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.

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1 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 Table of Contents Chapter 1 Fluid Properties Kinetic Theory of Gases Introduction 1 Fluid Deformation 1 Nonsimple Fluids 2 Simple Fluids The Macroscopic View of Matter 3 The Solid State 3 The Intermolecular Force Field The Liquid State 4 Melting 4 Evaporation 5 Viscosity On the Kinetic Theory of Gases Mass Density 7 Definition of Density 7 Concept of a Continuum Pressure in a Gas 9 Definition of Mean-Free-Speed 11 Equipartition of Energy 11 Equation of State The Molecular Mean-Free-Path Length 12 The Molecular Diameter 12 Mean-Free-Path Length 13 The Knudsen Number Relation between Fluid Viscosity and Mean-Free-Path Length 15 The Macroscopic Law 15 The Molecular Law On Heat Conduction 18 On the Ratio of Specific Heats of a Perfect Gas On the Units of Viscosity 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 The Heat Conduction Coefficient for Dilute Gases On the Viscosity of Liquids 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.

3 Chapter 2 Fluid Statics Introduction Equilibrium and Pressure in a Fluid Classification of Forces 39 Surface Forces 39 Field Forces The Fundamental Equation of Fluid Statics 39 Pressure Forces 40 Body Force 41 Vector Form 42 Potential Body Forces Equation of Fluid Statics for a Uniform Gravitational Field 43 Isobaric Surfaces 44 The Problem of Integrating the Fundamental Equation of Statics Pressure-Height Relation in a Liquid 44 Pressure Variation at an Interface Elementary Pressure-Measuring Instruments 47 The Barometer 47 The U-Tube Manometer 48 Gauge Pressure An Application in Manometry Forces and Moments on Submerged Surfaces 50 Force and Moment on a Flat Vertical Wall 50 Forces on Curved Surfaces 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 Forces on Submerged Bodies 61 Archimedes Principle The Concept of Static Stability 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 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

4 The Hydrostatic Moment 73 Equilibrium 74 The Stability Criterion 75 Discussion Fluid Statics of the Atmosphere 78 Description of the Atmosphere 79 Supplementary Shells U.S. Standard Atmosphere, 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 Concept of Geopotential Height Stability of the Atmosphere 88 Relation for an Adiabatic Displacement 89 The Effect of Moisture on Stability Buoyancy in the Atmosphere 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 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 Introduction Concepts from Kinematics 117 Velocity of a Particle 118 Acceleration of a Particle 119 Concept of an Inertial Observer Relating Observed Motion for Two Different Observers 122 Example Kinematics in a Flow Field 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.

5 3.6 Illustrative Examples 126 Example Example Example Involving Other Time Derivatives Concept of a One-Dimensional Flow 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 The Continuity Equation 133 Discussion 136 Example Example Example Invariance of Continuity Equation under Change of Observer 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 On Newton's Second Law 148 Example: Water Droplet Falling in a Vacuum On Concepts from Thermodynamics 150 Systems 150 Temperature 151 Equilibrium 151 Extensive and Intensive Variables The First Law of Thermodynamics 152 Enthalpy 154 First Law for an Arbitrary Observer 154 The Entropy Function 155 Isentropic Flow The Second Law of Thermodynamics The Energy Equation 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 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.

6 The Integrated Energy Equation 163 Flow Work 163 Discussion The Linear Momentum Equation 164 Momentum Equation for a Duct with Both Internal and External Pressure Fields Summary 167 Problems 168 Chapter 4 Applications in Constant-Density Flow Introduction, 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 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 Bernoulli's Equation Total Pressure Frictionless Flow out of a Pressurized Reservoir On Quasi-Steady Flow Flow Losses in Internal Inlets Elementary Flow-Metering Devices 187 The Venturi Meter 188 The Plate Orifice 191 A New Approach to Orifice Metering Two Flow Rate Examples 197 Venturi Meter 197 Orifice Pitot and Pitot-Static Tubes 200 Pitot-Static Tube in Nonuniform Flow 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

