gases 205

`m Mean-free path lengthm Molecular massM0 Molecular mass factorN Number densityp Pressureqi Ion charger Reaction radiusR Universal gas constantTv Characteristic gas temperatureT Temperatureu Internal energy per mass-unitU Internal energy densityz 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 theprevious discussion on the need to determine the units-conversion factors for bothtemperature and quantity. Further it will show how dimensional analysis can cor-rect an invalid assumption in formal analysis. The discussion is principally for thecharacteristics of ideal gases [28].

8.25 Mean-free Path Length in Gases

The mean-free path length is expected to depend upon the interception characteris-tics and so would then be a function of the mean speed, c, the mean spacing of themolecules measured as the number per volume unit, N , and a representative size ofthe molecule, a. Also it is expected that the application of Newton’s laws requirethe inclusion of the variables of the molecular mass, m and a force, Fˇ which isa measure of the inter-molecular force.

Accepting that the force, Fˇ follows an inverse square law, we define a coeffi-cient, Cˇ by,

Fˇ � Cˇ

r2(8.53)

so that Cˇ D constant for any particular gas.



`m c Na2 m Cˇ

L LT

1L

M ML3

T2

�

�

Cˇ

m

�

�

L3

T2

�

�

�

�

Cˇ

mc2

� � L

`mNa2�

�

�

�

�

�

CˇNa2

mc2

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 wayby either doubling the number of particles by pairing each one or by doubling thearea of each particle. In other words, the effect of doubling N so that N2 D 2N1 canbe cancelled by making a2

2 D 12 a2

1; 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:

`mNa2 D f

�Cˇ Na2

mc2

�

: (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:

`mNa2 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 sameargument 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 rotationalmotions of the molecules. This is represented by the parameter k! where k is a rep-resentative radius of gyration and ! is a representative angular velocity.

Then the pi-theorem gives Compact Solution 8.10.This gives:

Ua3

mc2D f

�

Na3;k!

c

�

: (8.57)

On a macroscopic scale, U / N so that,

U

Nmc2D f

�k!

c

�

:

The internal energy per unit mass, u, is given by, U D uAmM0N so that Equa-tion 8.57 becomes,

uM 0AmN

Nmc2 D f

�k!

c

�

:

Noting that,

m D AmM0


K U c N m a k!

MLT2

LT

1L3

M L LT

1 Um

�

�

1LT2

�

�

2 U

mc2

�

�

�

�

k!c

1L3

�

�

�

�

1

3 Ua3

mc2�

�

Na3

1�

�

�

�

1 � � �

k 3 1 1


then,

u

c2D f

�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

kBTD CV m

kBD constant :

Then also from Equation 8.58 for this ideal gas approximation,

k!

cD constant : (8.60)

From macroscopic thermodynamics and the more general case [28],

uM0

RD f .T / :

Therefore in Equation 8.58,

k!

cD f

�R

M0c2f .T /

�

: (8.61)

Using the later Equation 8.71, then,

k!

cD f

�1

Tf .T /

�

:

The right-hand side of this equation cannot satisfy the requirement of equality ofdimensions except for the case just considered of f .T / D T . A further variablehas 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!

cD f

�T

Tv

�

: (8.62)

8.27 The Pressure and Temperature 209


k! c T Tv

LT

LT

� �

k!c

�

�

1 �

�

�

TTv

�

�

� 1 �

Substituting this into Equation 8.58 gives,

u

c2D f

�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 surfacedrawn in the gas is from the equality of the interchange of molecules across thissurface to the incident and reflected motions at the solid surface: this implies thatthis equality means that the solid surface collisions are perfectly elastic. This is animportant 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:

p

mc2ND f

"

aN 1=3;k!

c;

Cˇ N 1=3

mc2

#

: (8.65)

As mN is the density, �, then,

p

�c2 D f

"

aN 1=3;k!

c;

Cˇ N 1=3

mc2

#

: (8.66)



K p c N m a k! Cˇ

MLT2

LT

1L3

M L LT

ML3

T2

1 p

m

�

�

Cˇ

m

1LT2

�

�

L3

T2

2 p

mc2�

�

�

�

k!c

Cˇ

mc2

1L3

�

�

�

�

1 L

3 p

mc2N

�

�

�

�

�

�

aN 1=3

1Cˇ

mc2a

1 � � � 1

k 3 1 1 3

Then as N ! 0,

p

�c2 D f

�k!

c

�

: (8.67)

The macroscopic equation of state for an ideal gas defining the temperature T is,

p

�TD R

M0(8.68)

so that comparison of Equations 8.67 and 8.68 gives,

T D M0c2

Rf

�k!

C

�

: (8.69)

From Equation 8.62,

T D M0c2

Rf

�T

Tv

�

: (8.70)

To retain the definition of temperature for a perfect gas by Equation 8.68 the lastterm of Equation 8.70 cannot be effective and so,

M0c2

RTD constant (8.71)

or, Equation 8.68 becomes,

p

� c2 D constant (8.72)

8.28 The Viscosity 211

which again is a result of kinetic theory.Also noting that,

m D M0Am with kB D RAm :

where the dimensional constant RAm is written as kB, the Boltzmann constant. Thenfrom Equation 8.71,

kBT / mc2

which is now a deduction of the form of Equation 2.16 and gives another result ofkinetic 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 uponthe values of m, c and N . Also, the motion concerned is related to that along themean free path, `m, which has a component perpendicular to the continuum shearso 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

mcD f

�N `3

m

: (8.73)


K � c N m `m

MLT

LT

1L3

M L

1 �

m

�

�

1LT

�

�

2 �

mc

�

�

�

�

1L2

�

�

�

�

3 �`2m

mc

�

�

N`3m �

�

�

�

1 � 1 � �

k 3 1


The viscosity is a measure of the momentum flux per unit area and so is proportionalto N . Thus Equation 8.73 reduces to:

�

Nmc`mD constant

or,

�

�c`mD constant (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 proportionalto the distance travelled by a molecule in a direction perpendicular to the shear andbetween 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 dimen-sions include one in temperature, so the dimensional constant, kB, must be intro-duced as a units conversion factor. The argument for preferring the variable, `m, onphysical 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 dimen-

sional analysis.Thus:

�`2m

kBcD f

�N `3

m

�: (8.75)

The thermal conductivity, �, is a measure of an energy interchange on a continuumscale along a temperature gradient. It is a measure of the transfer of internal energyand so of molecular kinetic energy through collisions. Thus on a macroscopic scaleit would be proportional to the total interchange and so to the value of N . ThusEquation 8.75 becomes:

�

N kBc`m :D constant

which is the result given by kinetic theory.

8.30 Diffusion 213


K � kB N `m c m

MLT3�

ML2

T2�1

L3L L

TM

1 �kB

�

�

mkB

1LT

�

�

T2�L2

2 �kBc

�

�

�

�

mc2

kB

1L2

�

�

�

�

�

3 �`2m

kBc� N `3

m � �

1 � 1 � �

k 3 1

8.30 Diffusion

Using the previous consideration of the physics, the diffusion coefficient, D, is writ-ten as,

D D f .m; c; N; `m/ :

Solution takes the form of Compact Solution 8.15 which gives,

D

c`mD f

�N `3

m

�: (8.76)

The diffusion coefficient is defined as a measure of the ratio of the rate of transportof N particles to the value of N . From this thermodynamic definition, D ¤ f .N /.


D m c N `m

L2

TM L

T1

L3L

Dc

�

�

L �

Dc`m

�

�

N`3m �

�

1 � 1 �


Thus Equation 8.76 reduces to,

D

c`mD constant : (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 balancebetween the electrostatic force on a particle of charge, qi, and the momentum inter-actions with the uncharged molecules. If Ohm’s law is invoked then this velocity isdefined as equal to kiE 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,

kicm

qiaD f

hNa3;

ai

a

i: (8.78)

As with pressure, on a continuum scale the phenomenon is a measure of a summa-tion of momentum effects. Then m and N are combined in the product mN so that


K ki qi m N c a ai

AT2

MAT M 1

L3LT

L L

1 kim �

AT2�

2 kim

qi

�

�

�

�

T � �

3 kimc

qi

�

�

�

�

�

�

L � � �

4 kimc

qia

�

�

�

�

Na3�

�

�

�

aia

1 � � 1 � � 1

k 4 1 1

8.31 Electrical Mobility 215

Equation 8.78 becomes:

kicmNa2

jzje D fhai

a

i(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,

kicm

jzje`mD f

hai

a

i(8.80)

and when ai / a this becomes,

kicm

jzje`mD constant (8.81)

which is the result given by kinetic theory.Further substitution from Equation 8.74 into Equation 8.80 gives,

ki�

jzjeN `2m

D fhai

a

i

or from Equation 8.55,

ki�Na4

jzje D fhai

a

i: (8.82)

In liquids, `m is replaced as a significant parameter by the mean molecular spacingsm [30] given by,

sm D 1=N 1=3 :

Thus Equation 8.82 becomes,

ki�a4

jzjes3m

D fhai

a

i:

If sm=a D constant, this further reduces to,

ki�ai

jzje D fhai

a

i: (8.83)

This is known as Walden’s rule for the mobility in liquids as it is related to theviscosity.

The standard derivation given in the literature invokes the use of Stokes’ lawfor the motion of a sphere at very low Reynolds number. However that law is fora continuum flow and so does not stand examination for this microscopic motion;


thus that standard form of derivation is fundamentally flawed. Here it is derivedquite satisfactorily from a rigorous use of dimensional analysis.

Some experimental evidence [31] shows that for ions above a certain size thenthe right hand side of Equation 8.83 is a constant so that,

ki�ai

jzje D constant : (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 kiE and the ionic force F being given by ejzjEthen,

v

FD ki

ejzjwhich from Equations 2.16, 8.77, and 8.81 gives,

v

F/ `m

mc;

/ D

mc2 ;

/ D

kBT:

This result is the Einstein relation.

8.33 Summarised Results

The present discussion provides a combination of the rigorous derivations of di-mensional analysis with some plausible assumptions of the physical character ofthe various phenomena. These derivations set out the physical difference betweenpressure and mobility on the one hand where the quantity .Nm/ is a significantparameter and the viscosity, diffusion and conductivity where `m is the significantparameter. In all cases the standard results of elementary kinetic theory are readilyobtained.

References 217

It is also of interest in that dimensional analysis produces the relation for mo-bility in a liquid which so far has required the unjustified use for this microscopicphenomenon of Stokes’ relation which is valid for only the low Reynolds numbercontinuum or macroscopic flow.


The first three phenomena discussed in this Chapter demonstrate cases of dimen-sional analysis succeeding where formal analysis is still not successful. It is notedthat all three phenomena involve flows that are turbulent. Further, for the second andthird examples there remain some doubts as to the physics of the events while in thethird case there are further doubts on the electrical-chemical behaviour. The fourthcase demonstrates the importance of clarifying the physics for a correct setting outof the independent variables and the dimensions to be needed. It also corrects anexisting basic fault in a derivation.

Exercises

8.1 An electrical charge exchange can occur when a liquid of very low conduc-tivity flows past a metal surface. This exchange is measured as a streamingcurrent. For a laminar flow in a pipe and with the notation of:

is Streaming currentl Representative sizeRe Reynolds number; .�U l/=�

U Reference velocity" Dielectric coefficient of liquid�0 Zero-charge conductivity of liquid� Viscosity of liquid� Density of liquid

obtain a relation for the streaming current. Then show that as the conductivityvalue tends to zero:

is / �0Re :

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 PhysicalLaboratory, Aero Rep. No. 1296, April 1969.


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. The-

sis, 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 airfoilfor 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, ConferenceSer. No. 48, Inst. Phys., pp. 145–160, London, 1979.

