A Physical Introduction to Fluid Mechanics Spring 2017 A Physical Introduction to Fluid Mechanics by Alexander J Smits Professor of Mechanical and Aerospace Engineering Princeton University Second Edition January 24, 2017 Copyright A.J Smits c 2017 Contents Preface xiii First Edition xiii Second Edition xiv Introduction 1.1 The Nature of Fluids 1.2 Units and Dimensions 1.3 Stresses in Fluids 1.4 Pressure 1.4.1 Pressure: direction of action 1.4.2 Forces due to pressure 1.4.3 Bulk stress and fluid pressure 1.4.4 Pressure: transmission through a 1.4.5 Ideal gas law 1.5 Compressibility in Fluids 1.6 Viscous Stresses 1.6.1 Viscous shear stresses 1.6.2 Viscous normal stresses 1.6.3 Viscosity 1.6.4 Measures of viscosity 1.6.5 Energy and work considerations 1.7 Boundary Layers 1.8 Laminar and Turbulent Flow 1.9 Surface Tension 1.9.1 Drops and bubbles 1.9.2 Forming a meniscus 1.9.3 Capillarity fluid Fluid Statics 2.1 The Hydrostatic Equation 2.2 Density and Specific Gravity 2.3 Absolute and Gauge Pressure 2.4 Applications of the Hydrostatic Equation 2.4.1 Pressure variation in the atmosphere 2.4.2 Density variation in the ocean 2.4.3 Manometers 2.4.4 Barometers 2.5 Vertical Walls of Constant Width 2.5.1 Solution using absolute pressures 2.5.2 Solution using gauge pressures 2.5.3 Moment balance v 10 11 11 12 13 14 15 16 17 18 19 20 21 21 23 25 25 27 28 30 30 31 31 32 34 35 35 36 vi CONTENTS 2.5.4 Gauge pressure or absolute pressure? 2.6 Sloping Walls of Constant Width 2.6.1 Horizontal force 2.6.2 Vertical force 2.6.3 Resultant force 2.6.4 Moment balance 2.7 Hydrostatic Forces on Curved Surfaces 2.7.1 Resultant force 2.7.2 Line of action 2.8 Two-Dimensional Surfaces 2.9 Centers of Pressure, Moments of Area 2.10 Archimedes’ Principle 2.11 Stability of Floating Bodies 2.12 Fluids in Rigid Body Motion 2.12.1 Vertical acceleration 2.12.2 Vertical and horizontal accelerations 2.12.3 Rigid body rotation 36 38 38 39 39 40 40 41 42 43 45 46 48 48 48 49 50 Equations of Motion in Integral Form 3.1 Fluid Particles and Control Volumes 3.1.1 Lagrangian system 3.1.2 Eulerian system 3.1.3 Small control volumes: fluid elements 3.1.4 Large control volumes 3.1.5 Steady and unsteady flow 3.1.6 Dimensionality of a flow field 3.2 Conservation of Mass 3.3 Flux 3.4 Continuity Equation 3.5 Conservation of Momentum 3.5.1 Forces 3.5.2 Flow in one direction 3.5.3 Flow in two directions 3.6 Momentum Equation 3.7 Viscous Forces and Energy Losses 3.8 Energy Equation 53 53 54 54 54 55 56 56 57 59 60 62 62 63 64 66 68 69 Kinematics and Bernoulli’s Equation 4.1 Streamlines and Flow Visualization 4.1.1 Streamlines 4.1.2 Pathlines 4.1.3 Streaklines 4.1.4 Streamtubes 4.1.5 Hydrogen bubble visualization 4.2 Bernoulli’s Equation 4.2.1 Force balance along streamlines 4.2.2 Force balance across streamlines 4.2.3 Pressure–velocity variation 4.2.4 Experiments on Bernoulli’s equation 4.3 Applications of Bernoulli’s Equation 4.3.1 Stagnation pressure and dynamic pressure 4.3.2 Pitot tube 73 