William В. Brower, Jr.

A PRIMER IN FLUID MECHANICS Dynamics of Flows in One Space Dimension

CRC Press Boca Raton London New York Washington, D.C.

Table of Contents

Chapter 1 Fluid Properties — Kinetic Theory of Gases 1 1.1 Introduction 1

Fluid Deformation 1 Nonsimple Fluids 2 Simple Fluids 2

1.2 The Macroscopic View of Matter 3 The Solid State 3 The Intermolecular Force Field 3

1.3 The Liquid State 4 Melting 4 Evaporation 5 Viscosity 6

1.4 On the Kinetic Theory of Gases 7 1.5 Mass Density 7

Definition of Density 7 Concept of a Continuum 7

1.6 Pressure in a Gas 9 Definition of Mean-Free-Speed 11 Equipartition of Energy 11 Equation of State 12

1.7 The Molecular Mean-Free-Path Length 12 The Molecular Diameter 12 Mean-Free-Path Length 13 The Knudsen Number 14

1.8 Relation between Fluid Viscosity and Mean-Free-Path Length 15 The Macroscopic Law 15 The Molecular Law 17

1.9 On Heat Conduction 18 On the Ratio of Specific Heats of a Perfect Gas 21

1.10 On the Units of Viscosity 22 1.11 On the Viscosity of Real Gases 23

Sutherland's Viscosity Law for Dilute Gases 25 On the Viscosity of Anomalous Gases 25 On the Viscosity of Dense Gases 25 Viscosity of Gas Mixtures 27

1.12 The Heat Conduction Coefficient for Dilute Gases 28 1.13 On the Viscosity of Liquids 28 1.14 Availability and Future Sources of Fluid Property Data .29 Problems 31

Chapter 2 Fluid Statics 37 2.1 Introduction 37 2.2 Equilibrium and Pressure in a Fluid 37 2.3 Classification of Forces 39

Surface Forces 39 Field Forces 39

2.4 The Fundamental Equation of Fluid Statics 39 Pressure Forces 40 Body Force 41 Vector Form 42 Potential Body Forces 43

2.5 Equation of Fluid Statics for a Uniform Gravitational Field 43 Isobaric Surfaces 44 The Problem of Integrating the Fundamental Equation of Statics 44

2.6 Pressure-Height Relation in a Liquid 44 Pressure Variation at an Interface 46

2.7 Elementary Pressure-Measuring Instruments 47 The Barometer 47 The U-Tube Manometer 48 Gauge Pressure 48

2.8 An Application in Manometry 49 2.9 Forces and Moments on Submerged Surfaces 50

Force and Moment on a Flat Vertical Wall 50 Forces on Curved Surfaces 54

2.10 Applications to Forces and Moments on Curved Surfaces 56 Example 1. The Forces on a Parabolic Surface 57 Example 2. The Moment on a Circular Arc Gate 59

2.11 Forces on Submerged Bodies 61 Archimedes Principle 63

2.12 The Concept of Static Stability 63 2.13 Application — Stability of a Buoy 64

Statement of the Problem 64 Equilibrium Condition 64 Moment Equation for an Arbitrary Rotation 65 The Stability Criterion 67 Discussion 67

2.14 On the Stability of a Floating Parabolic Segment 68 Historical Note 68 Description 69 Geometry of the Tilted Segment 70 Limiting Value for the Tilt Angle 71 Pressure Distribution on Segment Surface 72 The Differential Buoyancy Force 72

The Hydrostatic Moment 73 Equilibrium 74 The Stability Criterion 75 Discussion 76

2.15 Fluid Statics of the Atmosphere 78 Description of the Atmosphere 79 Supplementary Shells 79

2.16 U.S. Standard Atmosphere, 1976 80 The Composition of Air 80 Sea-Level Reference Conditions 80 Temperature-Height Relations 80 Temperature-Height Relation in the Troposphere 81 Pressure-Height Relation in the Troposphere 82 Other Standard Atmospheres 83 Four-Dimensional Global Reference Atmosphere Model 83

