Introductory Fluid Mechanics

Joseph Katz San Diego State University

ЩИ CAMBRIDGE Щ0 UNIVERSITY PRESS

Contents

Preface page xi

A Word to the Instructor xiii

1 Basic Concepts and Fluid Properties 1 1.1 Introduction 1 1.2 A Brief History 1 1.3 Dimensions and Units 3 1.4 Fluid Dynamics and Fluid Properties 4

1.4.1 Continuum 4 1.4.2 Laminar and Turbulent Flows 5 1.4.3 Attached and Separated Flows 6

1.5 Properties of Fluids 7 1.5.1 Density 7 1.5.2 Pressure 8 1.5.3 Temperature 8 1.5.4 Viscosity 9 1.5.5 Specific Heat 11 1.5.6 Heat Transfer Coefficient к 12 1.5.7 Surface Tension a 13 1.5.8 Modulus of Elasticity £ 16 1.5.9 Vapor Pressure 17

1.6 Advanced Topics: Fluid Properties and the Kinetic Theory of Gases 18

1.7 Summary and Concluding Remarks 21

2 The Fluid Dynamic Fquation 32

2.1 Introduction 32 2.2 Description of Fluid Motion 33 2.3 Choice of Coordinate System 34 2.4 Pathlines, Streak Lines, and Streamlines 36 2.5 Forces in a Fluid 37

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2.6 Integral Form of the Fluid Dynamic Equations 39 2.7 Differential Form of the Fluid Dynamic Equations 44 2.8 The Material Derivative 50 2.9 Alternative Derivation of the Fluid Dynamic Equations 52 2.10 Summary and Concluding Remarks 55

3 Fluid Statics 65

3.1 Introduction 65 3.2 Fluid Statics: The Governing Equations 65 3.3 Pressure Due to Gravity 66 3.4 Hydrostatic Pressure in a Compressible Fluid 70 3.5 "Solid-Body" Acceleration of Liquids 71

3.5.1 Linear Acceleration 72 3.5.2 Solid-Body Rotation of a Fluid 74

3.6 Hydrostatic Forces on Submerged Surfaces and Bodies 77 3.6.1 Hydrostatic Forces on Submerged Planar Surfaces 78 3.6.2 Hydrostatic Forces on Submerged Curved Surfaces 85

3.7 Buoyancy 87 3.8 Stability of Floating Objects 91 3.9 Summary and Conclusions 93

4 Introduction to Fluid in Motion - One-Dimensional (Frictionless) Flow I l l

4.1 Introduction 111 4.2 The Bernoulli Equation 112 4.3 Summary of the One-Dimensional Tools 113 4.4 Applications of the One-Dimensional Flow Model 114

4.4.1 Free Jets 114 4.4.2 Examples for Using the Bernoulli Equation 118 4.4.3 Simple Models for Time-Dependent Changes in a Control

Volume 119 4.5 Flow Measurements (Based on Bernoulli's Equation) 122

4.5.1 The Pitot Tube 122 4.5.2 The Venturi Tube 123 4.5.3 The Orifice 125 4.5.4 The Sluice Gate 126 4.5.5 Nozzles and Injectors 127

4.6 Summary and Conclusions 127

5 Viscous Incompressible Flow: Exact Solutions 142

5.1 Introduction 142 5.2 The Viscous Incompressible Flow Equations (Steady State) 142 5.3 Laminar Flow between Two Infinite Parallel Plates - The

Couette Flow 143 5.3.1 Flow with a Moving Upper Surface 144 5.3.2 Flow between Two Infinite Parallel Plates - The Results 145

5.3.3 Flow between Two Infinite Parallel Plates - The Poiseuille Flow 148

5.3.4 The Hydrodynamic Bearing (Reynolds Lubrication Theory) 151

5.4 Laminar Flow in Circular Pipes (The Hagen-Poiseuille Flow) 157

5.5 Fully Developed Laminar Flow between Two Concentric Circular Pipes 161

5.6 Flow in Pipes: Darcy's Formula 163 5.7 The Reynolds Dye Experiment, Laminar-Turbulent Flow in

Pipes 164 5.8 Additional Losses in Pipe Flow 166 5.9 Summary of One-Dimensional Pipe Flow 167

5.9.1 Simple Pump Model 170 5.9.2 Flow in Pipes with Noncircular Cross Sections 170 5.9.3 Examples for One-Dimensional Pipe Flow 172 5.9.4 Network of Pipes 177

5.10 Open Channel Flows 179 5.10.1 Simple Models for Open Channel Flows 179 5.10.2 Uniform Open Channel Flows 182 5.10.3 Hydraulic Jump 188 5.10.4 Flow Discharge through Sharp-Crested Weirs 192

5.11 Advanced Topics: Exact Solutions; Two-Dimensional Inviscid Incompressible Vortex Flow 194 5.11.1 Angular Velocity, Vorticity, and Circulation 197