73 73 74 75 75 76 76 78 79 80 81 82 83 84 CONTENTS 4.3.3 4.3.4 4.3.5 4.3.6 vii Venturi tube and Siphon Vapor pressure Draining tanks atomizer 86 87 89 89 Differential Equations of Motion 5.1 Rate of Change Following a Fluid Particle 5.1.1 Acceleration in Cartesian coordinates 5.1.2 Acceleration in cylindrical coordinates 5.2 Continuity Equation 5.3 Momentum Equation 5.3.1 Euler equation 5.3.2 Navier-Stokes equations 5.3.3 Boundary conditions 5.4 Rigid Body Motion Revisited 91 91 93 94 95 97 97 99 101 101 Irrotational, Incompressible Flows 6.1 Vorticity and Rotation 6.2 The Velocity Potential φ 6.3 The Stream Function ψ 6.4 Flows Where Both ψ and φ Exist 6.5 Summary of Definitions and Restrictions 6.6 Laplace’s Equation 6.7 Examples of Potential Flow 6.7.1 Uniform flow 6.7.2 Point source and sink 6.7.3 Potential vortex 6.8 Source and Sink in a Uniform Flow 6.9 Potential Flow Over a Cylinder 6.9.1 Pressure distribution 6.9.2 Viscous effects 6.10 Lift 6.10.1 Magnus effect 6.10.2 Airfoils and wings 6.10.3 Trailing vortices 6.11 Vortex Interactions 103 104 105 107 108 108 109 110 110 111 112 115 116 118 118 120 121 121 124 125 Dimensional Analysis 7.1 Dimensional Homogeneity 7.2 Applying Dimensional Homogeneity 7.2.1 Example: Hydraulic jump 7.2.2 Example: Drag on a sphere 7.3 The Number of Dimensionless Groups 7.4 Non-Dimensionalizing Problems 7.5 Pipe Flow Example 7.6 Common Nondimensional Groups 7.7 Non-Dimensionalizing Equations 7.8 Scale Modeling 7.8.1 Geometric similarity 7.8.2 Kinematic similarity 7.8.3 Dynamic similarity 127 128 130 130 132 136 138 139 141 142 144 144 145 145 viii CONTENTS Viscous Internal Flows 8.1 Introduction 8.2 Viscous Stresses and Reynolds Number 8.3 Boundary Layers and Fully Developed Flow 8.4 Transition and Turbulence 8.5 Poiseuille Flow 8.5.1 Fully developed duct flow 8.5.2 Fully developed pipe flow 8.6 Transition in Pipe Flow 8.7 Turbulent Pipe Flow 8.8 Energy Equation for Pipe Flow 8.8.1 Kinetic energy coefficient 8.8.2 Major and minor losses 8.9 Valves and Faucets 8.10 Hydraulic Diameter 8.11 Energy Equation and Bernoulli Equation 147 147 147 148 149 151 151 153 156 157 159 160 162 164 166 166 Viscous External Flows 9.1 Introduction 9.2 Laminar Boundary Layer 9.2.1 Control volume analysis 9.2.2 Blasius velocity profile 9.2.3 Parabolic velocity profile 9.3 Displacement and Momentum Thickness 9.3.1 Displacement thickness 9.3.2 Momentum thickness 9.3.3 Shape factor 9.4 Turbulent Boundary Layers 9.5 Separation, Reattachment and Wakes 9.6 Drag of Bluff and Streamlined Bodies 9.7 Golf Balls, Cricket Balls and Baseballs 9.8 Automobile Flow Fields 169 169 169 169 171 172 175 175 177 177 178 181 184 187 188 10 Open Channel Flow 10.1 Introduction 10.2 Small Amplitude Gravity Waves 10.3 Waves in a Moving Fluid 10.4 Froude Number 10.5 Breaking Waves 10.6 Tsunamis 10.7 Hydraulic Jumps 10.8 Hydraulic Drops? 