23. J C Gibbings, E T Hignett, Dimensional analysis of electrostatic streaming current, Elec-trochim. 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 flowthrough 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 flowingthrough 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 9Model Testing

– fifty Lockheed Hudson patrol bombers — were lost at seawithout trace. – thanks to that model ditching-technique workedout by D C McPhail – there was a rapid increase in the numberof Hudson crews rescued.R. Turnhill, A. Reed

Notation

A Cross-sectional areac Wetted circumferencecw Wave speedCd Drag coefficientCf Friction coefficientCF Force coefficientd Diameter; depth; grain sizeD Diffusion coefficient; drage Spring elasticityE Young’s modulusF ForceFr Froude numberg Acceleration from gravityh Water depthH Cylinder lengthI Second moment of areaj Current densityk Ratio of extensions; torsional radius of gyrationL Wave length` Representative sizem Mass; mass per unit lengthMa Mach numbern Rotational velocity; size ratio; Manning factorp Pressurepa Atmospheric pressurep0 Free-stream pressure; Poisson ratiopv Vapour pressureP PowerPm End load


220 9 Model Testing

Q Torquer Coordinate; scale ratiorv Vertical scale factorr� Slope factorR Ship resistance; cylinder radiusRe Reynolds numberR0 Equivalent radiust Time; thicknessV Velocityw Weight per unit length; stream widthPw Sediment mass-flow rate per unit width

W Weighty Deflexiony0 Asymmetrical offset

ˇ0 Units-conversion factor, angle� Ratio of specific heatsı Deflexion�f Torsional oscillation frequency" Dielectric coefficient; roughness height"f Flexural elastic modulus"t Torsional elastic modulus� Channel slope� Electrical conductivity� Viscosity˘ Non-dimensional group� Density�B Bridge ‘density’�g Grain density� Surface tension coefficient�0 Stress�w Wall shear stress' Electrical potential

9.1 The Application of Model Testing

Chapter 4 describes how the earliest applications of dimensional analysis were tothe testing of reduced size models of full-scale systems. The first two practical useswere to the flow in pipes and the aerodynamic forces on ropes in an airflow. Fromthen on tests of scale models of various components of aeroplanes became commonpractice 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 notbe 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 featuresand bridges invariably are now the subject of testing.

Yet still there have been cases where dimensional analysis has not been applied socausing considerable inconvenience. Once it was found that a large chemical plantwas constructed based upon a pilot plant which in turn was based on laboratoryexperiments. It was found that at full-scale a chemical reaction occurred which wasquite different from the one required and which latter had been obtained both in thelaboratory tests and then in the pilot plant.

Another example occurred when an international competition was held by a cityfor 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 andpoliticians was told by engineers that the shape was possibly the most favourableone to be the subject of serious aerodynamic oscillation in a wind: it was neverbuilt.

A contrasting example was reported by Chichester [1,3]. He had a yacht speciallybuilt for his epic around the world solo voyage. This famous boat, called Gipsy MothIV, was found to have two most serious and highly inconvenient faults. First, the keelwas of an inadequate size and weight and secondly, there was not a suitable balancebetween the mizzen sail and the headsails. After his triumphant return he foundthat a prototype of a commercial model of the yacht had revealed these two faultswhich were readily rectified by design changes to both model and full-scale boat.Understandably, Chichester reported his dismay that these model tests had not beenperformed 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 sailin the Round Pond before the boat was built! What an immense amount of trouble, worryand effort this would have saved me, by discovering Gipsy Moth’s vicious faults and curingthem before the voyage!’

A similar occasion occurred over the design of the Skylon artistic feature and thevery large Exhibition Dome building, both erected for the London 1951 Festival.Wind tunnel tests were made at a late stage resulting in extensive modifications tothese two structures that quite spoilt the designers’ planned appearances [2].

Model testing is a requisite of careful design being based upon this important useof 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 rela-tion, 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 Fig-ure 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 fol-lowing 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 havevalues in the test that are equal to these full-scale values at ‘D’. Then, in this test, bymeeting the last two relations of Equation 9.2, it will follow from Equation 9.1 thatalso 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 rangeof values of the pi-groups so covering a range of the full size behaviour. Someexamples are now described.

9.3 The Windmill

A piece of equipment, used especially in agricultural areas in North America, is thewindmill used to pump water. It is constructed with windmill blades formed fromsheet metal and with a complex supporting structure.

Following the discussion in Chapter 1 on the nature of the airflow past sharpedges, the numerical value of the air viscosity would be expected to have little or noeffect 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


P d V n �

ML2

T3L L

T1T

ML3

P�

�

�

L5

T3�

�

P�n3

Vn

�

�

�

�

L5 L � �

P�n3d 5

�

�

Vnd

�

�

�

�

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

�n3d 5D f

�V

nd

�

: (9.4)

Using a suffix ‘m’ to indicate the conditions of the model test, then from Equa-tion 9.4 the requirement is that,

Vm

nmdmD V

nd: (9.5)

It follows that:Pm

�mn3md 5

mD P

�n3d 5: (9.6)

This test might be set up in a wind tunnel. For a model of 1/10 th of the full-sizewindmill, then from Equation 9.5,

Vm

V� n

nmD dm

dD 0:1 : (9.7)

Running the wind tunnel so that, Vm D 3� V then,

nm

nD 3:10 D 30 : (9.8)

With the torque on the model windmill loaded so that it runs at 6020 rpm, whichis nm D 630:4 rad s�1, then the full-size windmill would run at 630:4=30 D21:0 rad s�1. With the corresponding torque on the model windmill so loaded asto 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 thefull size power would be given by,

P D Pm

�n

nm

�3�d

dm

�5

so that:

P D 599 � 30�3 � 105 D 2219 W :

There is one further important point to be made about such a test which will bereturned to later. It is the matter of the stress in the model windmill. With the torque,Q, we have that,

Stress / Q

d� 1

d 2/ Q

d 3/ P

nd 3:

For this example under load the ratio of the stresses is given by,

model stress

full-size stressD Pm

P� n

nm��

d

dm

�3

D 0:27:103

30D 9:0

This immediately reveals a problem in the use of suitable material to be used for theconstruction of the model windmill.

9.4 The Oil-insulated Transformer

Very large and very high voltage transformers rely on the use of extremely pure oilas an insulator. Because of their high degree of purity, these oils have an extremelylow electrical conductivity. Then problems arise over leakage current, over electri-cal field induced breakdown and over fires and explosions started by electrostaticdischarging especially during the process of filling. The use of small-scale modeltests are a clear requirement for these very large plants. Suppose a model is to beconstructed to investigate the steady state voltage potential in the oil. Then the po-tential, ', 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/ : (9.9)

Solving for application of the pi-theorem gives Compact Solution 9.2.This gives the result that,

'

'0D f

�r

`;

�`2

"D

�

: (9.10)

9.4 The Oil-insulated Transformer 225


' '0 r ` � " D

ML2

AT3ML2

AT3L L A2T3

ML3A2T4

ML3L2

T

'2" '20 " �

"

�

�

MLT2

MLT2

1T

�

�

'

'0

�

�

�

�

1 � �

�

�

�"D

�

�

�

�

�

�

1L2

�

�

�

�

�

�

r`

�

�

�`2

"D

�

�

�

�

� 1 � 1 � �

The non-dimensional group, r=` implies that measurements of ' on a model of thetransformer are to be made at relative positions corresponding to the full-scale ones.From the other independent group there is,

��`2

"D

�

mD�

�`2

"D

�

: (9.11)

Because the representative length appears in this group then in the model test oneor more of the electrical parameters would have to have a different value from thefull-scale one. Because the conductivity at full-scale is so extremely low it is readilypossible to increase considerably the value for the model test by adding a suitableelectrolyte. On the other hand, having the same values of " and D at model scale asat full scale makes the experimental conditions easy to set.

Supposing, for example, that it is arranged that �m D 100�, then from Equa-tion 9.11,

`m

`D�

�

�m

�1=2

D 0:1 :

That is, a 1/10 th scale model becomes feasible. The reference voltage, '0, in themodel test is at choice, a reduced value making experiment more convenient. ByEquation 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


j '0 ` � " D t

AL2

ML2

AT3L A2T3

ML3A2T4

ML3L2

TT

j 2

�

'20 � �

�

"�

MLT3

MLT3

�

�

T

j 2

'20 �2

�

�

�

�

1L2

�

�

�

�

�

�

�

�

�

�

"D�

t�"

� � � L2 1

j 2`2

'20 �2

�

�

�

�

�

�

�

�

"D�`2

1 � � � � 1

The solution is in Compact Solution 9.3.It thus follows that,

j`

'0�D f

�"D

�`2;

t�

"

�

: (9.12)

From the second independent non-dimensional group of Equation 9.12, for themodel test,

�t�

"

�

mD�

t�

"

�

:

For the present numerical example it follows that,

tm

tD �"m

�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 thesame 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-scalepotential, again easing experimental conditions. It follows from Equation 9.12 that,

�j`

'0�

�

mD�

j`

'0�

�

:

Thus for this example, jm=j D 100. Finally, this leakage current, which is pro-portional to j`2, will have the same value in the model test as at full-scale againassisting measurement in the model experiment.

9.6 Inapplicability of Hooke’s Law of Elasticity 227

Figure 9.2 Spring loadedrestraint under impact


m V0 e ı

M LT

T2

ML

me �

T2�

meV 20 � �

L2� �

meV 20

ı2 � � �

1 � � �

9.5 Collision Against a Spring Restraint

A spring restraint is used for protection against damage from an impact. The appli-cations 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 linearspring of elasticity, e; then the initial deflexion of the spring to a rest position, ı, thatis, before any rebound, is given as:

ı D f Œm; V0; e : (9.13)

Application of the pi-theorem leads to Compact Solution 9.4.Thus we have:

ı2

meV 20

D constant : (9.14)

A model test only requires a single set of measurements so as to determine the valueof 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 be-tween load and deflexion at the point of application of the load. This is because thereare many cases of purely elastic structures which contain elements that individually

228 9 Model Testing

Figure 9.3 Suspension byelastic strings

obey Hooke’s law but for which this does not apply for the structure as a whole. Thuscare must be taken when enhancing an answer from use of dimensional analysis byintroducing that proportionality between load and deflexion.

A simple example is that of an elastic string, initially straight, and suspended hor-izontally at each end. It is then loaded centrally with a weight of W giving a verticaldeflexion of ı. This is illustrated in Figure 9.3 and was described by Southwell [5].

Southwell derived the result that:

F D 2

kı

"

1 � `

.`2 C ı2/1=2

#

; (9.15)

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

k

ı3

`2C � � � (9.16)

so that, to first order and for small deflexions, the force is proportional to the cubeof 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 elasticbending of struts. The bending of a single strut is now considered. A real strut willhave some degree of eccentricity in both the shape of the cross-section and of theasymmetry of the end load: this is sketched in Figure 9.4. Denoting this off-set bythe amount of y0 and with a maximum sideways deflexion y then the governingdifferential equation is,

EId 2

dx2

�y

y0� 1

�

C Py

y0D 0 :

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 ofa strut


P E I ` y

y0

MLT2

MLT2

L4 L 1

PE

�

L2�

PE`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

y0D f

�P

E`2;

I

`4

�

: (9.18)

For small deflexions, there is elastic bending which is governed by the parameterEI . Thus Equation 9.18 reduces to:

y

y0D f

�P `2

EI

�

: (9.19)

230 9 Model Testing

This is the correct answer for, as Southwell showed [5], in this case the analyticalsolution gives:

y

y0D sec

"`

2

�P

EI

�1=2#

(9.20)

so that y is not proportional to the load. However, when the load is small thena series expansion leads to:

y � y0

y0

EI

P `2� constant (9.21)

so that the net deflexion is proportional to the load. This however is only valid forthis approximation.1 Also, from Southwell’s solution of Equation 9.20,

y ! 1 at`

2

�P

EI

�1=2

D

4

indicating a complete collapse before this value of P .In a model test then from Equation 9.19,

�P `2

EI

�

mD�

P `2

EI

�

(9.22)

and �y

y0

�

mD�

y

y0

�

: (9.23)

With,

I˛w4 :

Then,

I

ImD w4

w4m

:

Using the same material gives E D Em. Then from Equation 9.22 there is,

Pm

PD�

`

`m

�2Im

ID�

`

`m

�2�wm

w

�4:

1 The artificial case for which y0 is taken as zero gives the analytical result that P `2=EI Dconstant. This solution then gives no relation between the load and the deflexion. It is an exampleof unrealistic initial assessment of the physics.