2.17 Concept of Geopotential Height 86 2.18 Stability of the Atmosphere 88

Relation for an Adiabatic Displacement 89 The Effect of Moisture on Stability 90

2.19 Buoyancy in the Atmosphere 92 2.20 Application of the Inverse-Square Law of Gravitation: Pressure

at the Center of the Earth 94 Structure of Earth 96 Core Pressure of a Constant-Density Earth 96

2.21 Surface Tension 97 Introduction 97 A Method to Measure the Surface Tension Coefficient 100 Surface Tension at a Meniscus 102 Capillary Action within a Tube 103 On Energy Methods 105 On Vibrations in Liquids under Surface Tension 105

Problems 108

Chapter 3 The Equations of Fluid Dynamics 117 3.1 Introduction 117 3.2 Concepts from Kinematics 117

Velocity of a Particle 118 Acceleration of a Particle 119 Concept of an Inertial Observer 120

3.3 Relating Observed Motion for Two Different Observers 122 Example 122

3.4 Kinematics in a Flow Field 123 3.5 Time Derivatives and the Substantial Derivative 124

The Substantial Derivative 125

3.6 Illustrative Examples 126 Example 1 126 Example 2 127

3.7 Example Involving Other Time Derivatives 129 3.8 Concept of a One-Dimensional Flow 131 3.9 The Postulates of Fluid Dynamics 132

Conservation of Mass 132 Newton's Second Law 132 First Law of Thermodynamics 132 Second Law of Thermodynamics 133 Equations of State 133

3.10 The Continuity Equation 133 Discussion 136 Example 1 138 Example 2 138 Example 3 139

3.11 Invariance of Continuity Equation under Change of Observer 140 3.12 The Dynamic Equation of Motion 142

Surface Forces 143 Body Forces 144 The Dynamic Equation 145 Integrating the Dynamic Equation — Examples 145 The Centrifugal Force Equation 148

3.13 On Newton's Second Law 148 Example: Water Droplet Falling in a Vacuum 149

3.14 On Concepts from Thermodynamics 150 Systems 150 Temperature 151 Equilibrium 151 Extensive and Intensive Variables 152

3.15 The First Law of Thermodynamics 152 Enthalpy 154 First Law for an Arbitrary Observer 154 The Entropy Function 155 Isentropic Flow 156

3.16 The Second Law of Thermodynamics 156 3.17 The Energy Equation 157 3.18 Dissipation and the Role of Friction 158

Dissipation Due to Solid Friction 159 Nondissipative Work Performed by Friction Forces 159 Dissipation in a Fluid Flow — Fixed Wall 159 Dissipation in a Fluid Flow — Moving Wall 160 Conclusion 161

3.19 Work and the Integrated Form of the Energy Equation 161 Mechanical Work 161 Heat Added 162

The Integrated Energy Equation 163 Flow Work 163 Discussion 163

3.20 The Linear Momentum Equation 164 Momentum Equation for a Duct with Both Internal and External Pressure Fields 166

3.21 Summary 167 Problems 168

Chapter 4 Applications in Constant-Density Flow 173 4.1 Introduction , 173 4.2 A Catalog of Restrictions 174

Incompressible Flow 174 Constant-Density Flow 174 Inviscid Flow 175 Steady Flow 175 Uniform Flow 175 Source-Free Flow 175 Adiabatic Flow 176 Particle-Isentropic Flow 176 Isoenergetic Flow 176 Horizontal Flow 176 Gravitational Field Force 177

4.3 The Equations of Incompressible (Constant-Density) Flow 177 Continuity Equation 177 Dynamic Equation 177 First Law of Thermodynamics 178 Energy Equation 178 Linear Momentum Equation 179

