1 Introduction, Importance and Development of Fluid
Mechanics .................................................... 1
1.1 Fluid Flows and their Significance ...................... 1
1.2 Sub-Domains of Fluid Mechanics .......................... 4
1.3 Historical Developments ................................. 9
References .................................................. 14
2 Mathematical Basics ......................................... 15
2.1 Introduction and Definitions ........................... 15
2.2 Tensors of Zero Order (Scalars) ........................ 16
2.3 Tensors of First Order (Vectors) ....................... 17
2.4 Tensors of Second Order ................................ 21
2.5 Field Variables and Mathematical Operations ............ 23
2.6 Substantial Quantities and Substantial Derivative ...... 26
2.7 Gradient, Divergence, Rotation and Laplace Operators ... 27
2.8 Line, Surface and Volume Integrals ..................... 29
2.9 Integral Laws of Stokes and Gauss ...................... 31
2.10 Differential Operators in Curvilinear Orthogonal
Coordinates ............................................ 32
2.11 Complex Numbers ........................................ 36
2.11.1 Axiomatic Introduction to Complex Numbers ....... 37
2.11.2 Graphical Representation of Complex Numbers ..... 38
2.11.3 The Gauss Complex Number Plane .................. 39
2.11.4 Trigonometric Representation .................... 39
2.11.5 Stereographic Projection ........................ 41
2.11.6 Elementary Function ............................. 42
References .................................................. 47
3 Physical Basics ............................................. 49
3.1 SoUds and Fluids ....................................... 49
3.2 Molecular Properties and Quantities of Continuum
Mechanics .............................................. 51
3.3 Transport Processes in Newtonian Fluids ................ 55
3.3.1 General Considerations .......................... 55
3.3.2 Pressure in Gases ............................... 58
3.3.3 Molecular-Dependent Momentum Transport .......... 62
3.3.4 Molecular Transport of Heat and Mass in Gases ... 65
3.4 Viscosity of Fluids .................................... 69
3.5 Balance Considerations and Conservation Laws ........... 73
3.6 Thermodynamic Considerations ........................... 76
References .................................................. 81
4 Basics of Fluid Kinematics .................................. 83
4.1 General Considerations ................................. 83
4.2 Substantial Derivatives ................................ 84
4.3 Motion of Fluid Elements ............................... 85
4.3.1 Path Lines of Fluid Elements .................... 86
4.3.2 Streak Lines of Locally Injected Tracers ........ 90
4.4 Kinematic Quantities of Flow Fields .................... 94
4.4.1 Stream Lines of a Velocity Field ................ 94
4.4.2 Stream Function and Stream Lines of Two-
Dimensional Flow Fields ......................... 98
4.4.3 Divergence of a Flow Field ..................... 101
4.5 Translation, Deformation and Rotation of Fluid
Elements .............................................. 104
4.6 Relative Motions ...................................... 108
References ................................................. 112
5 Basic Equations of Fluid Mechanics ......................... 113
5.1 General Considerations ................................ 113
5.2 Mass Conservation (Continuity Equation) ............... 115
5.3 Newton's Second Law (Momentum Equation) ............... 119
5.4 The Navier-Stokes Equations ........................... 123
5.5 Mechanical Energy Equation ............................ 128
5.6 Thermal Energy Equation ............................... 130
5.7 Basic Equations in Different Coordinate Systems ....... 135
5.7.1 Continuity Equation ............................ 135
5.7.2 Navier-Stokes Equations ........................ 136
5.8 Special Forms of the Basic Equations .................. 142
5.8.1 Transport Equation for Vorticity ............... 143
5.8.2 The Bernoulli Equation ......................... 144
5.8.3 Crocco Equation ................................ 146
5.8.4 Further Forms of the Energy Equation ........... 147
5.9 Transport Equation for Chemical Species ............... 150
References ................................................. 151
6 Hydrostatics and Aerostatics ............................... 153
6.1 Hydrostatics .......................................... 153
6.2 Connected Containers and Pressure-Measuring
Instruments ........................................... 163
6.2.1 Communicating Containers ....................... 163
6.2.2 Pressure-Measuring Instruments ................. 166
6.3 Free Fluid Surfaces ................................... 168
6.3.1 Surface Tension ................................ 168