9.8 Impossibility of Scale Structural Modelling 231

If the model strut is of the full-size shape proportions, then,

w

`D wm

`m

so that,Pm

PD�

`m

`

�2

: (9.24)

In a model test, there is a need to determine both y0 and the value of the constant inEquation 9.21. From this equation:

y D y0 C `2K

EI� y0P :

Where K is the constant in Equation 9.21. A linear plot of experimental values ofy 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 havenow been modelled in Sections 9.5 and 9.7. There are commonly occurring caseswhere either impact or excess deformation results in plastic yield of structures. Thisplastic buckling is of considerable importance. Examples are for the collisions ofmotor cars and ships.

The plastic flow of metals under these deflexions beyond the elastic limit is com-plex. It is somewhat akin to the flow of non-Newtonian fluids. Three similarities arefrom:

a) The stress-strain relationship is of a complex form.b) There is a velocity effect upon the stress-strain relation.c) There is a temperature effect upon the stress-strain relation.

Currently reliance is upon empirically determined data which is used to determinefull-scale behaviour.

The work done in a deformation under load results in an increase, accordingto 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 thePoisson ratio, P0 D 0:29 and the internal energy is almost completely in the formof strain energy. In the plastic region behaviour is like that of rubber in the elasticregion where P0 D 0:5 and so the change in internal energy is all thermal causinga temperature rise [6]. This rise reduces the friction shear within the plastic regionso reducing the shear strain. For metal structures the thermal conductivity would bemostly through the metal adjacent to the plastic deformation.

232 9 Model Testing

Such is this complexity of behaviour that Jones lists twenty two independentvariables leading to eleven non-dimensional groups. The form of these groups pre-vents scale modelling and so forces a limitation to severely partial scale modelling:as Jones puts it, ‘–dimensionless retardation – is impossible to satisfy for geometri-cally similar scaling –’ [7].

As a result, for the impact of motor cars reliance is still placed upon full-sizetests. For ship collisions partial modelling of elements of structure has to be used.

Jones describes a comparatively simple model of the axial buckling compressionof a cylinder as is illustrated in Figure 9.5 [7]. This is reduced to four variables. Thisgives:

Pm D f .�0; H; R/ :

Where the notation is:

Pm End load�0 StressH Length of cylinderR Radius of cylinder.

The solution is in Compact Solution 9.6.The solution is thus:

Pm

�0R2D f

�H

R

�

:

Figure 9.5 Axial plastic buckling of a cylinder [7]

9.9 Limitations to Partial Modelling 233


Pm �0 H R

MLT2

MLT2

L L

Pm�0

�

�

L2�

Pm�0R2

�

�

HR

�

�

1 � 1 �

Jones gives the analytical solution as:

Pm

�0R2/�

H

R

�3=2

:

The result shown in Figure 9.5 shows how nominally identical cylinders can revealdifferent patterns of buckling. Jones describes this as a switching effect that canoccur during compression [7]. The phenomenon of plastic deformation then is socomplex that any scale modelling is subject to severe simplification of the represen-tation of the full-scale behaviour.

9.9 Limitations to Partial Modelling

A straightforward, but intractable, problem often arises when a model test is pro-posed. Experimental limitations can prevent some or all of the independent non-dimensional pi-groups being set to the same value in the model test as in the full-scale phenomenon. The engineer who requires data from experiment to enable com-petent design then has a limit set to this information.

Where this requirement of equality of the independent groups has, in part, to berelaxed, resort is made to partial modelling. Then only some of the non-dimensionalpi-groups can have identical numerical values between model and full scale.

This relaxation can sometimes be justified because a group represents the neg-ligible effect of a weak variable. Sometimes use can be made of analytical resultsto supplement the partial modelling and sometimes the limitation can be overcomeby extrapolation of data to the full scale criterion. Extrapolation is necessarily ofdoubtful accuracy when unsupported by extra information: the examples of Chap-ter 8 make this very clear. It is in the skill of using supplementary knowledge thatreliable 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 OrvilleWright [8]. When they started on the problem of aerial flight they assiduously sur-veyed all the then current aerodynamic data but came to realise its very doubtfulaccuracy and eventually discarded it all and started their own experimental pro-gramme.

They built a small wind tunnel and carried out a remarkably intensive test pro-gramme on aerofoils and other components [9]. They recognised the small scale oftheir 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 correspond-ing non-dimensional coefficient, CF defined by:

CF D F.�1

2�V 2`2

�

(9.25)

with,CF D f ŒRe : (9.26)

For their model tests the Reynolds number based on the wing chord was of the or-der of 104 whilst their full-size ‘Flier’ powered aircraft flew at a Reynolds numberof over 106. Having at that time no suitably extensive background of aerodynamicdata, a simple extrapolation of their wind-tunnel results would have been of highlydoubtful accuracy: in the light of present knowledge the error would have beenconsiderably greater than they could reasonably have estimated [10]. So they teth-ered one of their large man-carrying gliders to support itself in a steady wind andmeasured the forces on it that were transmitted through the supporting cables. Thecorrection 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 resultswith the obvious success of their final achievement of powered flight. As they saidwith justifiable pride; “– all our calculations were shown to have worked out withabsolute 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-scalevalue of the Reynolds number but where extrapolation is reliable because the nu-merical value of the Reynolds number has no effect. A case of this was mentionedand illustrated in Chapter 1. It makes possible perfectly valid tests in wind tunnelsof such things as very large buildings and suspension bridges both having sharpcorners.

In the case of suspension bridges, the phenomenon of particular concern is theoscillation in a wind though this can also occur with very large buildings of slenderproportions. 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 tooscillate up and down against the bending flexibility of the structure. The other is offar more serious consequence as it involves this bending combined with a torsionaloscillation. When these two modes are suitably coupled in phase an aerodynamicflutter can occur. This has been long understood in the design of aeroplane wingsand gives a particularly strong and oscillating forcing-load from the aerodynamiclift.

The latter bridge oscillation at a torsional angular frequency, �f with the corre-sponding 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 perunit area, and the torsional one, "t, the weight of the bridge per unit length, w witha 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:

ˇ0�f `

VD f

��`2

m;

"f `2

mV 2;

"t`2

mV 2;

w`

mV 2;

k

`

�

; (9.27)

where the value of the viscosity is excluded because of the negligible effect of theReynolds number.

Whilst the weight of the bridge affects its steady state deflexion and hence thetotal stress levels, the oscillation depends upon the mass as discussed in the exampleof Section 1.7. Thus excluding w as a variable and rearranging the groups reducesEquation 9.27 to:

ˇ0�f`

VD f

��`2

m;

"f

�V 2;

"t

"f;

k

`

�

: (9.28)


�f ˇ0 ` V � "f "t w m k

˛T

1˛

L LT

ML3

MLT2

MLT2

MT2

ML

L

�

m

"fm

"tm

wm

�

�

1L2

1T2

1T2

LT2

�

�

�fV

�

�

"fmV 2

"tmV 2

wmV 2

�

�

˛L

�

�

1L2

1L2

1L

�

�

�fˇ0V

�

�

�

�

�

�

1L

�

�

�

�

�

�

�fˇ0`

V

�

�

�

�

�

�

�`2

m

"f`2

mV 2"t`

2

mV 2w`

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`

VD f

��

�B

;"f

�V 2;

"t

"f;

k

`

�

: (9.29)

In a model test of scale n, and with the density at atmospheric value so that from thefirst independent non-dimensional group of Equation 9.29, the model ‘density’ hasto be the same as the full scale value or:

�Bm D �B : (9.30)

From the next two groups:"tm

"tD "fm

"fD V 2

m

V 2: (9.31)

From the last group:km

kD `m

`D 1

n: (9.32)

Then the dependent group gives:

�fm

�fD Vm`

V `mD n

Vm

V: (9.33)

From Equation 9.33, to avoid too high a frequency in the model test and to avoid tooextreme a loading on the model, the test can be run so that Vm < V . This reducesthe aerodynamic loading because the aerodynamic pressure is proportional to �mV 2

m.Then, from Equation 9.31, "fm < "f which can be achieved by using plastic for theconstruction of the model. This can require extra masses being attached to the modeland at suitable locations to satisfy Equations 9.30 and 9.32.

Thus there is considerable flexibility in the experimental design to enable achieve-ment of equality of all the above non-dimensional groups with the sole exceptionof a Reynolds number. Here a limitation has been found. In one test series it wasfound that a 1/100th scale model did not reproduce the expected behaviour whereasa 1/20th model gave acceptable results. The Reynolds number discontinuity was asseen in Figure 1.5.

The numerical value of the Reynolds number can be of no influence in othercases. These can be associated with flows past bluff bodies particularly having sharpcorners. Such a case is illustrated in Figure 9.6 (a) and (b). The figure (a) is a com-puter drawing of the super-critical flow, cavitation, hydraulic flume and water tunnelat 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 modelof an approaching helicopter. This rig was being used to ascertain the wind patternsin the region of the platform which have an important impact upon the piloting dur-ing landing. A laser system was used to map these flow patterns and the associatedwind velocities. Because of the bluff shapes of the ship there again is no markedinfluence 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 suspendedupside down from test section roof

238 9 Model Testing

this information has the important safety result that the data could be fed into thehelicopter flight characteristics which then would be entered into a full-scale flightsimulator so enabling pilots to practice a landing procedure in the difficult atmo-spheric 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:

CD � D.�1

2; �V 2`2

�

(9.34)

where � is the fluid density, V the stream velocity and ` a measure of the size ofthe aerofoil. With no heat transfer present, then following the discussion of Sec-tion 6.6.1,

CD D f ŒRe; Ma; � :

In the subsonic flight of aeroplanes, that is when the Mach number, Ma < 1 � 0, andwhen 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 theReynolds number achieved was quite unrepresentative of the aeroplane values.

The problem was overcome at that time by use of an analytical result derived longbefore 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 aerofoilto that in subsonic compressible flow past an aerofoil differing only in the thicknesscoordinate normal to the oncoming stream. The analysis then showed that, withincrease in the stream Mach number, the aerofoil pressure distribution correspondsto that of a thicker aerofoil in incompressible flow, and by a factor that depends ononly this stream Mach number. The analysis is not valid in the region of the roundedleading edge so the solution is limited in accuracy. Tests in incompressible flow ona series of aerofoils of varying values of this thickness gave data for high valuesof the Reynolds number which could then be used to determine the compressibilityeffect [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 fora non-viscous potential flow whereas the drag force is dictated by viscous effects.

Figure 9.7 The shape trans-form of an aerofoil betweenincompressible flow and com-pressible 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 pressuredistribution 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 andthe 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 tobe successful.

9.13 Partial Extrapolation Method

Partial modelling arises from the need to determine, at the design stage, the resis-tance of a large ship. The importance of accurate data has been shown in a recentstudy revealing estimates of the provision of engine power of up to 30 % high withvariation of 40 % in several predictions [14]. The flow past the ship which gives thisdrag involves several effects. The pressure stresses arise from inertia terms whichinvolves the density � and the velocity V and also from the work done against grav-ity 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 ten-sion coefficient � whilst cavitation can occur when the local pressure falls in valuefrom 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 atthe minimum pressure point which occurs at the surface [15]. So the vapour pres-sure, 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


R � V ` g � � pv � p0

MLT2

ML3

LT

L LT2

MLT

MT2

MLT2

R�

�

�

�

��

pv�p0�

L4

T2�

�

L2

TL3

T2L2

T2

R�V 2

�

�

�

�

g

V 2�

�V�

�V 2pv�p0

�V 2

L2�

�

�

�

1L

L L 1

R�V 2`2

�

�

�

�

�

�

g`

V 2�

�V `�

�V 2`

1 � � � 1 1 1

This then gives that:

R

�V 2`2D f

�V 2

`g;

�V `

�;

�

�V 2`;

pv � p0

�V 2

�

: (9.35)

The first independent group is a Froude number, Fr, the second is a Reynolds num-ber, Re.

