The book provides a summary of the historical evolution of dimensional analysis, and frames the problem of dimensions, systems of units and similarity in a vision dominated by the conventions that formalise even the exact sciences.The first four chapters address the definitions, with few dimensional analysis theorems and similarity criteria. There is also the analysis of self-similarity, both of first and second kind, with a couple of completely solved problems, framed within the group theory. From chapter 5 onward, the focus is on applications in some of the engineering sectors. The number of topics is necessarily limited, but, almost always, there are details, calculations and treatment of assumptions. The book contains descriptions of some of the experimental apparatuses currently used for the realisation of physical models, such as the wind tunnel, the shaking table, the centrifuge, and with the exclusion of many others, which can be found in specialist monographies. Measurement techniques and instrumentation and statistical data processing is also available in other books.
Some more specific notions, required by the context, are reported in the appendix, where appears also the description of numerous dimensionless groups, all of engineering interest, but with the exclusion of many others related to physical processes of electrical nature or physics of particles. A glossary lists the meaning of some specific terms typical of dimensional analysis and used in the book.
Author(s): Sandro G. Longo
Series: Mathematical Engineering
Publisher: Springer
Year: 2022
Language: English
Pages: 459
City: Cham
Preface
Acknowledgements
Introduction
Brief History of Dimensional Analysis: The Different Aims and Approaches
Overview of Methods and Application Scenarios
Perspectives and Further Developments
References
Contents
About the Author
Part I The Methods
1 Dimensional Analysis
1.1 Classification of Physical Quantities
1.2 The Systems of Units
1.2.1 Monodimensional Systems
1.2.2 Omnidimensional Systems
1.2.3 Multidimensional Systems
1.2.4 The Dimension of a Physical Quantity and the Transformation of the Units of Measurement
1.2.5 Some Writing Rules
1.3 Principle of Dimensional Homogeneity
1.3.1 The Arithmetic of Dimensional Calculus
1.4 The Structure of the Typical Equation …
1.4.1 The Method of Rayleigh
1.4.2 The Method of Buckingham (Π-Theorem, Buckingham's Theorem)
1.4.3 A Further Demonstration of Buckingham's Theorem
1.4.4 Interpretation Through Group Theory
1.4.5 The Generalised Proof of the Π-Theorem: Ipsen's Procedure Leading to Gibbing's Demonstration
1.4.6 The Criterion of Linear Proportionality
1.4.7 A Corollary of Buckingham's Theorem
References
2 Handling Dimensionless Groups in Dimensional Analysis
2.1 The Dimensional and Physical Relevance of Variables
2.1.1 Dimensionally Irrelevant Variables
2.1.2 Physically Irrelevant Variables
2.2 Reducing the Number of Dimensionless Groups
2.2.1 Vectorisation of Quantities
2.2.2 The Discrimination of Fundamental Quantities
2.2.3 The Process of Rationalisation with the Change in the Fundamental Quantities and the Grouping of the Variables
2.3 Formalisation of Matrix Methods
2.3.1 A Further Generalisation of the Matrix Technique for the Calculation of Nonzero-dimension Power Functions
2.3.2 The Number of Independent Solutions
2.4 A Recipe for Dimensionless Groups
2.4.1 Some Properties of Dimensional and Dimensionless Power Functions
References
3 The Structure of the Functions of the Dimensionless Groups, Symmetry and Affine Transformations
3.1 The Structure of the Functions of the Dimensionless Groups
3.1.1 The Structure of the Function of Dimensionless Groups is Necessarily a Power Function
3.1.2 The Structure of the Function of Dimensionless Groups is Necessarily a Non-power Function
3.1.3 The Structure of the Function of Dimensionless Groups is Possibly a Power Function
3.2 The Use of Symmetry to Specify the Expression of the Function
3.3 Group Theory and Affine Transformations for Self-similar Solutions
3.3.1 The Non-dimensionalisation of Algebraic Equations and Differential Problems
3.3.2 Methods of Identification of Self-similar Variables for Complete (First-Kind) Similarity
3.3.3 The Derivation for Incomplete (Second-Kind) Similarity
References
4 The Theory of Similarity and Applications to Models
4.1 Similarities
4.1.1 Geometric Similarity
4.1.2 Kinematic Similarity
4.1.3 Dynamic Similarity
4.1.4 Dynamic Similarity for Interacting Material Particle Systems
4.1.5 Dynamic Similarity for Rigid Bodies