6.3.2 Water Columns in Tubes and Between Plates ...... 172
6.3.3 Bubble Formation on Nozzles .................... 175
6.4 Aerostatics ........................................... 183
6.4.1 Pressure in the Atmosphere ..................... 183
6.4.2 Rotating Containers ............................ 187
6.4.3 Aerostatic Buoyancy ............................ 188
6.4.4 Conditions for Aerostatics: Stability of
Layers ......................................... 191
References ................................................. 192
7 Similarity Theory .......................................... 193
7.1 Introduction .......................................... 193
7.2 Dimensionless Form of the Differential Equations ...... 197
7.2.1 General Remarks ................................ 197
7.2.2 Dimensionless Form of the Differential
Equations ...................................... 199
7.2.3 Considerations in the Presence of Geometric
and Kinematic Similarities ..................... 204
7.2.4 Importance of Viscous Velocity, Time and
Length Scales .................................. 207
7.3 Dimensional Analysis and π-Theorem .................... 212
References ................................................. 219
8 Integral Forms of the Basic Equations ...................... 221
8.1 Integral Form of the Continuity Equation .............. 221
8.2 Integral Form of the Momentum Equation ................ 224
8.3 Integral Form of the Mechanical Energy Equation ....... 225
8.4 Integral Form of the Thermal Energy Equation .......... 228
8.5 Applications of the Integral Form of the Basic
Equations ............................................. 230
8.5.1 Outflow from Containers ........................ 230
8.5.2 Exit Velocity of a Nozzle ...................... 231
8.5.3 Momentum on a Plane Vertical Plate ............. 232
8.5.4 Momentum on an Inclined Plane Plate ............ 234
8.5.5 Jet Deflection by an Edge ...................... 236
8.5.6 Mixing Process in a Pipe of Constant Cross-
Section ........................................ 237
8.5.7 Force on a Turbine Blade in a Viscosity-Free
Fluid .......................................... 239
8.5.8 Force on a Periodical Blade Grid ............... 240
8.5.9 Euler's Turbine Equation ....................... 242
8.5.10 Power of Flow Machines ......................... 245
References ................................................. 247
9 Stream Tube Theory ......................................... 249
9.1 General Considerations ................................ 249
9.2 Derivations of the Basic Equations .................... 251
9.2.1 Continuity Equation ............................ 251
9.2.2 Momentum Equation .............................. 253
9.2.3 Bernoulli Equation ............................. 254
9.2.4 The Total Energy Equation ...................... 256
9.3 Incompressible, Flows ................................. 257
9.3.1 Hydro-Mechanical Nozzle Flows .................. 257
9.3.2 Sudden Cross-Sectional Area Extension .......... 258
9.4 Compressible Flows .................................... 260
9.4.1 Influences of Area Changes on Flows ............ 260
9.4.2 Pressure-Driven Flows Through Converging
Nozzles ........................................ 263
References ................................................. 273
10 Potential Flows ............................................ 275
10.1 Potential and Stream Functions ........................ 275
10.2 Potential and Complex Functions ....................... 280
10.3 Uniform Flow .......................................... 283
10.4 Corner and Sector Flows ............................... 284
10.5 Source or Sink Flows and Potential Vortex Flow ........ 288
10.6 Dipole-Generated Flow ................................. 291
10.7 Potential Flow Around a Cylinder ...................... 293
10.8 Flow Around a Cylinder with Circulation ............... 296
10.9 Summary of Important Potential Flows .................. 299
10.10Flow Forces on Bodies ................................. 302
References ................................................. 307
11 Wave Motions in Non-Viscous Fluids ........................ 309
11.1 General Considerations ................................ 309
11.2 Longitudinal Waves: Sound Waves in Gases .............. 313
11.3 Transversal Waves: Surface Waves ...................... 318
11.3.1 General Solution Approach ...................... 318
11.4 Plane Standing Waves .................................. 323
11.5 Plane Progressing Waves ............................... 325
11.6 References to Further Wave Motions .................... 329
References ................................................. 330
12 Introduction to Gas Dynamics ............................... 331
12.1 Introductory Considerations ........................... 331
12.2 Mach Lines and Mach Cone .............................. 335
12.3 Non-Linear Wave Propagation, Formation of Shock