4.4 Bernoulli's Equation 180 4.5 Total Pressure 181 4.6 Frictionless Flow out of a Pressurized Reservoir 182 4.7 On Quasi-Steady Flow 183 4.8 Flow Losses in Internal Inlets 185 4.9 Elementary Flow-Metering Devices 187

The Venturi Meter 188 The Plate Orifice 191 A New Approach to Orifice Metering 196

4.10 Two Flow Rate Examples 197 Venturi Meter 197 Orifice 199

4.11 Pitot and Pitot-Static Tubes 200 Pitot-Static Tube in Nonuniform Flow 202

4.12 Elementary Applications of the Momentum Equation 203 Steady Flow of a Liquid through a 90° Pipe Bend 203

Fire Nozzle 204 The Water Rocket 204 Steady Two-Dimensional Jet Impinging upon an Inclined Plate 206 Elementary Theory of a Propeller Treated as an Actuator Disk 208 Overall Efficiency 211 Propulsive Efficiency 211 Flow through a Ducted Fan 214

4.13 The Borda-Carnot Relation for a Sudden Enlargement 214 Theory 214 Experiment 216 Application 217 On Flow through a Contraction 217

4.14 The Borda Mouthpiece 218 Theory 218 Discussion 220

4.15 Forces on Vanes, Power Production 221 Fixed Vane 221 Moving Vane 222 The Moving Vane Considered from an Energy Viewpoint 223 Multivane Devices 224

4.16 The Rocket Sled Water Brake 225 Description 225 Quasi-Steady Solution 225 The Unsteady Solution 228 Discussion 231

4.17 The Water Sprinkler Problem 233 The Model 233 Theory 235 Calculation of the Rotational Speed 236 Discussion 237

Problems 238

Chapter 5 Dimensional Analysis, Dynamic Similitude, and Inspectional Analysis 249 5.1 Dimensions vs. Units 249

Basic Dimensions 249 Derived Dimensions 250 The Form of Derived Dimensions 251 Angular Measure 251

5.2 Standards in Science and Technology 251 The International Standards 252 The Mechanical, or Engineering, System of Units 254

5.3 Complete Physical Equations and Dimensional Homogeneity 255 Complete Physical Equations 255 Dimensional Homogeneity 256

5.4 A Primitive Example of Dimensional Analysis 256 5.5 The Role of Dimensionless Parameters 258

Example of Flow out of a Reservoir 258 Significance of Dimensionless Parameters for Correlation of Experimental Measurements 260

5.6 The Buckingham 7t-Theorem 261 5.7 Application of the к-Theorem 263

Force Coefficient 264 Advance Ratio 265 Geometric Similarity and Model Testing 265

5.8 Alternative Method for Determining the л-Ratios 266 Statement of the Problem 266 Interchanging the Roles of the Base and the Repeating Dimensions 267 Identification of the Significant л-Ratios 268

5.9 Example Where the Number of л-Ratios is Greater than m - n 269 5.10 Kinematic and Dynamic Similarity 270

Kinematic Similarity 270 Dynamic Similarity 271

5.11 On the Physical Significance of the Reynolds Number 272 5.12 Inspectional Analysis 272

Application to the Dynamic Equation 273 Discussion 274

5.13 Dynamic Similarity and Modeling 276 5.14 A Dimensional Analysis and an Inspectional Analysis Compared

with the Complete Solution 277 Dimensional Analysis 278 Inspectional Analysis 278 The Complete Solution 278 Application in Transonic Flow 279

Problems 280

Chapter 6 Flows in Pipes and Conduits 287 6.1 A Historical Note 287 6.2 The Experiments of Hagen and Poiseuille on Flow through

Capillary Tubes 288 6.3 Stokes' Solution for Hagen-Poiseuille Flow 290

Newton's Law of Resistance 291 Newton's Second Law Applied to the Cylinder 292 The Velocity Profile 293 Calculation of the Flow Rate 294 The Mean Flow Speed 294 The Skin-Friction Coefficient and the Friction Factor 295 On the Inspectional Analysis of Section 5.12 296