7 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 The Borda-Carnot Relation for a Sudden Enlargement 214 Theory 214 Experiment 216 Application 217 On Flow through a Contraction The Borda Mouthpiece 218 Theory 218 Discussion Forces on Vanes, Power Production 221 Fixed Vane 221 Moving Vane 222 The Moving Vane Considered from an Energy Viewpoint 223 Multivane Devices The Rocket Sled Water Brake 225 Description 225 Quasi-Steady Solution 225 The Unsteady Solution 228 Discussion 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 Dimensions vs. Units 249 Basic Dimensions 249 Derived Dimensions 250 The Form of Derived Dimensions 251 Angular Measure Standards in Science and Technology 251 The International Standards 252 The Mechanical, or Engineering, System of Units Complete Physical Equations and Dimensional Homogeneity 255 Complete Physical Equations 255 Dimensional Homogeneity 256

5.4 A Primitive Example of Dimensional Analysis 256 5.

8 5.4 A Primitive Example of Dimensional Analysis The Role of Dimensionless Parameters 258 Example of Flow out of a Reservoir 258 Significance of Dimensionless Parameters for Correlation of Experimental Measurements The Buckingham 7t-Theorem Application of the к-theorem 263 Force Coefficient 264 Advance Ratio 265 Geometric Similarity and Model Testing 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 Example Where the Number of л-ratios is Greater than m - n Kinematic and Dynamic Similarity 270 Kinematic Similarity 270 Dynamic Similarity On the Physical Significance of the Reynolds Number Inspectional Analysis 272 Application to the Dynamic Equation 273 Discussion Dynamic Similarity and Modeling 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 A Historical Note The Experiments of Hagen and Poiseuille on Flow through Capillary Tubes 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

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.

9 6.4 On the Correlation of Theory and Experiment The Darcy-Weisbach Equation for Head Loss in Pipe Flow 297 Definition of Head Loss in Pipe Flow 297 The Darcy-Weisbach Equation 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 The Application of Dimensional Analysis to Pipe Flow 304 On Pipe Roughness 304 Dimensional Analysis of Pipe Flow 305 The Correlation of Equation for Smooth-Wall Pipes in Turbulent Flow 307 Prandtl's Law for Smooth Pipes 309 Experimental Verification of Prandtl's Law Zagarola and the Superpipe 312 Results Flow in Artificially Roughened Pipes 317 Flow Losses 317 Discussion Flow Losses in Commercial Pipes 321 Moody's Correlation for Commercial Pipes 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 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 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 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.

10 Application to the Case of Laminar Flow 338 Solution of Equation Velocity and Area Distributions 339 Application 340 Problems 340 Chapter 7 Steady Compressible Flow Introductory Remarks 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 The Equations of Steady, One-Dimensional, Compressible Flow 353 Continuity Equation 353 Dynamic Equation 353 Energy Equation 353 Adiabatic Row 354 Constant-Area Flow On the Propagation of Disturbances 355 Disturbance Created by Impulsive Motion of a Piston The Speed of Sound On the Maximum Speed of a Fluid Expanding Into a Vacuum 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 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 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 7.10 Necessary Conditions for Accelerating a Flow in an Ideal Nozzle 371 Pressure Distribution 371 The Area-Mach Number Relation Isentropic Flow through a de Laval Nozzle 373 Area-Mach Number Relations 374 Example The Appearance of Shock Waves in a Convergent-Divergent Nozzle 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 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 A Numerical Example 392 Note on Experimental Studies 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 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 Discussion 406 Example Discussion Isothermal Flow 408 Discussion 410 Example 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.

12 Application Running Time for a Vacuum-Exhaust, Supersonic Wind Tunnel 415 Example 415 Problems 416 Chapter 8 Nonsteady Flow Introduction Analysis of Starting Flow in a Pipe Supplied by an Infinite Reservoir 427 Description : 427 Analysis 428 Sample Computation 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 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 The Inner Solution 448 Matching the Inner Solution to the Outer Solution 449 The Composite Solution 451 Discussion 452 Comparison with Experiment On the Notion of Characteristics 456 The Differential Equation for a Characteristic Line 458 Differentiating an Arbitrary Function along a Characteristic 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 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,

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

ENGINEERING FLUID MECHANICS. CHAPTER 1 Properties of Fluids

ENGINEERING FLUID MECHANICS. CHAPTER 1 Properties of Fluids CHAPTER 1 Properties of Fluids ENGINEERING FLUID MECHANICS 1.1 Introduction 1.2 Development of Fluid Mechanics 1.3 Units of Measurement (SI units) 1.4 Mass, Density, Specific Weight, Specific Volume, Specific