Waves ................................................. 338
12.4 Alternative Forms of the Bernoulli Equation ........... 341
12.5 Flow with Heat Transfer (Pipe Flow) ................... 344
12.5.1 Subsonic Flow .................................. 347
12.5.2 Supersonic Flow ................................ 347
12.6 Rayleigh and Fanno Relations .......................... 351
12.7 Normal Compression Shock (Rankine-Hugoniot
Equation) ............................................. 355
References ................................................. 360
13 Stationary, One-Dimensional Fluid Flows of
Incompressible, Viscous Fluids ............................. 361
13.1 General Considerations ................................ 361
13.1.1 Plane Fluid Flows .............................. 362
13.1.2 Cylindrical Fluid Flows ........................ 363
13.2 Derivations of the Basic Equations for Fully
Developed Fluid Flows ................................. 364
13.2.1 Plane Fluid Flows .............................. 364
13.2.2 Cylindrical Fluid Flows ........................ 366
13.3 Plane Couette Flow .................................... 366
13.4 Plane Fluid Flow Between Plates ....................... 369
13.5 Plane Film Flow on an Inclined Plate .................. 372
13.6 Axi-Symmetric Film Flow ............................... 376
13.7 Pipe Flow (Hagen-Poiseuille Flow) ..................... 379
13.8 Axial Flow Between Two Cylinders ...................... 383
13.9 Film Flows with Two Layers ............................ 386
13.10Two-Phase Plane Channel Flow .......................... 388
References ................................................. 391
14 Time-Dependent, One-Dimensional Flows of Viscous Fluids .... 393
14.1 General Considerations ................................ 393
14.2 Accelerated and Decelerated Fluid Flows ............... 397
14.2.1 Stokes First Problem ........................... 397
14.2.2 Diffusion of a Vortex Layer .................... 399
14.2.3 Channel Flow Induced by Movements of Plates .... 402
14.2.4 Pipe Flow Induced by the Pipe Wall Motion ...... 407
14.3 Oscillating Fluid Flows ............................... 414
14.3.1 Stokes Second Problem .......................... 414
14.4 Pressure Gradient-Driven Fluid Flows .................. 417
14.4.1 Starting Flow in a Channel ..................... 417
14.4.2 Starting Pipe Flow ............................. 422
References ................................................. 427
15 Fluid Flows of Small Reynolds Numbers ...................... 429
15.1 General Considerations ................................ 429
15.2 Creeping Fluid Flows Between Two Plates ............... 431
15.3 Plane Lubrication Films ............................... 433
15.4 Theory of Lubrication in Roller Bearings .............. 438
15.5 The Slow Rotation of a Sphere ......................... 443
15.6 The Slow Translatory Motion of a Sphere ............... 445
15.7 The Slow Rotational Motion of a Cylinder .............. 451
15.8 The Slow Translatory Motion of a Cylinder ............. 453
15.9 Diffusion and Convection Influences on Flow Fields .... 459
References ................................................. 461
16 Flows of Large Reynolds Numbers Boundary-Layer Flows ....... 463
16.1 General Considerations and Derivations ................ 463
16.2 Solutions of the Boundary-Layer Equations ............. 468
16.3 Flat Plate Boundary Layer (Blasius Solution) .......... 470
16.4 Integral Properties of Wall Boundary Layers ........... 474
16.5 The Laminar, Plane, Two-Dimensional Free Shear
Layer ................................................. 480
16.6 The Plane, Two-Dimensional, Laminar Free Jet .......... 481
16.7 Plane, Two-Dimensional Wake Flow ...................... 486
16.8 Converging Channel Flow ............................... 489
References ................................................. 492
17 Unstable Flows and Laminar-Turbulent Transition ............ 495
17.1 General Considerations ................................ 495
17.2 Causes of Flow Instabilities .......................... 501
17.2.1 Stability of Atmospheric Temperature Layers .... 502
17.2.2 Gravitationally Caused Instabilities ........... 505
17.2.3 Instabilities in Annular Clearances Caused
by Rotation .................................... 507
17.3 Generalized Instability Considerations (Orr-
Sommerfeld Equation) .................................. 512
17.4 Classifications of Instabilities ...................... 517
17.5 Transitional Boundary-Layer Flows ..................... 519
References ................................................. 522
18 Turbulent Flows ............................................ 523
18.1 General Considerations ................................ 523
18.2 Statistical Description of Turbulent Flows ............ 527
18.3 Basics of Statistical Considerations of Turbulent