6.4 On the Correlation of Theory and Experiment 296 6.5 The Darcy-Weisbach Equation for Head Loss in Pipe Flow 297

Definition of Head Loss in Pipe Flow 297 The Darcy-Weisbach Equation 299

6.6 Reynolds' Experiments on the Nature of Turbulence and the Discovery of a Criterion for the Transition from Laminar to Turbulent Flow 300

Reynolds' Dimensional Reasoning 300 The Experiments of Reynolds on Transition 301 The Criterion for Transition 302 A Summary of Reynolds' Conclusions 303

6.7 The Application of Dimensional Analysis to Pipe Flow 304 On Pipe Roughness 304 Dimensional Analysis of Pipe Flow 305 The Correlation of Equation 6.7-8 for Smooth-Wall Pipes in Turbulent Flow 307 Prandtl's Law for Smooth Pipes 309 Experimental Verification of Prandtl's Law 310

6.8 Zagarola and the Superpipe 312 Results 313

6.9 Flow in Artificially Roughened Pipes 317 Flow Losses 317 Discussion 319

6.10 Flow Losses in Commercial Pipes 321 Moody's Correlation for Commercial Pipes 321

6.11 Pumping Power Required to Maintain a Pipe Flow 323 Pumping Power in a Pipe 323 Horsepower Required 325 Power Requirements in Laminar vs. Turbulent Flow 326

6.12 Computation of Power Required in a Nonuniform Duct 326 Introduction 326 The Duct Geometry 326 Introduction of the Dynamic Equation 327 Integration of the Viscous Term 328 Flow Loss Coefficient 329 Flow Losses in Other Conduit Elements 330

6.13 Hydraulic Calculations for Simple Conduits and Flow Loops 330 Introduction 330 The Equations of Duct Flow 330 Determination of the Individual Loss Terms 332 Conversion of Head Loss to an Equivalent Length Pipe 333

6.14 Steady Flow through an Elastic Tube 334 The Pressure-Area Relation 334 The Continuity Equation 336 The Dynamic Equation 336

Application to the Case of Laminar Flow 338 Solution of Equation 6.14-14 338 Velocity and Area Distributions 339 Application 340

Problems 340

Chapter 7 Steady Compressible Flow 347 7.1 Introductory Remarks 347 7.2 Thermodynamics of Fluids 347

First Law 347 Specific Heats 348 Entropy 348 Exactness Criterion 349 Thermally Perfect Gas 349 Calorically Perfect Gas 351 Relations for Perfect Gases 351 Liquids 352

7.3 The Equations of Steady, One-Dimensional, Compressible Flow 353 Continuity Equation 353 Dynamic Equation 353 Energy Equation 353 Adiabatic Row 354 Constant-Area Flow 355

7.4 On the Propagation of Disturbances 355 Disturbance Created by Impulsive Motion of a Piston 355

7.5 The Speed of Sound 357 7.6 On the Maximum Speed of a Fluid Expanding Into a Vacuum 359 7.7 Discrepancies between Theory and Experiment 360

Integration of the Dynamic Equation 360 Mass Rate of Flow 362 Theory and Experiment Reconciled 362 The Exit Discharge Speed 363 Flow Geometry at the Exit 364

7.8 Total Conditions in a Compressible Flow — Introduction of Mach Number 365

Total Enthalpy 365 Total Temperature Dependence on Mach Number 366 Total Pressure and Total Density 366 Entropy Changes 367

7.9 Application — Temperature Rise at the Nose of a Reentry Vehicle 368 Problem 368 Solution 368 Discussion 369

7.10 Necessary Conditions for Accelerating a Flow in an Ideal Nozzle 371 Pressure Distribution 371 The Area-Mach Number Relation 372

