Dimensional analysis and similarity theory are essential in physics and engineering, particularly for designing and testing complex structures like airplanes, ships, and dams. These theories guide the conditions for model experiments and identify key parameters for fundamental effects and operations. Despite their simplicity and utility, they are often inadequately explained in textbooks and educational practices, leading to confusion and misconceptions.

The book highlights the importance of clear definitions of dimensional and dimensionless quantities and foundational concepts like the number of basic units of measurement. It critiques the superficial treatment of these topics in academia, which has occasionally led to paradoxes, such as misinterpretations in Rayleigh’s conclusions on heat emission.

Dimensional analysis is especially valuable when combined with broader physical principles, yielding significant insights in fields like turbulence, where a complete mathematical framework is lacking. The book includes new results in turbulence theory and provides detailed analyses of problems like turbulent fluid motion and Newton’s second law.

While many applications of dimensional analysis are not covered, the text aims to demonstrate standard methods and inspire the selection and formulation of new problems and experiments. The first half of the book is accessible to general readers, while the latter half requires some knowledge of hydromechanics.

Translated from the Russian by V. I. Kisin

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CONTENTS
Foreword to the First Russian Edition 7
Foreword to the Third Russian Edition 9
Foreword to the Sixth Russian Edition 10
Foreword to the Eighth Russian Edition 11
Foreword to the Ninth Russian Edition 11

CHAPTER I. General Dimensions Theory
§ 1. Introduction 13
§ 2. Dimensional and Dimensionless Quantities 14
§ 3. Fundamental and Derived Units of Measurement 15
§ 4. Dimensions Formulas 20
§ 5. On Newton’s Second Law 21
§ 6. Nature of the Functional Relations Between Physical Quantities 27
§ 7. Parameters Defining a Class of Phenomena 32
References 35

CHAPTER II. Similarity, Modelling, and Various Examples of the Application of Dimensional Analysis
§ 1. Motion of a Simple Pendulum 36
§ 2. Flow of a Heavy Fluid Through a Spillway 38
§ 3. Fluid Motion in Pipes 40
§ 4. Motion of a Body in a Fluid 44
§ 5. Heat Transfer from a Body in a Fluid Flow 51
§ 6. Dynamic Similarity and Modelling of Phenomena 54
§ 7. Steady Motion of a Solid Body in a Compressible Fluid 63
§ 8. Unsteady Motion in a Fluid 68
§ 9. Ship Motion 72
§ 10. Planing over the Water Surface 79
§ 11. Impact on Water 86
§ 12. Entry of a Cone and a Wedge at Constant Speed into a Fluid 93
§ 13. Small-Amplitude Waves on the Surface of an Incompressible Fluid 95
§ 14. Three-Dimensional Self-Similar Motions of Continuous Media 103
References 106

CHAPTER III. Applications to the Theory of Motion of a Viscous Fluid and to the Theory of Turbulence
§ 1. Diffusion of Vorticity in a Viscous Fluid 108
§ 2. Exact Solutions of the Equations of Motion of a Viscous Incompressible Fluid 110
§ 3. Boundary Layer in the Flow of a Viscous Fluid Past a Flat Plate 116
§ 4. Isotropic Turbulent Motion of an Incompressible Fluid 120
§ 5. Steady Turbulent Motion 151
References 163

CHAPTER IV. One-Dimensional Unsteady Motion of a Gas
§ 1. Self-Similar Motion of Spherical, Cylindrical, and Plane Waves in a Gas 166
§ 2. Ordinary Differential Equations and the Shock Conditions for Self-Similar Motions 175
§ 3. Algebraic Integrals for Self-Similar Motion 187
§ 4. Motions which Are Self-Similar in the Limit 196
§ 5. Investigation of the Family of Integral Curves in the (z, V) Plane 200
§ 6. The Piston Problem 208
§ 7. Problem of Implosion and Explosion at a Point 211
§ 8. Spherical Detonation 213
§ 9. Flame Propagation 220
§ 10. Collapse of an Arbitrary Discontinuity in a Combustible Mixture 225
§ 11. Problem of a Strong Explosion 229
§ 12. Point Explosion with Counterpressure Taken into Account 260
§ 13. On Modelling and on Formulas for the Peak Pressure and Impulse of Explosions 272
§ 14. Problem of a Strong Explosion in a Medium with a Variable Density 282
§ 15. Unsteady Motion of a Gas when the Velocity is Proportional to the Distance from the Centre of Symmetry 293
§ 16. On the General Theory of One-Dimensional Motion of a Gas 304
§ 17. Asymptotic Laws of Shock Wave Damping 317
References 325

CHAPTER V. Introduction to the Theory of Gas Engines
§ 1. On Averaging of Nonuniform Gas Flows in Ducts 334
§ 2. Similarity Conditions and Abstract Parameters Determining the Characteristics of Compressors 348
§ 3. On Flight Efficiency of an Ideal Propeller and an Ideal Air-Breathing Jet Engine 359
References 366

CHAPTER VI. Applications to Astrophysical Problems
§ 1. Some Observational Results 367
§ 2. On the Equations of Equilibrium and Motion of a Gaseous Mass Simulating a Star 377
§ 3. Theoretical Formulas Relating Luminosity with Mass, and Radius with Mass 382
§ 4. Some Simple Solutions of the System of Equations of Stellar Equilibrium 386
§ 5. On the Relation Between the Period of Variation of the Brightness and the Average Density for Cepheids 392
§ 6. On the Theory of the Flare-ups of Novae and Supernovae 395
References 417

Name Index 419
Subject Index 422