Flows ................................................. 528
18.3.1 Fundamental Rules of Time Averaging ............ 528
18.3.2 Fundamental Rules for Probability Density ...... 530
18.3.3 Characteristic Function ........................ 537
18.4 Correlations, Spectra and Time-Scales of Turbulence ... 538
18.5 Time-Averaged Basic Equations of Turbulent Flows ...... 542
18.5.1 The Continuity Equation ........................ 543
18.5.2 The Reynolds Equation .......................... 544
18.5.3 Mechanical Energy Equation for the Mean
Flow Field ..................................... 546
18.5.4 Equation for the Kinetic Energy of
Turbulence ..................................... 550
18.6 Characteristic Scales of Length, Velocity and Time
of Turbulent Flows .................................... 553
18.7 Turbulence Models ..................................... 557
18.7.1 General Considerations ......................... 557
18.7.2 General Considerations Concerning Eddy
Viscosity Models ............................... 560
18.7.3 Zero-Equation Eddy Viscosity Models ............ 565
18.7.4 One-Equation Eddy Viscosity Models ............. 573
18.7.5 Two-Equation Eddy Viscosity Models ............. 576
18.8 Turbulent Wall Boundary Layers ........................ 578
References ................................................. 585
19 Numerical Solutions of the Basic Equations ................. 587
19.1 General Considerations ................................ 587
19.2 General Transport Equation and Discretization
of the Solution Region ................................ 591
19.3 Discretization by Finite Differences .................. 595
19.4 Finite-Volume Discretization .......................... 598
19.4.1 General Considerations ......................... 598
19.4.2 Discretization in Space ........................ 600
19.4.3 Discretization with Respect to Time ............ 611
19.4.4 Treatments of the Source Terms ................. 613
19.5 Computation of Laminar Flows .......................... 614
19.5.1 Wall Boundary Conditions ....................... 615
19.5.2 Symmetry Planes ................................ 615
19.5.3 Inflow Planes .................................. 615
19.5.4 Outflow Planes ................................. 615
19.6 Computations of Turbulent Flows ....................... 616
19.6.1 Flow Equations to be Solved .................... 616
19.6.2 Boundary Conditions for Turbulent Flows ........ 620
References ................................................. 626
20 Fluid Flows with Heat Transfer ............................. 627
20.1 General Considerations ................................ 627
20.2 Stationary, Fully Developed Flow in Channels .......... 630
20.3 Natural Convection Flow Between Vertical Plane
Plates ................................................ 633
20.4 Non-Stationary Free Convection Flow Near a Plane
Vertical Plate ........................................ 637
20.5 Plane-Plate Boundary Layer with Plate Heating at
Small Prandtl Numbers ................................. 641
20.6 Similarity Solution for a Plate Boundary Layer with
Wall Heating and Dissipative Warming .................. 644
20.7 Vertical Plate Boundary-Layer Flows Caused by
Natural Convection .................................... 647
20.8 Similarity Considerations for Flows with Heat
Transfer .............................................. 649
References ................................................. 651
21 Introduction to Fluid-Flow Measurement ..................... 653
21.1 Introductory Considerations ........................... 653
21.2 Measurements of Static Pressures ...................... 656
21.3 Measurements of Dynamic Pressures ..................... 660
21.4 Applications of Stagnation-Pressure Probes ............ 662
21.5 Basics of Hot-Wire Anemometry ......................... 664
21.5.1 Measuring Principle and Physical Principles .... 664
21.5.2 Properties of Hot-Wires and Problems
of Application ................................. 667
21.5.3 Hot-Wire Probes and Supports ................... 672
21.5.4 Cooling Laws for Hot-Wire Probes ............... 676
21.5.5 Static Calibration of Hot-Wire Probes .......... 680
21.6 Turbulence Measurements with Hot-Wire Anemometers ..... 685
21.7 Laser Doppler Anemometry .............................. 694
21.7.1 Theory of Laser Doppler Anemometry ............. 694
21.7.2 Optical Systems for Laser Doppler
Measurements ................................... 701
21.7.3 Electronic Systems for Laser Doppler
Measurements ................................... 705
21.7.4 Execution of LDA-Measurements: One-Dimensional
LDA Systems .................................... 715
References ................................................. 717
Index ......................................................... 719
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