Dimensional Analysis Beyond the Pi Theorem (eBook)

Bahman Zohuri (Autor)

eBook Download: PDF

2016 | 1st ed. 2017
XIX, 266 Seiten
Springer International Publishing (Verlag)
978-3-319-45726-0 (ISBN)

Infos

Dimensional Analysis and Physical Similarity are well understood subjects, and the general concepts of dynamical similarity are explained in this book. Our exposition is essentially different from those available in the literature, although it follows the general ideas known as Pi Theorem.  There are many excellent books that one can refer to; however, dimensional analysis goes beyond Pi theorem, which is also known as Buckingham's Pi Theorem. Many techniques via self-similar solutions can bound solutions to problems that seem intractable.

A time-developing phenomenon is called self-similar if the spatial distributions of its properties at different points in time can be obtained from one another by a similarity transformation, and identifying one of the independent variables as time.  However, this is where Dimensional Analysis goes beyond Pi Theorem into self-similarity, which has represented progress for researchers.

In recent years there has been a surge of interest in self-similar solutions of the First and Second kind.  Such solutions are not newly discovered; they have been identified and named by Zel'dovich, a famous Russian Mathematician in 1956. They have been used in the context of a variety of problems, such as shock waves in gas dynamics, and filtration through elasto-plastic materials.

Self-Similarity has simplified computations and the representation of the properties of phenomena under investigation.  It handles experimental data, reduces what would be a random cloud of empirical points to lie on a single curve or surface, and constructs procedures that are self-similar.  Variables can be specifically chosen for the calculations.

Dr. Bahman Zohuri is founder of Galaxy Advanced Engineering, Inc. a consulting company that he formed upon leaving the semiconductor and defense industries after many years as a Senior Process Engineer for corporations including Westinghouse and Intel, and then as Senior Chief Scientist at Lockheed Missile and Aerospace Corporation. During his time with Westinghouse Electric Corporation, he performed thermal hydraulic analysis and natural circulation for Inherent Shutdown Heat Removal System (ISHRS) in the core of a Liquid Metal Fast Breeder Reactor (LMFBR). While at Lockheed, he was responsible for the study of vulnerability, survivability and component radiation and laser hardening for Defense Support Program (DSP), Boost Surveillance and Tracking Satellites (BSTS) and Space Surveillance and Tracking Satellites (SSTS). He also performed analysis of characteristics of laser beam and nuclear radiation interaction with materials, Transient Radiation Effects in Electronics (TREE), Electromagnetic Pulse (EMP), System Generated Electromagnetic Pulse (SGEMP), Single-Event Upset (SEU), Blast and, Thermo-mechanical, hardness assurance, maintenance, and device technology. His consultancy clients have included Sandia National Laboratories, and he holds patents in areas such as the design of diffusion furnaces, and Laser Activated Radioactive Decay. He is the author of several books on engineering and heat transfer.