7.11 Isentropic Flow through a de Laval Nozzle 373 Area-Mach Number Relations 374 Example 376

7.12 The Appearance of Shock Waves in a Convergent-Divergent Nozzle 377 7.13 Normal Shock Waves 378

Shock Relations for a General Substance 379 Perfect Gases 380 Conditions Across a Shock as a Function of M, 381 On Compression Shocks and Expansion Shocks 382

7.14 On the Structure of Shock Waves 385 The Basic Equations for Flow through a Shock Treated as a Flow Without a Discontinuity 385 Evaluation of the Heat-Conduction and Dissipation Functions in Terms of the Dynamic Flow Variables 386 Integration of Equation 7.14-9 389 A Numerical Example 392 Note on Experimental Studies 393

7.15 Analysis of Flow through a de Laval Nozzle with Shock Waves 394 Relation for Second-Throat Area 394 Example 396 Effect of Varying the Dump Tank Exit Pressure 396 Note on the Formation and Stability of Normal Shock Waves 397 On the Momentum Equation for Steady Compressible Flow 399 Application of the Momentum Equation to a Convergent-Divergent Nozzle 400

7.16 Fanno Processes 401 Working Equations 401 Mach Number Relations 402 The Influence of Friction 403 The Integrated Equations of Fanno Flow 404 On the Skin-Friction Coefficient 405 Example 1 405 Discussion 406 Example 2 407 Discussion 408

7.17 Isothermal Flow 408 Discussion 410 Example 410

7.18 The Inflow Problem 412 Application of Energy Balance for an Open System 412 Determination of Tank Pressure 414

Application — Running Time for a Vacuum-Exhaust, Supersonic Wind Tunnel 415 Example 415

Problems 416

Chapter 8 Nonsteady Flow 427 8.1 Introduction 427 8.2 Analysis of Starting Flow in a Pipe Supplied by an Infinite Reservoir 427

Description : 427 Analysis 428 Sample Computation 430

8.3 Nonsteady Liquid Flow through an Orifice in a Reservoir 430 Introduction 430 The Analytical Model 431 The Governing Differential Equation 433 The Quasi-Steady Solution 435 Discussion 435 Note on Previous Theoretical Work 437 Comparison with Experiment 438 Approximate Theory to Correct for Orifice Effects 439 Approximate Theory for a Reservoir with a Short Nozzle at the Outlet 441

8.4 The Draining of a Conical Reservoir 442 The Governing Differential Equation 442 The Quasi-Steady Solution for the Discharge Time 445 The Outer Solution of Equation 8.4-9 446 The Inner Solution 448 Matching the Inner Solution to the Outer Solution 449 The Composite Solution 451 Discussion 452 Comparison with Experiment 455

8.5 On the Notion of Characteristics 456 The Differential Equation for a Characteristic Line 458 Differentiating an Arbitrary Function along a Characteristic 459

8.6 Theory of Hyperbolic Equations for Functions of Two Independent Variables 460

Introduction 460 Basic Equations 460 The Notion of a Characteristic Direction 461 The Initial Data Line 464 Determination of the Characteristic Parameters 465 Note 467

8.7 The General Equations for One-Dimensional, Nonsteady Gas Flow in a Constant-Area Duct 467

The Governing Equations 467 Derivation of Equations Governing the Shape of the Characteristic Lines 470 Equations Governing the Dependent Variables (u, h) 471 The Equations for Nonsteady, Homentropic, Perfect Gas Flow 471

8.8 Impulsive Motion of a Piston in a Duct 473 Problem Statement 473 Preliminary Analysis 473 Solution 475 Application 477 Discussion 477

8.9 Propagation of an Isentropic Finite-Amplitude Compression Pulse 478 Introduction 478 The Pulse Definition 479 The Wave Diagram for a Pulse 480 Inception Time for Shock Formation 481 Discussion 482

8.10 An Approach to Acoustics 485 Introduction 485 Application 487 Wave Energy Transmission 489 Criteria for Sound Propagation in Air 490 Some Flow-Property Magnitudes in Acoustics 491

References 493 Index 501