5.12 Summary and Concluding Remarks 199

6 Dimensional Analysis and High-Reynolds-Number Flows 213

6.1 Introduction 213 6.2 Dimensional Analysis of the Fluid Dynamic Equations 213 6.3 The Process of Simplifying the Governing Equations 216 6.4 Similarity of Flows 217 6.5 Flow with High Reynolds Number 218 6.6 High-Reynolds-Number Flows and Turbulence 220 6.7 Summary and Conclusions 222

7 The (Laminar) Boundary Layer 227

7.1 Introduction 227 7.2 Two-Dimensional Laminar Boundary-Layer Flow over a Flat

Plate - (The Integral Approach) 228 7.3 Solutions Based on the von Kärmän Integral Equation 231 7.4 Summary and Practical Conclusions 238 7.5 Effect of Pressure Gradient 241 7.6 Advanced Topics: The Two-Dimensional Laminar

Boundary-Layer Equations 244

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7.6.1 Summary of the Blasius Exact Solution for the Laminar Boundary Layer 246

7.7 Concluding Remarks 248

8 High-Reynolds-Number Flow over Bodies (Incompressible) 254

8.1 Introduction 254 8.2 The Inviscid Irrotational Flow (and Some Math) 255 8.3 Advanced Topics: A More Detailed Evaluation of the Bernoulli

Equation 258 8.4 The Potential Flow Model 259

8.4.1 Methods for Solving the Potential Flow Equations 260 8.4.2 The Principle of Superposition 260

8.5 Two-Dimensional Elementary Solutions 261 8.5.1 Polynomial Solutions 261 8.5.2 Two-Dimensional Source (or Sink) 263 8.5.3 Two-Dimensional Doublet 265 8.5.4 Two-Dimensional Vortex 268 8.5.5 Advanced Topics: Solutions Based on the Green's Identity 270

8.6 Superposition of a Doublet and a Free Stream: Flow over a Cylinder 273

8.7 Fluid Mechanic Drag 277 8.7.1 The Drag of Simple Shapes 278 8.7.2 The Drag of More Complex Shapes 283

8.8 Periodic Vortex Shedding 287 8.9 The Case for Lift 289

8.9.1 A Cylinder with Circulation in a Free Stream 289 8.9.2 Two-Dimensional Flat Plate at a Small Angle of Attack (in

a Free Stream) 293 8.9.3 Note about the Center of Pressure 294

8.10 Lifting Surfaces: Wings and Airfoils 295 8.10.1 The Two-Dimensional Airfoil 296 8.10.2 An Airfoil's Lift 298 8.10.3 An Airfoil's Drag 300 8.10.4 An Airfoil Stall 301 8.10.5 The Effect of Reynolds Number 301 8.10.6 Three-Dimensional Wings 303

8.11 Summary and Concluding Remarks 313

9 Introduction to Computational Fluid Dynamics 324

9.1 Introduction 324 9.2 The Finite-Difference Formulation 325 9.3 Discretization and Grid Generation 327 9.4 The Finite-Difference Equation 328 9.5 The Solution: Convergence and Stability 331 9.6 The Finite-Volume Method 332 9.7 Example: Viscous Flow over a Cylinder 334

Contents

9.8 Potential Flow Solvers: Panel Methods 337 9.9 Summary 340

10 Elements of In viscid Compressible Flow 343

10.1 Introduction 343 10.2 Propagation of a Weak Compression Wave (the Speed

of Sound) 344 10.3 One-Dimensional Isentropic Compressible Flow 347

10.3.1 Critical Conditions 350 10.3.2 Practical Examples for One-Dimensional Compressible

Flow 352 10.4 Normal Shock Waves 355 10.5 Some Applications of the One-Dimensional Model 360

10.5.1 Normal Shock Wave ahead of a Circular Inlet 360 10.5.2 The Converging-Diverging Nozzle (de Laval Nozzle) 361 10.5.3 The Supersonic Wind Tunnel 364

10.6 Effect of Compressibility on External Flows 367 10.7 Concluding Remarks 370

11 Fluid Machinery 377

11.1 Introduction 377 11.2 Work of a Continuous-Flow Machine 380 11.3 Axial Compressors and Pumps (The Mean-Radius Model) 382

11.3.1 Velocity Triangles 385 11.3.2 Power and Compression-Ratio Calculations 387 11.3.3 Radial Variations 390 11.3.4 Pressure-Rise Limitations 392 11.3.5 Performance Envelope of Compressors and Pumps 394 11.3.6 Degree of Reaction 399

11.4 The Centrifugal Compressor (or Pump) 402 11.4.1 Torque, Power, and Pressure Rise 403 11.4.2 Impeller Geometry 405 11.4.3 The Diffuser 408 11.4.4 Concluding Remarks: Axial versus Centrifugal Design 410

11.5 Axial Turbines 411 11.5.1 Torque, Power, and Pressure Drop 412 11.5.2 Axial Turbine Geometry and Velocity Triangles 414 11.5.3 Turbine Degree of Reaction 415 11.5.4 Remarks on Exposed Tip Rotors (Wind Turbines and

Propellers) 423 11.6 Concluding Remarks 426

Appendix A: Conversion Factors 433

Appendix B: Properties of Compressible Isentropic Flow 435

Appendix C: Properties of Normal Shock Flow 431

Index 439