Inhalt

About the Author 7
Preface 9
Acknowledgments 12
About This Document 13
Contents 14
Chapter 1: Principles of the Dimensional Analysis 17
1.1 Introduction 17
Units of Force and Mass 21
1.2 Dimensional Analysis and Scaling Concept 23
1.2.1 Fractal Dimension 24
1.3 Scaling Analysis and Modeling 29
1.4 Mathematical Basis for Scaling Analysis 31
Lie Group 32
1.5 Dimensions, Dimensional Homogeneity, and Independent Dimensions 34
1.6 Basics of Buckingham´s pi (Pi) Theorem 36
Theory 39
1.6.1 Some Examples of Buckingham´s pi (Pi) Theorem 42
1.7 Oscillations of a Star 55
1.8 Gravity Waves on Water 55
1.9 Dimensional Analysis Correlation for Cooking a Turkey 57
1.10 Energy in a Nuclear Explosion 64
The Method of Least Squares 67
1.10.1 The Basic Scaling Argument in a Nuclear Explosion 70
Derivation of Eq.1.25 71
1.10.2 Calculating the Differential Equations of Expanding Gas of Nuclear Explosion 73
1.10.3 Solving the Differential Equations of Expanding Gas of Nuclear Explosion 75
1.11 Energy in a High Intense Implosion 78
Note 80
1.12 Similarity and Estimating 83
1.13 Self-Similarity 85
Blasius Boundary Layer 90
1.14 General Results of Similarity 94
1.14.1 Principles of Similarity 94
1.15 Scaling Argument 95
1.16 Self-Similar Solutions of the First and Second Kind 95
Note 97
1.17 Conclusion 98
References 99
Chapter 2: Dimensional Analysis: Similarity and Self-Similarity 101
2.1 Lagrangian and Eulerian Coordinate Systems 101
2.1.1 Arbitrary Lagrangian-Eulerian (ALE) Systems 107
2.2 Similar and Self-Similar Definitions 108
2.3 Compressible and Incompressible Flows 109
2.3.1 Limiting Condition for Compressibility 113
2.4 Mathematical and Thermodynamic Aspect of Gas Dynamics 117
2.4.1 First Law of Thermodynamics 117
2.4.2 The Concept of Enthalpy 118
2.4.3 Specific Heats 119
2.4.4 Speed of Sound 122
2.4.5 Temperature Rise 124
2.4.6 The Second Law of Thermodynamics 126
2.4.7 The Concept of Entropy 127
2.4.8 Gas Dynamics Equations in Integral Form 130
2.4.9 Gas Dynamics Equations in Differential Form 132
2.4.10 Perfect Gas Equation of State 133
2.5 Unsteady Motion of Continuous Media and Self-Similarity Methods 135
2.5.1 Fundamental Equations of Gas Dynamics in the Eulerian Form 138
2.5.2 Fundamental Equations of Gas Dynamics in the Lagrangian Form 140
2.6 Study of Shock Waves and Normal Shock Waves 141
2.6.1 Shock Diffraction and Reflection Processes 143
References 143
Chapter 3: Shock Wave and High-Pressure Phenomena 145
3.1 Introduction to Blast Waves and Shock Waves 145
3.2 Self-Similarity and Sedov-Taylor Problem 146
3.3 Self-Similarity and Guderley Problem 152
3.4 Physics of Nuclear Device Explosion 157
3.4.1 Little Boy Uranium Bomb 158
3.4.2 Fat Man Plutonium Bomb 161
3.4.3 Problem of Implosion and Explosion 162
3.4.4 Critical Mass and Neutron Initiator for Nuclear Devices 172
3.5 Physics of Thermonuclear Explosion 178
3.6 Nuclear Isomer and Self-Similar Approaches 184
3.7 Pellet Implosion-Driven Fusion Energy and Self-Similar Approaches 185
3.7.1 Linear Stability of Self-Similar Flow in D-T Pellet Implosion 193
3.8 Plasma Physics and Particle-in-Cell Solution (PIC) 194
3.9 Similarity Solutions for Partial and Differential Equations 195
3.10 Dimensional Analysis and Intermediate Asymptotic 196
3.11 Asymptotic Analysis and Singular Perturbation Theory 200
3.12 Regular and Singular Perturbation Problems 201
3.13 Eigenvalue Problems 201
3.14 Quantum Mechanics 202
3.15 Summary 206
References 208
Chapter 4: Similarity Methods for Nonlinear Problems 210
4.1 Similarity Solutions for Partial and Differential Equations 210
4.2 Fundamental Solutions of the Diffusion Equation Using Similarity Method 213
4.3 Similarity Method and Fundamental Solutions of the Fourier Equation 216
4.4 Fundamental Solutions of the Diffusion Equation: Global Affinity 222
Hermite Function or Polynomial Definition 227
4.5 Solution of the Boundary-Layer Equations for Flow over a Flat Plate 228
4.6 Solving First-Order Partial Differential Equations Using Similarity Method 234
Definition 1 235
4.6.1 Solving Quasilinear Partial Differential Equations of First-Order Using Similarity 240
4.6.2 The Boundary Value Problem for a First-Order Partial Differential Equation 246
4.6.3 Statement of the Cauchy Problem for First-Order Partial Differential Equation 247
4.7 Exact Similarity Solutions on Nonlinear Partial Differential Equations 251
4.8 Asymptotic Solutions by Balancing Arguments 253
Note 256
References 257
Appendix A: Simple Harmonic Motion 259
A.1 We Start with Hooke´s Law 259
Appendix B: Pendulum Problem 263
B.1 Definition 263
Appendix C: Similarity Solution Methods for Partial Differential Equations (PDEs) 268
C.1 Self-Similar Solutions by Dimensional Analysis 268
C.2 Similarity Solutions by Stretching Transformation 270
C.3 Similarity Solution for the Rayleigh Problem 274
Index 277

Produkt-Details

• Erscheint lt. Verlag: 2.11.2016
• Zusatzinfo: XIX, 266 p. 78 illus., 36 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Bauwesen
• Schlagworte: Buckingham Theorem • Laminar and turbulent flow • Magneto-Hydro-Dynamics • perturbation theory • Self-Similarity Methods
• ISBN-10: 3-319-45726-8 / 3319457268
• ISBN-13: 978-3-319-45726-0 / 9783319457260