If tests are proposed on a model of size 1=n of the full-size ship, then from thefirst independent group, and with g D gm:

Vm

VD�

`m

`

�1=2

D 1=p

n (9.36)

and the test speed is to be correspondingly reduced from the full-size one which isexperimentally convenient.

From the last non-dimensional group in Equation 9.35:

pv � p0m

pv � p0D V 2

m

V 2D 1

n: (9.37)

This relation can be satisfied by use of a low value of the model stream pressureusing a super-critical cavitation flume.

From the second independent group, and obeying Equation 9.36,

.�m=�m/

.�=�/D V `

Vm`mD n

pn D n3=2 :

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 replacingwater at 15 ıC with mercury at 100 ıC gives this ratio as only 12 � 6. A preciserepresentation is not possible. Looking at the third independent group shows therequirement of:

.�m=�m/

.�=�/D V 2

m`m

V 2`D 1

n

1

nD 10�4

and again an impossible requirement is set. Faced with this situation, resort is madeto partial modelling.

First, experience shows that the capillarity has no significant effect upon the valueof R and so the third independent group can be neglected. It is only effective with thevery small toy boats that were propelled by a crystal of camphor thereby changinglocally the surface tension. It is also important for the propulsion of some floatinginsects.

Secondly, the resistance is composed of pressure and viscous stresses upon thehull. The former are dominated by the influence of the first independent group ofEquation 9.35: the latter by the second. An example of this division of forces isshown in Figure 9.9 [16]. The upper curve is for the total resistance as can be mea-sured by a force balance. The component due to friction and separation at the sterncan 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 wasalso derived from analysis of the measured wave patterns.

At high values of the Reynolds number, this viscous component of the resistanceis a ‘well behaved’ function of the second independent group and so lends itself toreasonably accurate extrapolation based upon supplementary information.

On a model the breakdown of the flow in the surface boundary layer from lam-inar to turbulent conditions can occur well back along the model hull whereas atfull scale this occurs at a comparably equal downstream length which places it rightat the bow. Thus the full scale friction force is virtually all due to the turbulentboundary layer whereas the model would be a mixture of laminar and turbulent.This would make extrapolation of the resistance from model to full scale somewhatproblematical. A testing technique to avoid this is to attach carefully assessed rough-ness at the nose of the model to trip the laminar layer to a turbulent one [17] so thatthe model pressure and viscous drags also are virtually all due to turbulent flow inthe boundary layer. The effective model shape is changed in this partial modellingboth by the attachment of the roughness trips and by the different relative thicknessof the boundary layer.

The combination of tests over the experimentally attainable range of the Reynoldsnumber together with a wealth of experience gained from both model testing andfull-scale trials gives reasonable confidence in extrapolations of model tests to thefull-scale values of the second independent group.

Further refinements to the experimental technique are required giving successto this partial modelling method. Where precision of design estimate can be lessprecise is, for example, in the case of very-large crude-carriers whose viscous resis-tance extrapolated to very high values of the Reynolds number is the greater portionof the whole.

The same problem arises in the case of compressible flow where the drag dueto the presence of shock waves in regions of supersonic flow add to the pressuredrag. Experimental separation of the wave drag component is rather rare. It hasbeen 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 ofthe 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 ofSection 6.6.1, a force F exerted by the gas stream is represented by:

F

�V 2`2D f ŒRe; �; Ma : (9.38)

9.14 The Range Limitation Method 243

It is readily shown that where viscous effects are absent in the flow, the densitychange can be written, for M 2

a � 2=.� � 1/, in the series [19]:

�s � �

�sD�

1

2M 2

a C � � ��

(9.39)

Here the suffix s refers to stagnation values. The pressure differences are given interms of the pressure coefficient, Cp , by:

Cp � ps � p12 �V 2

D�

1 C 1

4M 2

a C � � ��

(9.40)

Up to Ma D 0 � 14 there is only a 1 % error in neglecting the density change anda 0 � 5 % error in determination of the pressure coefficient: then the flow can beconsidered 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 ofthe Reynolds number needs to be reproduced. With a ratio of full size dimension tomodel value of `=`m D n this requires:

�m

�

Vm

V

�

�mD n : (9.41)

The viscosity is a function of the particular gas and, except for great extremes ofpressure, only the gas temperature. To vary this property in practice involves un-due complexity in the test rig and instrumentation with only limited control of theReynolds number.

The left-hand side of Equation 9.41 can be suitably adjusted by running themodel test at n times the full-scale velocity. This procedure can be limited by theneed to avoid the above mentioned compressibility effect as well as by the largepowers required of the test rig which, in general terms, increases as the cube ofthe test velocity. Alternatively, the test channel can be pressurised, a density oftwenty-five times that of the atmospheric value having been used in wind-tunneltests. This leads to large stresses on the model as an alternative dependent variableis .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 mod-elling is the rule, experiment being limited to reproduction of the full-scale Machnumber. As with ship model testing the influence of Reynolds number is extrap-olated using background knowledge and an extensive body of prior experimentaldata. This still has its hazards when high accuracy is required. It has been said that atthe onset of the use of jet engines to propel civil airliners one of the great companieswon the market against a rival company largely because the formers aerodynami-cists made the more precise estimate of the extrapolation of wind-tunnel model teststo the full-scale Reynolds number.

244 9 Model Testing


v � R0 � d g t

LT

ML3

L MLT

L LT2

T

�

�

�

�

�

�

L2

T

�

�

�

�

�

�v

g

v2vt

�

�

�

�

L 1L

L

�

�

�

�

�

�

�

�vR0

dR0

gR0v2

vtR0

� � � 1 1 1 1

9.15 The Distortion Method

Model reproduction of the flow in rivers and estuaries requires partial modelling ofseveral shape factors and several compromises in balancing conflicting requirementsin the model.

The flow is one under gravity and against friction, and can be unsteady as forexample when floods occur or when waves travel along the stream. The mean ve-locity, 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 R2

0. The pitheorem solution is in Compact Solution 9.9.

The result is:�vR0

�D f

�d

R0;

v2

R0g;

�t

�R20

�

: (9.42)

The first and dependent group is a Reynolds number, the second is a shape factoron the river cross-section, the third is a Froude number, and the last one is a non-dimensional time group.

Starting with the steady flow in a straight parallel channel, of constant cross-sectional area and shape, inclined at the angle � to the horizontal, the mean velocity,v, is expressed by [20]:

v2 D 2g sin �

Cf:A

c(9.43)

Here A is the cross-sectional area, c is the wetted circumference and Cf is the frictioncoefficient defined by:

Cf � �w=

�1

2�v2

�

(9.44)

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]:

c2w D gL

2tan h

2h

L:

When h � L then:

c2w D gh :

Two requirements are set for the flow through a model. First, the full-scale flowwill be turbulent, and over a rough surface and well into the region where, for therandom irregularity of the roughness, the friction coefficient does not vary greatlywith the Reynolds number if at all. Secondly, for smooth river flow away from suchas weirs and waterfalls the flow velocity will be sub-critical, that is below the abovewave speed corresponding to the Froude number:

v2

gh< 1:0 : (9.45)

The first criterion presents the greater difficulty for two reasons. First, the Reynoldsnumber has to be greater than a limiting value, which is not precisely known for theparticular nature of the surface roughness, in order that the flow is in the fully-roughregion [22]. Secondly, practicality requires the surface roughness to be modelled byarbitrarily chosen roughness forms and locations. To achieve the necessary valueof the Reynolds number the cross-section of the real flow is distorted in the modelby an extension of the depth scale in order that the flow can attain the fully-roughcondition. Further, the slope of the flow, � , is increased in the model by distortingthe vertical scale of the water level.

Thus for a model reduced in plan view by the factor 1=n, the vertical depth scaleof the cross-section is extended by the factor rv and the slope is extended by thefactor 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

d� w

wmD rv

n� n D rv : (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

cD 1

n

�1 C 2rv.d=w/

1 C 2.d=w/

�

and this relation is used in the following example. Also for this illustration, the fol-lowing relations from Nikuradse’s experiments upon the flow in sand-rough circularpipes 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:

v2m

dmg� 1

and:

2�vm

�

�2Am

�1=2

4:64 � 103 .R0="/5=6m :

Combining these gives:

dmg v2m

4:64 � 103�

2�

!2

2Am

�R0

"

�5=3

m:

Then using the scaling relations, Equations 9.46 and 9.47 leads to:

r2v

n3 w

A2

4:64 � 103�

2�

!2

2g

�R0

"

�5=3

m:

9.16 Complexity of Modelling 247

With �=� D 1:14 � 10�6 m2 s�1 for water and g D 9:81 m s�2 then the limitingcriterion is given by:

r2v

n3 1:12 � 10�6 w

A2

�R0

"

�5=3

m:

There are several parameters that can be set for the model tests. With full-scalevalues of A and w being set, the choice of n gives R0 and with an upper limit set torv 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 appliedwith care for real river flows [23].

The reproduction of the full-scale roughness effect presents a particular diffi-culty. Equation 9.49 was developed from extensive testing upon pipe flows and, asrepresenting satisfactorily the roughness effect, has been generally accepted as be-ing applicable also to channel flows. In comparison, Manning gave an empiricallyderived relation which was [24]:

v D 1

n

�A

c

�2=3

:sin1=2� (9.50)

In this relation, n, is Manning’s friction factor, v, A and c are measured in SI unitsand � is measured in radians. As,

n�T

L� L2=3 D T

L1=3

it is seen that Manning’s factor has the fundamental disadvantage of not being non-dimensional. It was introduced well before dimensional analysis came to be usedin engineering. Yet it is still used universally and large tabulations exist of valuesfor various types of surface without any reference being made to the correspondingvalues of g and of A=c. This is seen when from Equations 9.43 and 9.50 there is:

n2 D Cf

2g

�R0

2

�1=3

:

9.16 Complexity of Modelling

The flow of sediment in river and estuary flows is of considerable importance. Be-cause 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 followingsources of these difficulties.

a) Even after the grain size and the distribution of this size together with theturbulence character of this flow have been reproduced, at least four non-dimensional groups remain to be satisfied.

248 9 Model Testing

b) 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 ofsizes exist in the full-scale flows.

c) There is a complexity of the full-scale flow patterns especially when the flowis both channel flow combined with wave motion as in estuaries.

d) There is enormous complexity of the full-scale sedimentation movements [26].e) Because of (a) above, some non-dimensional groups cannot be reproduced in

value at model scale so quite considerable partial modelling is necessarily re-sorted to.

f) There are difficulties in measuring sediment movement whilst the flow is run-ning so that much data is limited to the overall change from the beginning tothe end of a test run.

