This book deals with these two tools, with a focus on the Rayleigh method and the Vaschy-Buckingham method. It deals with the homogeneity of the equations and the conversion between the systems of units SI and CGS, and presents the dimensional analysis approach, before addressing the similarity of flows.
Dimensional Analysis and Similarity in Fluid Mechanics proposes a scale model and presents numerous exercises combining these two methods. It is accessible to students from their first year of a bachelor?s degree.
Table of Contents
Foreword ix
Preface xi
Introduction xiii
Chapter 1. Homogeneity of Relationships and Conversion of Units 1
1.1. Introduction 1
1.2. Definitions of the basic SI units 2
1.2.1. Definition of the meter as adopted in 1983 2
1.2.2. Definition of the kilogram 2
1.2.3. Definition of the second adopted in 1967 3
1.2.4. Definition of the ampere adopted in 1948 4
1.2.5. Definition of Kelvin adopted in 1967 4
1.2.6. Definition of a mole 5
1.2.7. Definition of the candela adopted in 1979 5
1.3. Additional quantities and SI derived quantities 5
1.4. Rules for the use of units 7
1.4.1. Unit name 7
1.4.2. Unit symbols 8
1.4.3. Compound symbols 8
1.5. Exercises 9
1.5.1. Exercise 1: calculation of dimensions 9
1.5.2. Exercise 2: homogeneity of relationships 15
1.5.3. Exercise 3: dimension of the constants of an equation 22
1.5.4. Exercise 4: equation for perfect gases 23
1.5.5. Exercise 5: unit conversions 24
Chapter 2. Dimensional Analysis: Rayleigh Method and Vaschy-Buckingham Method 29
2.1. Introduction 29
2.2. Definition of dimensional analysis 30
2.3. The Rayleigh method 31
2.3.1. Example of application: the period of the swinging of a pendulum 31
2.4. Vaschy-Buckingham method or method of π 34
2.4.1. The Vaschy-Buckingham theorem 35
2.4.2. Formation of terms in π 36
2.4.3. Application example: linear pressure drop calculation 37
2.5. Exercises: homogeneity method or Rayleigh method 41
2.5.1. Exercise 1: Reynolds number 41
2.5.2. Exercise 2: the Weber number 44
2.5.3. Exercise 3: capillary number 46
2.5.4. Exercise 4: power of a propeller 47
2.5.5. Exercise 5: flow through an orifice with thin walls 50
2.5.6. Exercise 6: a linear pressure drop along a horizontal pipe 52
2.5.7. Exercise 7: force exerted by a fluid on a body 57
2.5.8. Exercise 8: oscillation of a liquid in a U-shaped tube 59
2.5.9. Exercise 9: a falling ball 61
2.5.10. Exercise 10: implosion time of an air bubble 66
2.5.11. Exercise 11: vibration of a drop of water 68
2.5.12. Exercise 12: drag force of water on a ship 70
2.6. Exercises: Vaschy-Buckingham method or method of π 72
2.6.1. Exercise 13: pressure drop in a pipe of circular cross-section 72
2.6.2. Exercise 14: friction forces on a flat plate 75
2.6.3. Exercise 15: drag force exerted on a sphere 79
2.6.4. Exercise 16: hydraulic jump 84
2.6.5. Exercise 17: flow through a thin-walled spillway with a horizontal crested 86
2.6.6. Exercise 18: flow through a triangular weir 89
2.6.7. Exercise 19: volume of a bubble 92
2.6.8. Exercise 20: flow through an orifice 94
2.6.9. Exercise 21: sudden narrowing of a section 98
2.6.10. Exercise 22: capillary tube 102
2.6.11. Exercise 23: deformation of a bubble 106
2.6.12. Exercise 24: laminar dynamic boundary layer on a flat plate 108
2.6.13. Exercise 25: power of a stirrer 115
Chapter 3. Similarity of Flows 119
3.1. Definition and principle of similarity 119
3.1.1. Geometric similarity 119
3.1.2. Kinematic similarity 120
3.1.3. Dynamic similarity 121
3.1.4. Similarity conditions for viscous, incompressible, non-heavy fluids (Reynolds similarity) 124
3.1.5. Similarity conditions for non-viscous, incompressible, heavy fluids (Reech-Froude similarity) 124
3.1.6. Similarity requirements for non-viscous, non-compressible, heavy fluids 125
3.1.7. Conditions of similarity of turbulent flows 126
3.1.8. Distortion of the model 127
3.2. Exercises: similarity of flows 127
3.2.1. Exercise 1: similarity between ships 127
3.2.2. Exercise 2: similarity of centrifugal pumps. 130
3.2.3. Exercise 3: volumetric pumps with small dimensions 136
3.2.4. Exercise 4: characteristics of a centrifugal pump 138
3.2.5. Exercise 5: test of an automobile in a wind tunnel 140
3.2.6. Exercise 6: power ratios (p model / p prototype) of a pump 142
3.2.7. Exercise 7: flow in a pipe 145
3.2.8. Exercise 8: viscous force on a rotating disk 146
3.2.9. Exercise 9: development study of a hydroelectric gallery 151
3.2.10. Exercise 10: movement of solid matter by a water current 155
3.2.11. Exercise 11: a tapered body 159
3.2.12. Exercise 12: model of a seaplane 162
3.2.13. Exercise 13: tide study 164
3.2.14. Exercise 14: transient gas flow 168
3.2.15. Exercise 15: model of a torpedo 170
3.2.16. Exercise 16: movement of a ball in a fluid 174
3.2.17. Exercise 17: similarity of the movement of an airship 177
3.2.18. Exercise 18: resistance to the movement of a ship 180
3.2.19. Exercise 19: mixing tank 185
3.2.20. Exercise 20: friction on a prototype probe 192
Appendices 195
Appendix 1. Some Dimensionless Numbers Used in Fluid Mechanics 197
Appendix 2. Coefficients of Conversion to the International System or to the English System 201
References 205
Index 207
