Dimensional Analysis Beyond the Pi Theorem

ISBN-13: 9783319833590 / Angielski / Miękka / 2018 / 266 str.

Bahman Zohuri

Spis treści

Table of Contents

About the Author

Preface

Acknowledgment

CHAPTER ONE: Principles of the Dimensional Analysis

1.1 Introduction

1.2 Dimensional Analysis and Scaling Concept

1.2.1 Fractal Dimension

1.3 Scaling Analysis and Modeling

1.4 Mathematical Basis for Scaling Analysis

1.5 Dimensions, Dimensional Homogeneity, and Independent Dimensions

1.6 Basics of Buckingham’s π (Pi) Theorem

1.6.1 Some Examples of Buckingham’s π (Pi) Theorem

1.7 Oscillations of a Star

1.8 Gravity Waves on Water

1.9 Dimensional Analysis Correlation for Cooking a Turkey

1.10 Energy in a Nuclear Explosion

1.10.1 The Basic Scaling Argument in a Nuclear Explosion

1.10.2 Calculating the Differential Equations of Expanding Gas of Nuclear Explosion

1.10.3 Solving the Differential Equations of Expanding Gas of Nuclear Explosion1.11 Energy in a High Intense Implosion

1.12 Similarity and Estimating

1.13 Self-Similarity

1.14 General Results of Similarity

1.14.1 Principles of Similarity

1.15 Scaling Argument

1.16 Self-Similarity Solutions of the First and Second Kind

1.17 Conclusion

1.18 References

CHAPTER TWO: Dimensional Analysis: Similarity and Self-Similarity

2.1 Lagrangian and Eulerian Coordinate Systems

2.1.1 Arbitrary Lagrangian Eulerian (ALE) Systems

2.2 Similar and Self-Similar Definitions<

2.3 Compressible and Incompressible Flows

2.3.1 Limiting Condition for Compressibility

2.4 Mathematical and Thermodynamic Aspect of Gas Dynamics

2.4.1 Gas Dynamics Equations in Integral Form

2.4.2 Gas Dynamics Equations in Differential Form

2.4.3 Perfect Gas Equation of State

2.5 Unsteady Motion of Continuous Media and Self-Similarity Methods

2.5.1 Fundamental Equations of Gasdynamics in the Eulerian Form

2.5.2 Fundamental Equations of Gasdynamics in the Lagrangian Form

2.6 Study of Shock Waves and Normal Shock Waves

2.6.1 Shock Diffraction and Reflection Processes

2.7 References

CHAPTER THREE: Shock Wave and High Pressure Phenomena

3.1 Introduction to Blast Waves and Shock Waves

3.2 Self-Similarity and Sedov - Taylor Problem

3.3 Self-Similarity and Guderley Problem

3.4 Physics of Nuclear Device Explosion

3.4.1 Little Boy Uranium Bomb

3.4.2 Fat Man Plutonium Bomb

3.4.3 Problem of Implosion and Explosion

3.4.4 Critical Mass and Neutron Initiator for Nuclear Devices

3.5 Physics of Thermonuclear Explosion

3.6 Nuclear Isomer and Self-Similar Approaches

3.7 Pellet Implosion Driven Fusion Energy and Self-Similar Approaches

3.7.1 Linear Stability of Self-Similar Flow in D-T Pellet Implosion

3.8 Plasma Physics and Particle-in-Cell Solution (PIC)

3.9 Similarity Solutions for Partial and Differential Equations

3.10 Dimensional Analysis and Intermediate Asymptotic

3.11 Asymptotic Analysis and Singular Perturbation Theory

3.12 Regular and Singular Perturbation Problems

3.13 Eigenvalue Problems

3.14 Quantum Mechanics

3.15 Summary

3.16 References

CHAPTER FOUR: Similarity Methods for Nonlinear Problems

4.1 Similarity Solutions for Partial and Differential Equations

4.2 Fundamental Solutions of the Diffusion Equation Using Similarity Method

4.3 Similarity Method and Fundamental Solutions of the Fourier Equation

4.4 Fundamental Solutions of the Diffusion Equation; Global Affinity

4.5 Solution of the Boundary-Layer Equations for Flow over a Flat Plate

4.6 Solving First Order Partial Differential Equations using Similarity Method

4.6.1 Solving Quasilinear Partial Differential Equations of First Order using Similarity

4.6.2 The Boundary Value problem for a First Order Partial Differential Equation

4.6.3 Statement of the Cauchy Problem for First Order Partial Differential Equation

4.7 Exact Similarity Solutions on Nonlinear Partial Differential Equations

4.8 Asymptotic Solutions by Balancing Arguments

4.9 References

APPENDIX A: Simple Harmonic Motion

APPENDIX B: Pendulum Problem

APPENDIX C: Similarity Solutions Methods for Partial Differential Equations (PDEs)

C-1 Self-Similar Solutions by Dimensional Analysis

C-2 Similarity Solutions by Stretching Transformation

C-3 Similarity Solution for the Rayleigh Problem

INDEX

O autorze

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.

Blurb

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.