With the following notation:

d Grain sizeg Acceleration due to gravity` Scale sizePw Sediment weight-flow rate per unit widthV Reference flow velocity

� Fluid viscosity� Fluid density�g Grain density

Then the functional relation is:

Pw D f�d; g; `; V; �; �; �g

: (9.51)

The pi-theorem solution is as in Compact Solution 9.10. The result is that:

Pw�gV 3

D f

�gd

V 2;

`

d;

�

�gVd;

�

�g

�


Pw d g ` V � � �g

MT3

L LT2

L LT

MLT

ML3

ML3

Pw�g

�

�g

�

�g

�

�

L3

T3L2

T1 �

�

Pw�gV 3

g

V 2�

�

�

�gV

�

�

1 1L

�

�

L �

�

�

�

gd

V 2`d

�

�

�

�gVd

�

�

� 1 1 � 1 �

9.17 Model Testing in Engineering Design 249

or,

Pw�gV 3

D f

�V 2

g`;

d

`;

�V `

�;

�

�g

�

:

In this equation the first independent group is the Froude number and the third is theReynolds 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. Thislatter is important in that it would control the development of the vortices of turbu-lence which spring from the boundary and which can lift sediment into the flow: thisis the behaviour seen with atmospheric whirlwinds. However progress can be madeby the use of surface roughness to develop the necessary level of turbulence. Whenthe last independent group is satisfied then there is a problem in obtaining a modelgrain size sufficiently small.

Yet despite these difficulties some most useful agreements of data between modeland full-scale results have been achieved [25]. Extensive discussion is given by Yalinin 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 itfits 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 de-termine the aerodynamic behaviour of the flow around the proposed aeroplane atthe landing configuration together with the corresponding calculation of the aero-dynamic loading. This would form the first stage of the design. Such computationshave become markedly improved in accuracy in recent times from the rapid growthin the operating capacity and speed of computers. Yet, as is illustrated in the dis-cussion of turbulence in flows given in Chapter 8, such calculations in detail stillrequire 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 sothat 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 config-uration as it would be flight tested with extensive on-board instrumentation. Com-parison of flight-test results with the computer output and the model tests enablesthe latter two to be further refined.

In this way engineers hone their design skills and build up experience and exper-tise so essential to successful design developments.

250 9 Model Testing

Figure 9.11 Airbus 380 aeroplane, landing configuration (a) Computer output, (b) Wind tunneltest, (c) Full-size aeroplane

9.18 Assessment of the Physics

These examples show that a good understanding of the physics is required beforeundertaking the use of dimensional analysis to order an experiment: a considerableunderstanding is needed when partial modelling is undertaken both for the designof the experiment and the interpretation of the resulting data.

This latter is the engineering approach in which some answer must be obtainedfor design or operational needs even though partial modelling may be somewhatapproximate in its output. The skill in engineering is in using scientific knowledgeand background experience to minimise the approximation. But this is no more thanthe 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 ofthat wherein lies the skill.

Exercises

9.1 The spin of an aeroplane can be studied in a spinning tunnel which has a testsection containing a stream of air flowing vertically upward. A model aero-plane can have the controls set so that it spins steadily within this stream, theair velocity being adjusted so that the vertical aerodynamic force balancesthe weight of the model. Obtain the requirements of a model test. Considerboth the case of steady spinning and that of recovery from the spin when thecontrol surfaces are operated by radio control.

9.2 Tests are to be made on a model of the pressurised cabin of an airliner tofind the effect of sudden failure of a window upon the subsequent trajectoryof the passengers. Demonstrate that for dynamical similarity the absolutetemperature of the air within the model must be proportional to `; where` is a representative length. Also show that if the external pressure is heldconstant then the density of the dummy passengers must be inversely pro-portional to `. [Univ. Liverpool., 1958.].

9.3 Extend the discussion of Section 9.7 to take account of the masses containedwithin the full-scale shell structure such as, for a car the engine and othermasses.

9.4 A safety valve, which is illustrated in Figure 9.12, operates in the followingmanner. A piston of mass m rests on a seal A and is restrained vertically bya 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 . Ini-tially the outlet from this chamber at C is constricted and so the full releasepressure now acts upon the whole piston area. The piston rises rapidly until itclears the constriction at C rising to a maximum height where it is held andso releasing the air to the atmosphere. Tests are to be performed upon a smallscale model of this valve to determine this vertical movement. Air at the full-scale pressure is used in the container and this exhausts to the atmosphere.Neglecting viscous and heat conduction effects in the flow, demonstrate thatfull dynamical similarity can only be obtained if the absolute temperatureof the enclosed gas is proportional to a typical length `. Also show that the

Figure 9.12 Design ofa safety release pressurepiston

252 9 Model Testing

weight of the piston and the pre-tension force are to be proportional to thesquare of the size; and that the spring rate (in units of force per length) mustbe proportional to size. Show also the importance of retaining the full-sizeoverall pressure ratio. (Univ. Liverpool)

9.5 A very large vacuum vessel is to be set up out of doors. Set in the bottomis to be a circular safety bursting disc. There is concern for the safety ofpersons in the vicinity of this disc should it fail and a person be suckedinto the vessel. A scaled down model test, in which the external pressureis equal to the full-scale atmospheric pressure, is to be set up to determinea safe approach distance. Show that the density of a dummy figure should beinversely proportional to the size and that the pressure in the vessel must bein a fixed proportion to the outside pressure. Show that this latter conditioncan be relaxed when the inside pressure is sufficiently low so that then theoutside temperature must be proportional to the model size. Show that alsothe velocity of the dummy would be proportional to the square root of thesize. [Univ. Liverpool. 1988].

9.6 An aerofoil wing is tested in a wind tunnel at a Mach number of 1:35 anda Reynolds number of 7:8 � 106 giving a drag coefficient, based on the wingplan area, of 0.0087. Calculate the friction drag coefficient, using the relationgiven in Exercise 9.9 below, at a full-scale Reynolds number of 8:2�108. Thencalculate the full scale total drag.

9.7 A design proposal has been made for a container to hold survival equipmentwhich is to be dropped into the sea from an aeroplane and be just buoyantwhen stationary. It is to have a small drogue parachute to hold the orientationat impact. Cavitation, which occurs when the local fluid pressure in the flowdrops to the local vapour pressure, is possible at entry. Model tests are to bemade to determine the maximum depth of immersion and the impact load onthe nose of the container. Set out the requirements of a model test to a scaleof 1/20 in which the atmospheric pressure can be varied to suit.

9.8 A proposal for a ship is for a length of 180 m with a propeller of 2.55 mdiameter rotating at 85 rpm just below the sea surface. The design is to betested using a model of scale 1 : 120 in test flume using fresh water. Calculatethe diameter and the rotational speed of the model propeller. If the ship issailing in sea water of density 1.035 that of fresh water, calculate the ratioof the power to that determined in the model test. Determine also the size ofthe model ship. Exclude viscous and cavitation effects.

9.9 A ship is planned having a length, `, of 165 m and a wetted area, S , of5445 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 m�3 and viscosity, � of 1:14 �10�3 kg m�1 s�1. A test on a model of length 1:2 m at the appropriate speedin fresh water gave a total resistance of 0:94 N. The frictional drag, is givenby

D D CD1

2�V 2S

References 253

Figure 9.13 Flow down a spillway

and the drag coefficient can be estimated from

CD D 0 � 455hlog10

�_V `

�

�i2�58

Prepare the estimate. Calculate the drag due to friction as a percentage of thewhole resistance at both model and full scale and consider the implicationsfor the precision of your estimate.

9.10 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 modelof 2.0 m width. The vertical scale of the model is 1 : 9.3 and the model flowrate is measured to be 0.47 m3 s�1. What is an estimate of the full-scale flowrate?

9.11 Model tests are required on a harbour for which the outer breakwater wallis subject to waves of 1.5 m height travelling at 10.2 m s�1. A scale modelof 1/300 full-size is chosen. Deduce the size and speed of the waves in themodel. Tides occur at intervals of 12 h. What is this tidal period to be in themodel test?

9.12 A section of a river has a width of 100 m and a mean depth of 7 m. The meanslope of the river is 2 � 10�4 rad. Determine the dimensions of a model whenthe horizontal scale is 1 : 2000 with a vertical enhancement of 40. Take themodel 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 experi-ments,: 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 dis-

placement 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 super-

sonic speeds with an analysis of their drag characteristics, Br. Aer. Res. Council, Current PaperNo. 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 10Assessing Experimental Correlations

– an experimental reading is not to be trusted until itsbackground has been thoroughly investigated.F. Drabble

Notation

a, b, c Constants; Equation 10.19A, B Dimensional productsC Dimensional constantc1,� � � , c6 Regression coefficientsd Paddle, fan diameterD Container, pump diameterD12 Diffusion coefficientDC, D� Ion diffusivitye Paddle clearanceE1, E2 ErrorsF Fraction of cream to milkg Acceleration due to gravityH Head of liquidi0, i1 Electrical current, zero, infinite timej0, j1 Electrical current density, zero, Infinite timej Electrical current densityK1 Parameter` Scale sizen, m Non-dimensional indicesn Rotational speedP PowerPQ Volume flow-rateRe Reynolds number; .�!d 2/=�

t Timew Paddle widthx Variable, control settingxt True value

ˇ0 Units-conversion factor for angle


256 10 Assessing Experimental Correlations

" Permittivity, error�0 Electrical conductivity� Liquid viscosity˘ Non-dimensional group� Liquid density� Efficiency� Torque'0 Electrical potential! Angular velocity of paddle.!t/c Computed regression value

10.1 Interpretation of Dimensionless Correlations

The power of dimensional analysis in enabling greatly simplified correlation of ex-perimental data has been demonstrated in Chapter 1. Care is needed on occasions ininterpreting the precision of these correlations. Again, a clear understanding of boththe physics of the phenomenon and of the physical significance and composition ofeach non-dimensional group is required.

10.2 Interpretation of Experimental Error

In a phenomenon involving a functional relation between only two non-dimensionalgroups˘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 formof products, we can write,

˘1 D Axn ;

˘2 D Bxm : (10.2)

It is common practice to judge the level of the random error in experimental resultsby measuring the scatter about the mean correlating curve whether the latter is de-rived in an analytical form or as a hand-drawn curve on a plot of the results showingthe 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 truevalue, 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,

xn D xnt .1 C "/n

D xnt .1 C n"C 1

2n.n � 1/"2 C � � � /so that,

xn � xnt .1 C n"/ : (10.3)

Then, using Equations 10.2 with 10.3, the corresponding error on˘1 denoted byE1

is given by:

E1 � ˘1 �˘1t

˘1t

D Axn � Axnt

Axnt

D xnt .1 C n"/� xn

t

xnt

so that,

E1 D n" : (10.4)

Similarly, the error on˘2 is,E2 D m" : (10.5)

A plot of ˘1 versus˘2 can be misleading in estimating the error as is illustrated inthe sketch of Figure 10.1. With the true value at point ‘A’ in this diagram and withn > 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 inthe first case a cursory glance at the experimental plot would mislead as to a highaccuracy and the second case would equally mislead as to a poor one. Other casesfor a negative value of the slope of the correlation curve and for other values of nand m are readily sketched.

There are two important lessons. First, from the above example a correlationcurve, for which the values of the non-dimensional groups are equally straddledabout 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 fromplots of non-dimensional groups and from statistical analyses. Clearly discrepanciesare enhanced by high values of the indices n and m. This can often happen. Forexample, the non-dimensional group for the power of a propeller is,

P

�n3d 5

so that the error in measuring n is trebled and that in d is increased five-fold.


Figure 10.1 Illustration ofassessment of experimentalerror

Massey has called this matter a ‘spurious correlation’ [1] but this can be a lim-iting expression. The more appropriate use for this term would be as used later inthis chapter. A suitable term for the present difficulty would be ‘misleading error-estimation’.

Again referring to Figure 10.1, the ‘true’ error arising from error in only x, isreadily assessed, once the correlating curve is acceptable, by drawing the line fromthe plotted point ‘B’ at the slope of � given by:

tan � D ˘1 �˘1t

˘2 �˘2tD ˘1tn"

˘2tm"��n

m

˘1

˘2

�

to intercept the correlated curve giving the ‘true’ data point at ‘A’. This is the equiv-alent of measuring the vertical discrepancy between an experimental point and thecorrelation curve when, for example there is a plot between two variables with anerror assigned to only one.