4.1.6 Affine Transformations of Trajectories and Conditions of Distorted Similarity
4.1.7 The Constitutive Similarity and the Other Criteria of Similarity
4.2 The Condition of Similarity on the Basis of Dimensional Analysis
4.3 The Condition of Similarity on the Basis of Direct Analysis
4.4 An Extension of the Concept of Similarity: Some Scale Laws in Biology
4.4.1 A Derivation of the Exponent of Kleiber's Law
References
Part II The Applications
5 Applications in Fluid Mechanics and Hydraulics
5.1 The Dimensionless Groups of Interest in Fluid Mechanics
5.1.1 The Linear Momentum Balance Equation
5.1.2 Boundary Conditions
5.2 The Conditions of Similarity in Hydraulic Models
5.2.1 Reynolds-Euler Similarity
5.2.2 Froude Similarity
5.2.3 Mach Similarity
5.2.4 Similarity in Filtration in the Darcy and Forchheimer Regimes
5.3 Geometrically Distorted Hydraulic Models
5.4 Scale Effects in Hydraulic Models
5.5 Analogue Models
References
6 Applications to Heat Transfer Problems
6.1 The Relevant Dimensionless Groups
6.1.1 The Heat Exchanger
6.1.2 Heat Transfer in Nanofluids
6.1.3 Heat Exchange in the Presence of Vapours
6.1.4 The Heat Exchange of a Homogeneous Body
6.2 Heat Transfer in Fractal Branching Networks
References
7 Applications to Problems of Forces and Deformations
7.1 Classification of Structural Models
7.2 Similarity in Structural Models
7.3 Statically Loaded Structures
7.3.1 Scale Ratios in Undistorted Structural Similarity for Static Elastic Models
7.3.2 The Plastic Behaviour
7.3.3 Models of Reinforced or Pre-compressed Concrete Structures
7.3.4 The Bending of a Beam Made of Ductile Material
7.3.5 The Phenomenon of Instability
7.3.6 The Plastic Rotation of a Reinforced Section
7.4 Dynamically Loaded Structures
7.4.1 The Action of a Periodic Force
7.4.2 The Action of an Impulsive Force: Impact Phenomena
7.5 Structures Subject to Thermal Loads
7.6 The Vibrations of the Elastic Structures
7.7 Aeroelastic and Hydroelastic Models
7.8 Models with Explosive Loads External to the Structure
7.9 Dynamic Models with Earthquake Action
7.10 Scale Effects in Structural Models
References
8 Applications in Geotechnics
8.1 The Shaking Table
8.1.1 Conditions of Similarity for a Model on a Shaking Table
8.2 The Centrifuge
8.2.1 Scales in Centrifuge Models
8.2.2 Scale Effects and Anomalies in Centrifuges
8.2.3 Contaminant Transport Models in Centrifuges
8.2.4 The Similarity in Dynamic Models in Centrifuges
8.2.5 Similarity in Tectonic Processes
8.3 Some Applications for the Solution of Classic Problems
8.4 Dimensional Analysis of Debris Flows
8.4.1 The Physical Process of Cliff Recession
References
9 Applications in Wind Tunnel Technology
9.1 Classification of Wind Tunnels
9.2 Aeronautical and Automobile Wind Tunnels
9.2.1 Environmental Wind Tunnels
9.3 Scale Effects in Wind Tunnels
9.4 Models for Multiphase Flows: An Application to Wind Waves
References
10 Physical Models in River Hydraulics
10.1 Similarity for a Non-prismatic Stationary (and Non-uniform) Stream
10.1.1 Distorted Models of Rivers and Canals in the Gradually Varied Flow Regime
10.1.2 The Scale Ratio of the Friction Coefficient and Roughness
10.1.3 Distorted Models of Rivers and Canals in the Generic Flow Regime
10.2 Models in the Unsteady Flow Regime
10.3 Inclined Physical Models
References
11 Physical Models with Sediment Transport
11.1 Conditions of Similarity in Rivers in the Presence of a Movable Bed
11.1.1 The Undistorted Models: Reynolds Number for the Sediments rightarrowinfty
11.1.2 The Undistorted Models: Reynolds Number for the Sediments <70
11.2 Hypothesis of Sediment Transport Independent of the Depth of the Water Stream
11.2.1 Hypothesis of Sediment Transport Independent of the Depth of the Water Stream and Reynolds Number for the Sediments rightarrowinfty
11.3 The Bottom in the Presence of Dunes, Ripples and Other Bedforms: The Calculation of the Equivalent Roughness
11.3.1 The Conditions of Similarity for Sediment and Water Streams in the Presence of Bedforms
11.4 Time Scales in Distorted Movable-Bed Models
11.5 Localised Phenomena
11.6 The Modelling of Sediment Transport in the Presence of Waves
11.6.1 The Similarity of Sediment Transport Forcing Actions (Waves and Currents)
11.6.2 The Hypothesis of a Dominant Bed Load
11.6.3 Hypothesis of Dominant Suspended Load
References
Appendix A Homogeneous Functions and Their Properties
Appendix B Relevant Dimensionless Parameters (or Groups or Numbers)
Appendix Glossary
Appendix General Bibliography
Author Index
Subject Index