10.9 Surges and Bores 10.10 Flow Through a Smooth Constriction 10.10.1 Subcritical flow in contraction 10.10.2 Supercritical flow in contraction 10.10.3 Flow over bumps 195 195 195 197 198 199 200 201 205 205 206 210 211 212 CONTENTS ix 11 Compressible Flow 11.1 Introduction 11.2 Pressure Propagation in a Moving Fluid 11.3 Regimes of Flow 11.4 Thermodynamics of Compressible Flows 11.4.1 Ideal gas relationships 11.4.2 Specific heats 11.4.3 Entropy variations 11.4.4 Speed of sound 11.4.5 Stagnation quantities 11.5 Compressible Flow Through a Nozzle 11.5.1 Isentropic flow analysis 11.5.2 Area ratio 11.5.3 Choked flow 11.6 Normal Shocks 11.6.1 Temperature ratio 11.6.2 Velocity ratio 11.6.3 Density ratio 11.6.4 Pressure ratio 11.6.5 Mach number ratio 11.6.6 Stagnation pressure ratio 11.6.7 Entropy changes 11.6.8 Summary: normal shocks 11.7 Weak Normal Shocks 11.8 Oblique Shocks 11.8.1 Oblique shock relations 11.8.2 Flow deflection 11.8.3 Summary: oblique shocks 11.9 Weak Oblique Shocks and Compression Waves 11.10 Expansion Waves 11.11 Wave Drag on Supersonic Vehicles 12 Turbomachines 12.1 Introduction 12.2 Angular Momentum Equation for a Turbine 12.3 Velocity Diagrams 12.4 Hydraulic Turbines 12.4.1 Impulse turbine 12.4.2 Radial-flow turbine 12.4.3 Axial-flow turbine 12.5 Pumps 12.5.1 Centrifugal pumps 12.5.2 Cavitation 12.6 Relative Performance Measures 12.7 Dimensional Analysis 12.8 Propellers and Windmills 12.9 Wind Energy Generation 213 213 214 217 218 218 218 220 221 222 223 223 226 227 228 229 229 229 229 230 230 231 232 232 233 234 235 236 236 238 239 241 241 243 244 246 246 248 249 249 250 252 254 255 257 261 x CONTENTS 13 Environmental Fluid Mechanics 13.1 Atmospheric Flows 13.2 Equilibrium of the Atmosphere 13.3 Circulatory Patterns and Coriolis Effects 13.4 Planetary Boundary Layer 13.5 Prevailing Wind Strength and Direction 13.6 Atmospheric Pollution 13.7 Dispersion of Pollutants 13.8 Diffusion and Mixing Appendices A Analytical Tools A.1 Rank of a Matrix A.2 Scalar Product A.3 Vector Product A.4 Gradient Operator ∇ A.5 Divergence Operator ∇· A.6 Laplacian Operator ∇2 A.7 Curl Operator ∇× A.8 Div, Grad, and Curl A.9 Integral Theorems A.10 Taylor-Series Expansion A.11 Total Derivative and the Operator A.12 Integral and Differential Forms A.13 Gravitational Potential A.14 Bernoulli’s Equation A.15 Reynolds Transport Theorem 265 265 266 268 270 271 272 273 275 279 V·∇ 281 281 282 282 283 283 284 284 285 286 286 287 288 289 290 290 B Conversion Factors 293 C Fluid and Flow Properties 295 Index 314 ... Appendices A Analytical Tools A. 1 Rank of a Matrix A. 2 Scalar Product A. 3 Vector Product A. 4 Gradient Operator ∇ A. 5 Divergence Operator ∇· A. 6 Laplacian... when a particular snowflake lands The fluid that carries the snowflake on its path experiences similar contortions, and generally the velocity and acceleration of a particular mass of fluid vary... what sometimes makes fluid mechanics difficult to learn and understand: physical insight takes time and familiarity to develop, and the reasons for adopting certain assumptions or approximations