It is also common practice when computing a regression curve to assume theerror to be only in the ordinate values which are conventionally chosen as measuringthe dependent variable. In the present example, if there is error only in x then thereare 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 teston 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 fora range of values of the Reynolds number

torque, � , from a shaft rotating at n rps on a machine of reference diameter,D. Thewater of density, � and viscosity,� is flowing through the turbine nozzle at a volumerate, PQ, under a head of water of height, H; so that the applied pressure differenceis �gH . The flow nozzle for the water jet has a needle valve to adjust the nozzlecross-sectional area with a value of the control setting, x.

It is noted that:PQ D f .gH; x;D/ (10.6)

so that PQ and gH are alternative variables.Then with the function of:

� D f� PQ; �;�; x;D; n; ˇ0

�:

The pi-theorem solution is in Compact Solution 10.1.Thus the pi theorem leads to:

�D

� PQ2D f

��D

� PQ ;x

D;ˇ0nD

3

PQ�

or:�

�ˇ20n

2D5D f

"�ˇ0nD

2

�;x

D;

PQˇ0nD3

#

: (10.7)

A convenient procedure is to fix H , x and hence PQ and then vary � by varyingthe output load on the turbine and measuring n. This gave the curves shown in



� PQ � � x D n ˇ0

ML2

T2L3

TML3

MLT

L L ˛T

1˛

ˇ0n �

1T

�

�

��

�

�

�

�

�

�

L5

T2�

�

L2

T�

�

�

� PQ2 � �

�

� PQ

ˇ0nPQ

�

1L

�

�

�

�

1L

1L3

�

�

�D

� PQ2 � �

�D

� PQ

xD

�

ˇ0nD3

PQ�

1 � � 1 1 � 1 �

Figure 10.2. This would seem unsatisfactory as the value of both the second and thefourth groups in Equation 10.7 vary along each curve. The groups can be rearrangedso that:

�

�ˇ20n

2D5D f

"x

d;

PQˇ0nD3

;PQ��D

#

(10.8)

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 ofinterpretation. The question is, by bearing in mind Equation 10.8, does the distinc-tion between the four curves give a measure of changes in the second or in the fourthgroups of Equation 10.8 or of both?

An alternative formulation is obtained by replacing the variable, PQ with the hy-draulic head term, gH so that the third group in Equation 10.8 becomes .gH/=

�ˇ2

0

n2D2�. This represents the square of the ratio of the velocity of the water jet to

the peripheral speed of the deflecting buckets of the turbine. Also, the fourth groupcan correspondingly be replaced by

��2gHD2

�=�2. Further, the efficiency of the

turbine, �, which is given by:

� � ˇ0n�

�gH PQcan be alternatively used as the dependant variable so that:

� D f

�x

D;

gH

ˇ20n

2D2;�2gHD2

�2

�

: (10.9)


Figure 10.3 Pelton wheel experimental data: efficiency versus pressure-head group: codes as inFigure 10.2

It is known that a significant effect of Reynolds number, represented in Equa-tion 10.9 by the last group, will show up particularly in values of the maximumefficiency. Plots of the experimental data are shown in Figure 10.3. There is a com-mon curve for the sets of data at the three highest values of the Reynolds numberwith an indication of a common maximum value of the efficiency for all four flowrates. This suggests that the divergence between the first three and the lowest fourthflow rate, at the lower value of the second independent group of Equation 10.9, isan indication of the effect of x=D and that there may well be no Reynolds numbereffect for all the results.

To confirm this latter conclusion, experiment would have to be done with varia-tion 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 electrostaticproblems. The evidence is that the charge carriers are impurity particles of unde-termined composition because of their extremely small size [3]. With these liquidsthere is an interaction with electrodes that gives rise to the electrical boundary layerpreviously described in Section 7.8. Thus a problem arises in the determination ofthe very low conductivity because of the absence of a single value of the field. Fur-ther, above a limiting applied potential difference across the electrodes convection



j '0 t �0 " DC

D�

`

AL2

ML2

AT3T A2T3

ML3A2T4

ML3L2

TL2

TL

'0�0 �

�

"�0

AL

�

�

T

j

'0�0

�

�

�

�

1L

�

�

�

�

�

�

�

�

�

�

"�0t

DC

t D�

t

� � � 1 L2 L2

j`

'0�0

�

�

�

�

�

�

DC

t

`2D

�

t

`2�

�

1 � � � 1 1 �

of the liquid can take place and so this effect has to be excluded by limiting theupper 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 atzero charge density, �0, the liquid permittivity, ", the diffusion coefficients of thepositive 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`

�0'0D f

�"

�0t;DCt`2

;D�t`2

�

: (10.10)

This is rewritten as:

j`

�0'0D f

��0t

";DC"�0`2

;DCD�

�

: (10.11)

Before application of the potential there is a charging current [4]. This current is verysmall compared with those to be measured in a conductivity cell so the distortion ofthe charge distribution and hence of the field distribution within the liquid is takenas being negligible. Thus, at the instant of application of the potential, the field is


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 correspond-ing to infinite time. From Equation 10.11 and for a fixed liquid at t D 1 then theratio of these two current values, i1=i0 is given by:

i1i0

D j1j0

D j1`'0�0

: (10.12)

It then follows from Equation 10.11. and for a fixed electrolyte that,

i1i0

D f

�DC"�0`2

�

: (10.13)

This physical modelling of the phenomenon is confirmed by the experimental corre-lation shown in Figure 10.4 [3,5]. These experiments were carried out after a carefulcheck confirmed that there was no electrically generated convection of this liquidelectrolyte. The value of DC and of " were not determined in this experiment. Aseach would have a constant value then the values of �0`

2 are representative of theindependent group of Equation 10.13. This correlation in terms of non-dimensionalgroups shows clearly the importance of the diffusion in the measurement of conduc-tivity of these liquids, an effect that increases with increase in the diffusion coeffi-cient.

Figure 10.4 Experimentalvalues of electrical currentversus the diffusion group [3]


This example again illustrates how much understanding can be gained when in-spection of the form of the non-dimensional groups is combined with inspectionof the experimentally derived form of the functional relation between the non-dimensional 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 anal-ysis of experimental data will now be demonstrated in detail through a particularexample.

The statistical analysis of experimental data is common practice by scientists andengineers. In industry and research establishments, and when these units are verylarge, the group doing the statistical analysis can be separate from those who haveperformed the experiment. The value of incorporating dimensional analysis with thestatistical analysis has been remarked upon elsewhere, including advocacy by statis-ticians [6–8]. Not to do this can result in a false correlation especially when neitherthe experimental group nor the statistics group has applied dimensional analysisbefore 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 ofthe 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 ofa stirring paddle needed to just disperse a layer of cream that had been separated upto the top of a container of milk by gravity forces. The experimental rig is shownsketched in Figure 10.5. This shows the various shape variables which are; D, thevessel diameter; d , the paddle diameter; w, the paddle width; e, the paddle clear-ance; H , the liquid depth and the paddle angular velocity, !. The flow would beone of viscous rotating motion within the vessel and of turbulent separation fromthe sharp edges of the paddle. This motion would contain vortices in combinationwith diffusion. The liquid variables would be the proportion by volume of cream tomilk, F ; the viscosity, �; the density, � and the diffusivity,D12 where the diffusioncoefficient is common between the cream and the milk [13].

Two other variables that Miller considered were the difference of the densitybetween cream and milk and the acceleration due to gravity. These would enter intoa buoyancy effect which would be significant in the original settling out of the creamThis would take a time that would be several orders of magnitude greater than thestirring time and so could be neglected in the latter operation.

10.5 A Mixing Experiment 265

Figure 10.5 Sketch illustrat-ing 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:

!t D f

�d

D;w

D;e

D;H

D;�!D2

�;D12

!D2; F

�

:


t D d w e H ! � � D12 F

T L L L L L 1T

MLT

ML3

L2

T1

�

�

�

�

�

�

TL2

�

�

�

�

�!

�

D12!

�

�

�

�

1L2

L2

!t

1�

�

dD

wD

eD

HD

�

�

�

�

�!D2

�

D12!D2

� 1 1 1 1 � � 1 1


It is convenient for the analysis of the experimental results to rewrite this as:

!t D f

�d

D;w

D;e

D;H

D;�!d 2

�;�D12

�;F

�

: (10.15)

Miller excluded the variables � and D12. But if the density is excluded in Equa-tion 10.14 then, as seen in the above tabulation of dimensions in Compact Solu-tion 10.3, this equation cannot be made dimensionally homogeneous because therewould be only the variable � containing the dimension of mass.

In liquids containing a small concentration of the solute, by Walden’s ruleD12 / � and so the penultimate group in Equation 10.15 would be constant invalue. Also, in the experiment the third and the fifth group were held constant. ThusEquation 10.15 reduces to:

!t D f

�d

D;e

D;�!d 2

�;F

�

: (10.16)

10.6 The Regression Function

The third independent group in Equation 10.16 is the Reynolds number. Experiencesuggests that for this type of flow the two variables, Re and d=D might be theprinciple ones. An appeal to the physics of this fluid flow indicates that it wouldcomprise a combination of separated flow from the sharp edges of the paddles anda viscous mixing flow from both the vortices generated by this separation and thegeneral rotating motion in the container. The former might not show a Reynoldsnumber 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 inde-pendent of the shape parameter d=D. This plot gives a first estimate of the value ofthis index as �1:35. Thus Equation 10.16 becomes:

.!t/R1:35e D f

�d

D;e

D;F

�

: (10.17)

There are indications in Figure 10.6 that the parameter d=D has a greater influencethan that of e=D. Consequently Figure 10.7 is a plot of .!t/ � R1:35

e against d=D.This indicates a further power law relation which, because of the constant slopes ofabout 2.0 in Figure 10.7, is a simple multiplying one so that Equation 10.17 reducesto:

K1 � .!t/R1:35e =.d=D/2 D f

h e

D;Fi: (10.18)

Figure 10.6 indicates that there is a better correlation for d=D D 0:228 and 0.380than 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 ofgroup product versus diameterratio

Equation 10.18 leads to the further plot shown in Figure 10.8. It is seen that asthe correlation is being built up, so that the non-dimensional variables successivelytaken into account are becoming weaker in their influence, then doubtful experi-mental points become revealed. There are now six of these marked in the plot of


Figure 10.8 Correlation of K1 versus paddle clearance ratio

Figure 10.9 Correlation ofcoefficient of Equation 10.18versus the cream ratio

Figure 10.8. This figure suggests a correlation in the form of a set of parabolas ofthe 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 pointsmarked in Figure 10.8 are now excluded. Thus the final form of the function ofEquation 10.16 is obtained as:

.!t/Rc1e

.d=D/c2D c3 C c4F C c5.e=D/C c6.e=D/

2 : (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, multiplevariable, regression technique [14]. Using that quoted technique, a least-squares fitto Equation 10.20 was obtained, excluding the points marked in Figure 10.8. Theregression was on the actual values of each variable. The values of the coefficientswere 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 wereexcluded from the regression curve are now plotted in this figure. They reveal twoadmirable fits to straight lines which straddle the presently accepted curve. Thisraises a possibility that somewhere in the experiments some fixed variable was er-roneously reported.

Figure 10.10 Final correlation for the time group


Inspection of the original data and of Figure 10.6 does not reveal the cause ofthese divergences in what is, for this type of experiment, an otherwise pleasinglyaccurate one. In terms of the variables in Equation 10.14 only the variables w andH were excluded. Inspection of Figure 10.10 suggests that the divergences camefrom two systematic causes.

The statisticians who analyzed Miller’s results adopted the Rayleigh assumptionof a simple power product for the form of the function. They used their study toremove some of the variables in Equation 10.14 resulting in the simple power lawrelation of:

t D C!ad b : (10.21)

Further, they did a simplified logarithmic linear regression curve fit. These approxi-mations 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 beimportant. One noteworthy omission is the viscosity. Secondly, the simple power-law assumption is seen to be not acceptable being unacceptably restrictive. Thirdly,the present analysis has revealed what appears to have been a systematic divergencein reporting some of the data.

This case study shows that Miller’s experiment was greatly more precise than hewas given to understand. It has revealed the great advantages of both a study of thephysics of a phenomenon and the application of dimensional analysis before usingstatistical analysis to determine the form of the regression function that representsexperimental data. These matters have been considered in detail elsewhere ( [2],Ch. 9).


The great value of dimensional analysis in synthesizing experimental data has beendemonstrated both here and in Chapters 1 and 7. The discussion here shows alsohow care has to be taken in interpreting results in both their significance and theiraccuracy.

Exercises

10.1 Tests are to be made upon the trajectory of artillery shells by firing identicalshells at a set of values of the muzzle velocity and the muzzle inclinationto the horizontal. Consider how the test results may be expressed in termsof non-dimensional groups and how some of these groups may be assignedfixed values in the test.

10.2 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. Liv-

erpool, 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 non-

linear functions, J. App. Stat., Vol. 20, No. 1, pp. 57–67, 1993.

Chapter 11Similar Systems

So scientists have reproached him for having sometimeslavished his calculus on physical hypotheses, or even onmetaphysical principles, of which he had not sufficientlyexamined the likelihood and solidity. [Eloge de M. Euler, 1783]R. Giacomelli, E. Pistolesi

Notation

c Mean wing chordC Bulk compressibility modulous; ion concentrationD Diffusion coefficient; leg widthE Electric fieldfa Aerodynamic frequency parameterF Force; Faraday constantg Gravity accelerationh Heat transfer coefficientis Electrostatic streaming currentk1, k2 CoefficientsL Leg length` Scale sizen Beat frequencyp PressurePQ Heat rate

Re Reynolds numberT TemperatureU VelocityV Velocityx Joint location measurey Bone lengths

˛ Coefficientˇ Coefficient of volume expansion" Dielectric coefficient� Thermal conductivity� Viscosity� Density� Surface tension coefficient; charge density


274 11 Similar Systems

11.1 The Concept of Similitude

The concept of similitude and its applications does not appear to be universallydescribed. Palacios [1] refers to it by writing “– the Principle of Similitude on whichare based the experiments with reduced models –”. Thus his idea is the use of theresults of dimensional analysis for designing model tests as has been described herein Chapter 9. Duncan had a similar approach for he wrote that: “The determinationof the quantitative conditions for similarity of behaviour is an essential part of thestudy of physical similarity and these are most conveniently found by the techniqueof dimensional analysis.” [2].

Again, Isaacson and Isaacson link similarity to the relationships derived fromdimensional analysis that are used to validate model testing and which can enablescaling parameters to be determined [3]. Massey adopts the same idea of the appli-cation of the idea of similarity to model testing [4].

In contrast, Kline advocates its use in place of dimensional analysis by consid-ering instead certain physical quantities that govern a phenomenon [5]. However,initially he limits these to mechanical forces and length ratios. He specifically ex-cludes generalised forces “– such as in irreversible thermodynamics.”. He continueshis discussion by introducing forms of energy and also factors defining the thermo-dynamic properties of a system. As mentioned earlier, this idea of similar systemsbased upon kinematics and dynamics was first used by Newton. This was eventuallyovertaken by the growing use by Rayleigh of dimensional analysis.

Here the idea of similitude is used for two purposes. First, to gain physical insightinto a solution that has been obtained using dimensional analysis. This is becausethe latter form of solution, as presented here, is both rigorous and straightforwardespecially 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 phenomenoncannot be stated in terms of functional relations so that the very basis of dimensionalanalysis is lacking. This, for example, provides a different application to anatomicalstudies 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 toa 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 justa 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 theproduct of a mass and an acceleration; the former is represented by �`3, the latterby u � du=ds which is represented by u.u=`/. The inertial force is then representedby �`3u2=` D �u2`2. A viscous force is represented by the product of an area anda shear stress or by `2�.u=`/. The ratio of these two forces is .�V `/=� which iscalled the Reynolds number.

This non-dimensional product has been named after Osborne Reynolds eventhough Stokes first demonstrated the dependence of a flow on this parameter [6]some 35 years before Reynolds famous paper [7]. This Reynolds number is giventhe symbol Re – Duncan says that Re is an etymological abomination as that shouldmean .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 asnearly four decades ago, lists 282 of these [4]; others abound. The result is duplica-tion, 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 repre-sentation of the inertia force is,

Body force

Inertia forceD F

� U 2`2:

This is the non-dimensional group introduced in Section 1.6. It has been called theEuler 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

Gravity forceD �U 2`2

�g`3D U 2

g`:

This ratio is named the Froude number which then represents this ratio of these twoforces in a fluid flow.

A surface tension force can be represented by �`; that is the surface tension timesa length. Dividing this into the inertia force above gives:

Inertia force

Surface tension forceD �U 2`2

�`D �U 2`

�:

This ratio of forces is called the Weber number.


The buoyancy force in a liquid resulting from a temperature difference is repre-sented by �g`3ı�. With the coefficient of volume expansion given by:

ˇ � � 1

�

@�

@T:

Then the buoyancy force is represented by:

g�`3ˇ�T :

Multiplying this force by the representation of the inertial one and then dividing bythe square of that of the viscous force from above gives:

Inertia force � Buoyancy force

.Viscous force/2D �U 2`2g�ˇ�T `3

.�U `/2D �2gˇ`3�T

�2

which is named the Grashof number. This non-dimensional group is now seen torepresent a combination of three forces.

In a compressible flow of a gas the stress is represented by ıp and the strainby ı�=� which latter is non-dimensional. The bulk compressibility modulus, C , isdefined as:

C � 1

�

d�

dp

so that:

Stress

StrainD 1

C:

The stress force is then represented by:

Stressforce D Strain`2

C :

Then we have that:

Inertia force

Stress force=StrainD �U 2`2C

`2D U 2 d�

dp

which is the Mach number.Considering now the phenomenon of heat transfer we have that the convection

heat rate is given by:

PQconv D h`2�T

11.2 Physical Significance of Non-dimensional Groups 277

and the conductivity heat rate is given by

PQcond D �`2 :.�T=`/

The ratio of these two heat rates is then:

PQconv

PQcondD h`2�T

�`2.�T=`/D h`

�

which is called the Nusselt number.We now turn to the phenomenon in electro-chemistry where there is an electro-

static streaming current in a fluid flow [8]. The result of the application of dimen-sional analysis is given in Section 8.16. It is as follows:

i2s

�U 4"`2D f

�"U

�0`;

D"

�0`2;

DCD�

;�U `

�

�

: (11.1)

The convection current is the charge per unit time which can be written as:

is D Charge density � Volume

Unit time

D �`3

`=U(11.2)

D �`2U :

The electrical force is given by:

Field force D Field � Charge

D E�`3 :

The Poisson relation is represented by:

�

"D E

`:

Thus:

Field force D �2`4

"

D i2s

U 2":

Dividing this by the inertia force gives:

Field force

Inertia forceD i2

S

�U 4"`2


which is the first non-dimensional group in Equation 11.1.The conduction current is represented by:

Conduction current D �`2E

D ��`3

"

so that:

Convection current

Conduction currentD "U

�`

which is the second non-dimensional group in Equation 11.1.The diffusion current is represented by:

Diffusion current D �F .DCrCC � D�rC�/ `2

D �FD` .CC � C�/

D �D�` :

Thus:

Diffusion current

Conduction currentD � D"

�`2:

This is the third non-dimensional group in Equation 11.1. The fourth group is justa measure of the ratio of the mobility of the positive and negative ions whilst thefifth 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 physicalsignificance they do not give a direct numerical measure or even of the order of theratios 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 whichthe viscous force dominates is Re � 4 whilst that for the case where the flow isindependent of the numerical value of the viscosity is about 104. Whilst this givesa physical meaning to this non-dimensional product it is nowhere near a measureof the order of magnitude of this ratio. Again, for example, on a slender aerofoilat a low angle to an oncoming flow, when the value of this non-dimensional groupis of the order of 108, the force due to the pressures can be of the order of onlyone 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 11.1are typically of the order of 10�14.1

11.4 The Use of Similarity

When the physics of a phenomenon is known there have been attempts to replacedimensional analysis by adopting concepts of similarity. This is done by specify-ing the existence of a group through its physical representation as, for example, bygroups such as considered in Section 11.2. above. This procedure can be acceptablein problems in simple kinematics and Newtonian mechanics.

It does, however, raise considerable difficulties in more complex phenomenonsuch as with compressible flow. Whilst groups such as the Reynolds, Mach, andGrashof numbers can be deduced in this way there is no similar argument that willbring in the ratio of the specific heats. There is an even greater difficulty in that thereis no argument in terms of such non-dimensional ratios which will deduce the needfor inclusion of any units-conversion factors.

Thus such an approach based upon similarity is very limited in these sorts ofapplication.

11.5 Similarity in Anatomy

D’Arcy Thompson, starting in the nineteenth century, made great use of similarityin anatomy [9, 10].

One interesting example that he reported was on the measurements that he madeof 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 thathe reported were as given in Table 11.1.

Here values of x are numerical counters for the joints on a scale of 0–1. Valuesof y are measured lengths on a scale of 0–100. Values from this Table are shownplotted in Figure 11.2. Here the separate curves become a function of L=D. Forcorrelation, 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 k1k2.�1/.1 � y/.k2�1/dy :

1 This demonstrates the remark made in the Preface that dimensional analysis cannot give numer-ical values. Yet there have been proposals that all non-dimensional groups have a numerical valueof the order of unity.


Figure 11.1 Reproduction ofleg proportions measured byD’Arcy Thompson (see [9])

Table 11.1 (See [9])

Joint indicators (Figure 11.1) a b c dNumerical locations; x 0.25 0.5 0.75 1.00Length values; y

Ox; 0 18 27 42 100Sheep; 0 10 19 36 100Giraffe; 0 5 10 24 100

Animal; Ox Sheep GiraffeL=D 2.5 3.7 6.4

Figure 11.2 Plot of the pro-portions of the leg elements:symbol codes; ı, Ox; �,Sheep; �, Giraffe

11.5 Similarity in Anatomy 281

Figure 11.3 Correlation of curve slopes

At y D 0:�

dy

dx

�

0D 1

k1k2

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 sothat from Equation 11.4 we have that:

k2 D ˛L

D:

Thus Equation 11.3 becomes:

ln.1 � x/ D ˛L

Dln.1 � y/ : (11.5)

This correlation is confirmed in Figure 11.4. The regression straight line gives ˛ D0:923. In Figure 11.5 the plot is of values computed from Equation 11.5, denotedby yc and the values in Table 11.1. This Figure shows the final correlation accuracywhere values of y are compared with those calculated from Equation 11.5. Thecalculated 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 verydisparate 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-


Figure 11.4 Correlation plot

Figure 11.5 Final correlationplot

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 frequencyparameter, n is the beat frequency, c is the mean wing chord and U is the flightspeed [13].

A review of the use of non-dimensional groups in assessing biological phenom-ena has been given by Vogel [14]. This demonstrates the wide range of applicationsto these phenomena. A further general contribution was made in 1977 [15].

References 283


It is seen that similitude is particularly useful when there is no way of starting dimen-sional analysis with the statement of an equation forming a functional relationship.Thus it is particularly applicable to the study of animal structure and behaviour. Likedimensional analysis it does not give a numerical answer but it enables the obtainingof correlations between variables.

Exercises

11.1 It has been proposed that the comfortable walking speed of humans accordswith the legs swinging as if they were pendulums at their natural frequencyof swing. Show that then the walking speed is proportional to the square rootof a person’s height

11.2 The medical profession measures the level of obesity in human beings bythe value of a ‘Body-mass index’, B . This is defined by B D m=h2 wherem is the body mass in kilograms and h is the height in metres. Comment onthe validity of this criterion as used for this purpose. Derive a valid one fromthe assumption that obesity is measured as a ratio of the mass of fat contentto the total mass. A criterion that can be readily measured is required.

11.3 For a member of a rowing eight, assume that the force exerted by the musclesis proportional to the square of the rower’s size and the distance over whichthis force is exerted is proportional to the size. Then assume that for sucha slender boat the wave resistance is negligible so that the resistance thenis proportional to the square of the velocity and the square of the boat sizerelated 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.


8. J C Gibbings, E T Hignett. Dimensional analysis of electrostatic streaming current, Elec-trochim. 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 ADerivation of Dimensions of Quantities

Notation

A AreaB Magnetic flux densityc ConcentrationC Electrical capacitanceCV Coefficient of specific heatD Electric displacementDi Diffusion coefficientE Electric field; illuminationF Force; luminous fluxh Specific enthalpyH Magnetic field strengthI Luminous intensityj Electric current densityk Thermal conductivityL Electrical inductance; luminancem Massn Normal to isothermsPni Diffusion flux rateP Magnetic pole strength; pressureq Electrical chargePQ Heat rateR Electrical resistanceS Specific entropyT Temperatureu Specific internal energyU Internal energyW WorkPW Power


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 mechanicalquantities 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 ; (A.1)

r �D D q

L3; (A.2)

r �H D j C @D

@t; (A.3)

B D �H ; (A.4)

r � B D 0 ; (A.5)

r �E D �@B@t: (A.6)

(A.7)

2. Permittivity:

F D q2

"r2I "�

A2T2

L2 � T2

MLD A2T4

ML3

3. Magnetic pole strength:

W D 4�ipI p�ML2

T2 � 1

AD ML2

AT2

4. Permeability:

F D p2

�`2I ��

�ML2

AT2

�2

� 1

L2� T2

MLD ML

A2T2

A.2 Magnetic Units 287

5. Electrical potential:

i� D W�ML2

T3I ��

ML2

AT3

6. Electrical field:

E D @�

@n�

ML

AT3

7. Electrical conductivity:

j D �EI ��A

L2 � A2T3

MLD A2T3

ML3

8. Electrical capacitance:

C D q

��

AT

1� AT3

ML2 D A2T4

ML2

9. Electrical resistance:

R D �

i�

ML2

AT3

1

AD ML2

A2T3

10. Electrical inductance:

L D � �

di=dt�

ML2

AT3� T

AD ML2

A2T2

11. From Equation A.1:

D D "E�A2T4

ML3� ML

AT3D AT

L2

12. From Equation A.2:

r �D D q

`3D AT

L3

A.2 Magnetic Units

13. Magnetic field strength:

H

`D j C D

t�

A

L2 C A

L2 I H�A

L

288 A Derivation of Dimensions of Quantities

14. Magnetic flux density.

B D �H�ML

A2T2� A

LD M

AT2

A.3 Diffusion

15. Diffusivity.

Pni D �cDi

�@

@n

�ci

c

��

IDi�1

L2TL3L D L2

T

A.4 Illumination units

16. Luminous intensity, I , cd:

I�C

17. Luminous flux, F , lm:

F�I˝�C�

18. Luminance,L:

L�I

`2�

C

L2

19. Illumination,E, lux D lm m�2:

E D dF

dA�

C�

L2

20. Exposure (camera),H , lux sec:

H D Et�C�T

L2

21. Mechanical equivalent of light, P0, watts lm�1:

F D 1

P0� dE

dt

P0�ML2

T3 � 1

C�D ML2

T3C�

A.6 Mechanical Units 289

A.5 Thermal Units

22. Specific internal energy, u.

u D U

m�

ML2

T2

1

MD L2

T2

23. Thermal conductivity, k.

PQA

D k@T

@nI k�

ML2

T3

1

L2

L

D ML

T3I

24. Coefficient of specific heat, CV :

CV � u

T�

L2

T2

25. Specific enthalpy, h:

h ��p

�C u

�

�

�M

LT2

L3

MC L2

T2

�

D L2

T2

26. Specific entropy, s:

s D 1

m

Q

T�

1

M

ML2

T2

1

D L2

T2

A.6 Mechanical Units

27. Viscosity, �, kg m�1 s�1:

� D �du

dn

��ML

T2

1

L2

L

1

T

LD M

LT

Name Index

A

Acosta A J 82, 254Advisory Committee for Aeronautics 93Al-Shukri S M 218Annual Report of the Aeronautical Society of

Great Britain 93Anon 254Arnold G C 271Augustine (Saint) 53

B

Bacon F (Lord Verulam) 53Bacon R H 93Bairstow L 93Baker A G 271Barr D I H 82Bearman P W 217Benjamin T B 82Birkhoff G 82Booth H 93Brennen C 284Bridgman P W 53, 82, 93, 114, 147, 149Brokaw C J 284Bronowski J 54, 92Brook E J 271Brooke Benjamin T 114Buckingham E 23, 53, 81, 93

C

Cardwell D S L 92Cheers F 254Chichester F 253Chow V T 254Churchill S W 146

Crowe C T 82

D

D’Alembert 179Davies P T 271Drabble F 255Duncan W J 1, 283Dunn J F 53Dyer K R 254

E

Esnault-Pelterie R 53, 82Estermann I 218

F

Finney D J 271Focken G M 81, 114Fourier J B J 53, 92Fox L 54

G

Gessler J 114Giacomelli R 93, 273Gibbings J C 23, 53, 81, 93, 114, 147, 177,

217, 218, 254, 271, 284Goksel O T 254Goldstein S 23, 82, 92, 147, 218, 254Goodwin J E 218Gray V H 218Green S L 54

H

Hall D J 254Hawkes N 53

291

292 Name Index

Hawking S 54Hignett E T 218, 284Huntley H E 114

I

Inglis C (Sir) 55Ipsen D C 82Isaacson E de St Q 82, 283Isaacson M de St Q 82, 283

J

Jeffreys H 23, 53John J E A 218Jones N 254Jones R V 95

K

Keenan J H 53, 81Kelvin Lord 23Kline S J 54, 218, 283Kroon R P 54Kuethe A M 147

L

Lamb H 254Langhaar H L 82Lei H 218Liepmann H W 147, 254Lighthill M J 284

M

Macagno E O 92Mackey A M 177, 271Madadnia J 218Manning R 254Massey B S 82, 114, 271, 283Masuda S 218Maxwell J C 23, 53, 93McFarland M W 93, 254Melville-Jones B 93Miller E J 271Mironer A 254Morrison L V 53Munk M M 114

N

Newton I (Sir) 92

P

Palacios J 283Pankhurst R C 54, 147Piercy N A V 23, 93, 218Pistolesi E 93Plotinus 53Pomerantz M A 92Prandtl L 93, 115, 146, 218Puckett A E 147, 254

R

Rayleigh J W S (Lord) 81, 93, 147Reed A 219Reynolds O 25, 93, 283Riabouchinsky D 93, 147Roberson J A 82Robinson R A 218Rott N 83, 92, 93Rowe P N 271Russell B 53, 114

S

Sabersky P H 82, 254Sage W 218Saluja G S 177, 271Schetzer J D 147Schmitz F W 254Schon G 218Sedov L I 53Shapiro A H 254Southwell R V 54, 254Staicu C I 82Stokes G G 92, 283Stokes R H 218Szucs E 92

T

Taylor E S 53, 82Thompson D’Arcy W 92, 284Thomson J 92Thomson W (Sir Lord Kelvin) 23Tilly G P 218Truscott G F 218Turnhill R 219

V

Van Driest E R 82Vaschy A 93Vogel S 284Von Glahn U H 218Von Karman, Th 254

Name Index 293

W

Waismann F 53Wakefield G L 53Wang L-Z 218Washizu M 218White M 146Wilkie D 271Williams W 114Wilson W 53

Wu T Y-T 284Wu Z-N 218

Y

Yalin M S 177, 254

Z

Zemansky M W 53

Subject Index

A

analytical results 179anatomy, similarity 279angle, units-conversion factor 74

dimensions 106

B

beam, bending 116boundary layer, transition fixing 242

C

collisions, spring restraint 227compact solutions 15complete equations 28, 37compressible flow, thermal convection 124concepts, derived 28, 39

measure 27nature 27primary 27types 43

consistent equation 5

D

data, synthesis 174points 7

definition, concepts 28design, model testing 249dimensional analysis, benefits 19, 150

statistics 264dimensional system 5dimensionless groups, non effective 234dimensions 3dimensions, angle 106

choice of 96electrical 107mass and force 97mass and quantity 104mass and volume 100number reduction 45physical quantities 6symbolism 85symbols 5temperature and quantity 102vectorial 109

dissipation function 126, 128drag coefficient 12drag, fluid 18

particle 192wave 242

E

Einstein relation 216electrical conductivity, liquids 261electrical dimensions 107electromagnetic field energy 64electrostatic charging, fluids 199

experimental comparison 202physics 200

equality, dimensional 33equations, complete 37

uncoupled 121errors, dimensionless 256Euler number 275experiment, errors 256

number of readings 19range of application 156validity criterion 167

experimental data 6experimental limits 10

295

296 Subject Index

experimental results, interpretation 270extension 29extrapolated solution 120extrapolation, partial 239

F

flow patterns, atmospheric 236fluid mechanic force 16force, definition 32frictional resistance 87Froude number 240, 275function, general 11functional relationship 11

operations, limit 36

G

gases, kinetic theory 204Stokes law inapplicability 215

Grashof number 142, 276gravitation law 119

H

heat exchanger 66history, dimensional analysis 83

first stage 84similitude 84

Hooke’s lawinapplicability 227

hydraulic turbine, deduction of results 258

J

jet flow 190impact of 63

K

Kelvin (Lord) 3kinetic theory of gases 204

diffusion 213Einstein relation 216electrical mobility 214internal energy 207mean-free path 205pressure, temperature 209thermal conductivity 212viscosity 211

L

length 29

linear scales 28logical steps 27

M

Mach number 128, 276magnitude, constant relative 33, 35

pi-groups 121mass, oscillating 13, 62

non-linear 62Maxwell 5measurement 3

limitations 162mixing length, turbulence 186mixing processes 264model testing, application 220

engineering design 249essence of 221

modelling, analytical input 238complexity 247distorted 244full-scale comparison 234

N

non-dimensional groups, numerical signifi-cance 278

P

partial modelling, limits 233particle, abrasion 191

fragmentation 198impact classes 197shape 199

physics, assessment 250pi theorem 15, 57

general results 76generalised 59previous proofs 67transformation 60

pi-groups 11magnitude 121

pipe flow, liquid 72Reynolds’ experiment 85, 90

planetary motion 118Prandtl number 127pressure coefficient 243properties, thermodynamic 129

Q

quantity, definition 32

Subject Index 297

R

Rayleigh–Riabouchinsky problem 134regression analysis, non-dimensional 266relative magnitude 28results, experimental 258Reynolds number 12, 127

physical significance 274

S

scale modelling, partial limit 233scales, linear 28similarity 273

anatomy 279similitude 274solutions, asymptotic 120

extrapolated 120statistics, non-dimensional variables 264structural frame 75structures, scale model limits 231struts, deformation 228supplementation 116symbolism, dimensions 85synthesis of data 174Systeme International d’Unites 5

T

thermal convection, compressible flow 124incompressible flow 131natural 139

thermodynamic properties 129time, definition 30transformer, electrical modelling 225turbulence, complexity 180

jet flow 190log-law 188mixing length 186physical nature 180power law 184

U

uncoupled equations 121units 3units, reference measure 27units-conversion factors 3, 28, 34, 47

angle 74dimensionless groups 48dimensions 44inclusion 47

universal constants 47

V

validity criterion 167variables, choice of 72

dependent, alternate 155effectiveness 165influence of 159missing 157reduction of 151superfluous 157

vibration, stretched wire 111

W

wear rate, particle abrasion 195Weber number 275Windmill, model test 222

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